[Note: the content of this post is standard number theoretic material that can be found in many textbooks (I am relying principally here on Iwaniec and Kowalski); I am not claiming any new progress on any version of the Riemann hypothesis here, but am simply arranging existing facts together.]

The Riemann hypothesis is arguably the most important and famous unsolved problem in number theory. It is usually phrased in terms of the Riemann zeta function , defined by

for and extended meromorphically to other values of , and asserts that the only zeroes of in the critical strip lie on the critical line .

One of the main reasons that the Riemann hypothesis is so important to number theory is that the zeroes of the zeta function in the critical strip control the distribution of the primes. To see the connection, let us perform the following formal manipulations (ignoring for now the important analytic issues of convergence of series, interchanging sums, branches of the logarithm, etc., in order to focus on the intuition). The starting point is the fundamental theorem of arithmetic, which asserts that every natural number has a unique factorisation into primes. Taking logarithms, we obtain the identity

for any natural number , where is the von Mangoldt function, thus when is a power of a prime and zero otherwise. If we then perform a “Dirichlet-Fourier transform” by viewing both sides of (1) as coefficients of a Dirichlet series, we conclude that

formally at least. Writing , the right-hand side factors as

whereas the left-hand side is (formally, at least) equal to . We conclude the identity

(formally, at least). If we integrate this, we are formally led to the identity

or equivalently to the exponential identity

which allows one to reconstruct the Riemann zeta function from the von Mangoldt function. (It is instructive exercise in enumerative combinatorics to try to prove this identity directly, at the level of formal Dirichlet series, using the fundamental theorem of arithmetic of course.) Now, as has a simple pole at and zeroes at various places on the critical strip, we expect a Weierstrass factorisation which formally (ignoring normalisation issues) takes the form

(where we will be intentionally vague about what is hiding in the terms) and so we expect an expansion of the form

and hence on integrating in we formally have

and thus we have the heuristic approximation

Comparing this with (3), we are led to a heuristic form of the *explicit formula*

When trying to make this heuristic rigorous, it turns out (due to the rough nature of both sides of (4)) that one has to interpret the explicit formula in some suitably weak sense, for instance by testing (4) against the indicator function to obtain the formula

which can in fact be made into a rigorous statement after some truncation (the von Mangoldt explicit formula). From this formula we now see how helpful the Riemann hypothesis will be to control the distribution of the primes; indeed, if the Riemann hypothesis holds, so that for all zeroes , it is not difficult to use (a suitably rigorous version of) the explicit formula to conclude that

as , giving a near-optimal “square root cancellation” for the sum . Conversely, if one can somehow establish a bound of the form

for any fixed , then the explicit formula can be used to then deduce that all zeroes of have real part at most , which leads to the following remarkable amplification phenomenon (analogous, as we will see later, to the tensor power trick): any bound of the form

can be automatically amplified to the stronger bound

with both bounds being equivalent to the Riemann hypothesis. Of course, the Riemann hypothesis for the Riemann zeta function remains open; but partial progress on this hypothesis (in the form of zero-free regions for the zeta function) leads to partial versions of the asymptotic (6). For instance, it is known that there are no zeroes of the zeta function on the line , and this can be shown by some analysis (either complex analysis or Fourier analysis) to be equivalent to the prime number theorem

see e.g. this previous blog post for more discussion.

The main engine powering the above observations was the fundamental theorem of arithmetic, and so one can expect to establish similar assertions in other contexts where some version of the fundamental theorem of arithmetic is available. One of the simplest such variants is to continue working on the natural numbers, but “twist” them by a Dirichlet character . The analogue of the Riemann zeta function is then the https://en.wikipedia.org/wiki/Multiplicative_function, the equation (1), which encoded the fundamental theorem of arithmetic, can be twisted by to obtain

and essentially the same manipulations as before eventually lead to the exponential identity

which is a twisted version of (2), as well as twisted explicit formula, which heuristically takes the form

for non-principal , where now ranges over the zeroes of in the critical strip, rather than the zeroes of ; a more accurate formulation, following (5), would be

(See e.g. Davenport’s book for a more rigorous discussion which emphasises the analogy between the Riemann zeta function and the Dirichlet -function.) If we assume the generalised Riemann hypothesis, which asserts that all zeroes of in the critical strip also lie on the critical line, then we obtain the bound

for any non-principal Dirichlet character , again demonstrating a near-optimal square root cancellation for this sum. Again, we have the amplification property that the above bound is implied by the apparently weaker bound

(where denotes a quantity that goes to zero as for any fixed ). Next, one can consider other number systems than the natural numbers and integers . For instance, one can replace the integers with rings of integers in other number fields (i.e. finite extensions of ), such as the quadratic extensions of the rationals for various square-free integers , in which case the ring of integers would be the ring of quadratic integers for a suitable generator (it turns out that one can take if , and if ). Here, it is not immediately obvious what the analogue of the natural numbers is in this setting, since rings such as do not come with a natural ordering. However, we can adopt an algebraic viewpoint to see the correct generalisation, observing that every natural number generates a principal ideal in the integers, and conversely every non-trivial ideal in the integers is associated to precisely one natural number in this fashion, namely the norm of that ideal. So one can identify the natural numbers with the ideals of . Furthermore, with this identification, the prime numbers correspond to the prime ideals, since if is prime, and are integers, then if and only if one of or is true. Finally, even in number systems (such as ) in which the classical version of the fundamental theorem of arithmetic fail (e.g. ), we have *the fundamental theorem of arithmetic for ideals*: every ideal in a Dedekind domain (which includes the ring of integers in a number field as a key example) is uniquely representable (up to permutation) as the product of a finite number of prime ideals (although these ideals might not necessarily be principal). For instance, in , the principal ideal factors as the product of four prime (but non-principal) ideals , , , . (Note that the first two ideals are actually equal to each other.) Because we still have the fundamental theorem of arithmetic, we can develop analogues of the previous observations relating the Riemann hypothesis to the distribution of primes. The analogue of the Riemann hypothesis is now the Dedekind zeta function

where the summation is over all non-trivial ideals in . One can also define a von Mangoldt function , defined as when is a power of a prime ideal , and zero otherwise; then the fundamental theorem of arithmetic for ideals can be encoded in an analogue of (1) (or (7)),

which leads as before to an exponential identity

and an explicit formula of the heuristic form

in analogy with (5) or (10). Again, a suitable Riemann hypothesis for the Dedekind zeta function leads to good asymptotics for the distribution of prime ideals, giving a bound of the form

where is the conductor of (which, in the case of number fields, is the absolute value of the discriminant of ) and is the degree of the extension of over . As before, we have the amplification phenomenon that the above near-optimal square root cancellation bound is implied by the weaker bound

where denotes a quantity that goes to zero as (holding fixed). See e.g. Chapter 5 of Iwaniec-Kowalski for details.

As was the case with the Dirichlet -functions, one can twist the Dedekind zeta function example by characters, in this case the Hecke characters; we will not do this here, but see e.g. Section 3 of Iwaniec-Kowalski for details.

Very analogous considerations hold if we move from number fields to function fields. The simplest case is the function field associated to the affine line and a finite field of some order . The polynomial functions on the affine line are just the usual polynomial ring , which then play the role of the integers (or ) in previous examples. This ring happens to be a unique factorisation domain, so the situation is closely analogous to the classical setting of the Riemann zeta function. The analogue of the natural numbers are the monic polynomials (since every non-trivial principal ideal is generated by precisely one monic polynomial), and the analogue of the prime numbers are the irreducible monic polynomials. The norm of a polynomial is the order of , which can be computed explicitly as

Because of this, we will normalise things slightly differently here and use in place of in what follows. The (local) zeta function is then defined as

where ranges over monic polynomials, and the von Mangoldt function is defined to equal when is a power of a monic irreducible polynomial , and zero otherwise. Note that because is always a power of , the zeta function here is in fact periodic with period . Because of this, it is customary to make a change of variables , so that

and is the renormalised zeta function

We have the analogue of (1) (or (7) or (11)):

which leads as before to an exponential identity

analogous to (2), (8), or (12). It also leads to the explicit formula

where are the zeroes of the original zeta function (counting each residue class of the period just once), or equivalently

where are the reciprocals of the roots of the normalised zeta function (or to put it another way, are the factors of this zeta function). Again, to make proper sense of this heuristic we need to sum, obtaining

As it turns out, in the function field setting, the zeta functions are always rational (this is part of the Weil conjectures), and the above heuristic formula is basically exact up to a constant factor, thus

for an explicit integer (independent of ) arising from any potential pole of at . In the case of the affine line , the situation is particularly simple, because the zeta function is easy to compute. Indeed, since there are exactly monic polynomials of a given degree , we see from (14) that

so in fact there are no zeroes whatsoever, and no pole at either, so we have an exact prime number theorem for this function field:

Among other things, this tells us that the number of irreducible monic polynomials of degree is .

We can transition from an algebraic perspective to a geometric one, by viewing a given monic polynomial through its roots, which are a finite set of points in the algebraic closure of the finite field (or more suggestively, as points on the affine line ). The number of such points (counting multiplicity) is the degree of , and from the factor theorem, the set of points determines the monic polynomial (or, if one removes the monic hypothesis, it determines the polynomial projectively). These points have an action of the Galois group . It is a classical fact that this Galois group is in fact a cyclic group generated by a single element, the (geometric) Frobenius map , which fixes the elements of the original finite field but permutes the other elements of . Thus the roots of a given polynomial split into orbits of the Frobenius map. One can check that the roots consist of a single such orbit (counting multiplicity) if and only if is irreducible; thus the fundamental theorem of arithmetic can be viewed geometrically as as the orbit decomposition of any Frobenius-invariant finite set of points in the affine line.

Now consider the degree finite field extension of (it is a classical fact that there is exactly one such extension up to isomorphism for each ); this is a subfield of of order . (Here we are performing a standard abuse of notation by overloading the subscripts in the notation; thus denotes the field of order , while denotes the extension of of order , so that we in fact have if we use one subscript convention on the left-hand side and the other subscript convention on the right-hand side. We hope this overloading will not cause confusion.) Each point in this extension (or, more suggestively, the affine line over this extension) has a minimal polynomial – an irreducible monic polynomial whose roots consist the Frobenius orbit of . Since the Frobenius action is periodic of period on , the degree of this minimal polynomial must divide . Conversely, every monic irreducible polynomial of degree dividing produces distinct zeroes that lie in (here we use the classical fact that finite fields are perfect) and hence in . We have thus partitioned into Frobenius orbits (also known as *closed points*), with each monic irreducible polynomial of degree dividing contributing an orbit of size . From this we conclude a geometric interpretation of the left-hand side of (18):

The identity (18) thus is equivalent to the thoroughly boring fact that the number of -points on the affine line is equal to . However, things become much more interesting if one then replaces the affine line by a more general (geometrically) irreducible curve defined over ; for instance one could take to be an ellpitic curve

for some suitable , although the discussion here applies to more general curves as well (though to avoid some minor technicalities, we will assume that the curve is projective with a finite number of -rational points removed). The analogue of is then the coordinate ring of (for instance, in the case of the elliptic curve (20) it would be ), with polynomials in this ring producing a set of roots in the curve that is again invariant with respect to the Frobenius action (acting on the and coordinates separately). In general, we do not expect unique factorisation in this coordinate ring (this is basically because Bezout’s theorem suggests that the zero set of a polynomial on will almost never consist of a single (closed) point). Of course, we can use the algebraic formalism of ideals to get around this, setting up a zeta function

and a von Mangoldt function as before, where would now run over the non-trivial ideals of the coordinate ring. However, it is more instructive to use the geometric viewpoint, using the ideal-variety dictionary from algebraic geometry to convert algebraic objects involving ideals into geometric objects involving varieties. In this dictionary, a non-trivial ideal would correspond to a proper subvariety (or more precisely, a subscheme, but let us ignore the distinction between varieties and schemes here) of the curve ; as the curve is irreducible and one-dimensional, this subvariety must be zero-dimensional and is thus a (multi-)set of points in , or equivalently an effective divisor of ; this generalises the concept of the set of roots of a polynomial (which corresponds to the case of a principal ideal). Furthermore, this divisor has to be *rational* in the sense that it is Frobenius-invariant. The prime ideals correspond to those divisors (or sets of points) which are irreducible, that is to say the individual Frobenius orbits, also known as closed points of . With this dictionary, the zeta function becomes

where the sum is over effective rational divisors of (with being the degree of an effective divisor ), or equivalently

The analogue of (19), which gives a geometric interpretation to sums of the von Mangoldt function, becomes

thus this sum is simply counting the number of -points of . The analogue of the exponential identity (16) (or (2), (8), or (12)) is then

and the analogue of the explicit formula (17) (or (5), (10) or (13)) is

where runs over the (reciprocal) zeroes of (counting multiplicity), and is an integer independent of . (As it turns out, equals when is a projective curve, and more generally equals when is a projective curve with rational points deleted.)

To evaluate , one needs to count the number of effective divisors of a given degree on the curve . Fortunately, there is a tool that is particularly well-designed for this task, namely the Riemann-Roch theorem. By using this theorem, one can show (when is projective) that is in fact a rational function, with a finite number of zeroes, and a simple pole at both and , with similar results when one deletes some rational points from ; see e.g. Chapter 11 of Iwaniec-Kowalski for details. Thus the sum in (22) is finite. For instance, for the affine elliptic curve (20) (which is a projective curve with one point removed), it turns out that we have

for two complex numbers depending on and .

The Riemann hypothesis for (untwisted) curves – which is the deepest and most difficult aspect of the Weil conjectures for these curves – asserts that the zeroes of lie on the critical line, or equivalently that all the roots in (22) have modulus , so that (22) then gives the asymptotic

where the implied constant depends only on the genus of (and on the number of points removed from ). For instance, for elliptic curves we have the *Hasse bound*

As before, we have an important amplification phenomenon: if we can establish a weaker estimate, e.g.

then we can automatically deduce the stronger bound (23). This amplification is not a mere curiosity; most of the *proofs* of the Riemann hypothesis for curves proceed via this fact. For instance, by using the elementary method of Stepanov to bound points in curves (discussed for instance in this previous post), one can establish the preliminary bound (24) for large , which then amplifies to the optimal bound (23) for all (and in particular for ). Again, see Chapter 11 of Iwaniec-Kowalski for details. The ability to convert a bound with -dependent losses over the optimal bound (such as (24)) into an essentially optimal bound with no -dependent losses (such as (23)) is important in analytic number theory, since in many applications (e.g. in those arising from sieve theory) one wishes to sum over large ranges of .

Much as the Riemann zeta function can be twisted by a Dirichlet character to form a Dirichlet -function, one can twist the zeta function on curves by various additive and multiplicative characters. For instance, suppose one has an affine plane curve and an additive character , thus for all . Given a rational effective divisor , the sum is Frobenius-invariant and thus lies in . By abuse of notation, we may thus define on such divisors by

and observe that is multiplicative in the sense that for rational effective divisors . One can then define for any non-trivial ideal by replacing that ideal with the associated rational effective divisor; for instance, if is a polynomial in the coefficient ring of , with zeroes at , then is . Again, we have the multiplicativity property . If we then form the twisted normalised zeta function

then by twisting the previous analysis, we eventually arrive at the exponential identity

in analogy with (21) (or (2), (8), (12), or (16)), where the *companion sums* are defined by

where the trace of an element in the plane is defined by the formula

In particular, is the exponential sum

which is an important type of sum in analytic number theory, containing for instance the Kloosterman sum

as a special case, where . (NOTE: the sign conventions for the companion sum are not consistent across the literature, sometimes it is which is referred to as the companion sum.)

If is non-principal (and is non-linear), one can show (by a suitably twisted version of the Riemann-Roch theorem) that is a rational function of , with no pole at , and one then gets an explicit formula of the form

for the companion sums, where are the reciprocals of the zeroes of , in analogy to (22) (or (5), (10), (13), or (17)). For instance, in the case of Kloosterman sums, there is an identity of the form

for all and some complex numbers depending on , where we have abbreviated as . As before, the Riemann hypothesis for then gives a square root cancellation bound of the form

for the companion sums (and in particular gives the very explicit Weil bound for the Kloosterman sum), but again there is the amplification phenomenon that this sort of bound can be deduced from the apparently weaker bound

As before, most of the known proofs of the Riemann hypothesis for these twisted zeta functions proceed by first establishing this weaker bound (e.g. one could again use Stepanov’s method here for this goal) and then amplifying to the full bound (28); see Chapter 11 of Iwaniec-Kowalski for further details.

One can also twist the zeta function on a curve by a multiplicative character by similar arguments, except that instead of forming the sum of all the components of an effective divisor , one takes the product instead, and similarly one replaces the trace

by the norm

Again, see Chapter 11 of Iwaniec-Kowalski for details.

Deligne famously extended the above theory to higher-dimensional varieties than curves, and also to the closely related context of *-adic sheaves* on curves, giving rise to two separate proofs of the Weil conjectures in full generality. (Very roughly speaking, the former context can be obtained from the latter context by a sort of Fubini theorem type argument that expresses sums on higher-dimensional varieties as iterated sums on curves of various expressions related to -adic sheaves.) In this higher-dimensional setting, the zeta function formalism is still present, but is much more difficult to use, in large part due to the much less tractable nature of divisors in higher dimensions (they are now combinations of codimension one subvarieties or subschemes, rather than combinations of points). To get around this difficulty, one has to change perspective yet again, from an algebraic or geometric perspective to an -adic cohomological perspective. (I could imagine that once one is sufficiently expert in the subject, all these perspectives merge back together into a unified viewpoint, but I am certainly not yet at that stage of understanding.) In particular, the zeta function, while still present, plays a significantly less prominent role in the analysis (at least if one is willing to take Deligne’s theorems as a black box); the explicit formula is now obtained via a different route, namely the Grothendieck-Lefschetz fixed point formula. I have written some notes on this material below the fold (based in part on some lectures of Philippe Michel, as well as the text of Iwaniec-Kowalski and also this book of Katz), but I should caution that my understanding here is still rather sketchy and possibly inaccurate in places.

** — 1. l-adic sheaves and the etale fundamental group — **

From the point of view of applications to analytic number theory, one can view Deligne’s theorems as providing bounds of square root cancellation type for various sums of the form

where is some (quasi-projective) curve (or possibly a higher dimensional variety), and is a certain type of “structured” function on the set of -rational points on , such that is not entirely degenerate (e.g. constant). In particular, Deligne’s results allow one to obtain square root cancellation bounds of the form

for certain non-degenerate structured functions defined on except at a few points where is “singular”, and the restricted sum denotes a sum over the non-singular points of .

The class of functions that can be treated by Deligne’s machinery is very general. The Weil conjectures for curves (and twisted curves) already allows one to obtain bounds of the shape (29) for several useful classes of functions , such as phases

of rational functions, multiplicative characters , or products of the two. Deligne’s results enlarge this class of functions to include Fourier transforms of existing structured functions, such as

in fact the class of structured functions is closed under a large number of operations, such as addition, multiplication, convolution or pullback, making it an excellent class to use in analytic number theory applications. The situation here is not dissimilar to that of characters of finite-dimensional representations of some group , in that the class of characters is also closed under basic operations such as addition and multiplication (which correspond to tensor sum and tensor product of representations). Indeed, the formal definition of a structured function will involve such a finite-dimensional representation, but with two technical details: the vector space is not exactly defined over the complex numbers, but instead defined over the -adic numbers for some prime coprime to , and also the group is going to be the étale fundamental group of .

We first describe (in somewhat vague terms) what the étale fundamental group of a connected variety (or more generally, a connected Noetherian scheme) defined over a field . Crucially, we do not require the underlying field to be algebraically closed, and in our applications will in fact be the finite field . The étale fundamental group is the common generalisation of the (profinite completion of the) topological fundamental group (applied to, say, a smooth complex variety), and of the absolute Galois group of a field . (This point of view is nicely presented in this recent book of Szamuely.) To explain this, we first consider the *topological* fundamental group of a smooth connected manifold at some base point (the choice of which is not too important if one is willing to view the fundamental group up to conjugacy). This group is conventionally defined in terms of loops in based at , but the notion of a loop does not make much sense in either Galois theory or algebraic geometry. Fortunately, as observed by Grothendieck, there is an alternate way to interpret this fundamental group as follows. Let be any covering space of , with covering map ; then above the base point there is a discrete fibre , and given any point in this fibre, every loop based at lifts by monodromy to a path starting at and ending at another point in the fibre . The endpoint is not affected by homotopy of the path, so this leads to an action of the fundamental group (or more precisely, of the opposite group to this fundamental group, but never mind this annoying technicality) on the fibre above of any covering space. For instance, if we take the -fold cover of the unit circle by itself formed by multplying by a natural number , then the fibre may be identified with , and the fundamental group acts on this fibre by translation.

The actions of the fundamental group are natural in the following sense: given a morphism between two covering spaces , of (so that ), then the action of the fundamental group is intertwined by , thus for any and . Conversely, every collection of actions on fibres that is natural in the above sense arises from a unique element of the fundamental group ; this can be easily seen by working with the universal cover of , of which all other (connected) covers are quotients, and on whose fibre the fundamental group acts freely and transitively. Thus, one could *define* the fundamental group as the group of all possible collections of isomorphisms on the fibres above which are natural in the above sense. (In category-theoretic terms, is the group of natural isomorphisms of the fibre functor that maps covers to fibres .)

There is an analogous way to view the absolute Galois group of a field . For simplicity we shall only discuss the case of perfect fields (such as finite fields) here, in which case there is no distinction between the separable closure and the algebraic closure , although the discussion below can be extended to the general case (and from a scheme-theoretic viewpoint it is in fact natural to not restrict oneself to the perfect case). The analogue of the covering spaces of the manifold are the finite extensions of . Here, one encounters a “contravariance” problem in pursuing this analogy: for covering spaces, we have a map from the covering space to the base space ; but for field extensions, one instead has an inclusion from the base field to the extension field. To make the analogy more accurate, one has to dualise, with the role of the covering space being played not by the extension , but rather by the set of all field embeddings of into the algebraic closure (cf. the Yoneda lemma). (There is nothing too special about the algebraic closure here; any field which is in some sense “large enough” to support lots of embeddings of would suffice.) This set projects down to , which has a canonical point , namely the standard embedding of in ; the fibre of at is then the set of field embeddings of to that fix . It is a basic fact of Galois theory that if is an extension of of degree , then this fibre is a finite set of cardinality (for perfect fields, this can be easily deduced from the primitive element theorem). The Galois group then acts on these fibres by left-composition, and one can verify that the action of a given Galois group element is natural in the same category-theoretic sense as considered previously. Conversely, because the algebraic closure of can be viewed as the direct limit of finite extensions of , one can show that every natural isomorphism of these fibres (or more precisely, the fibre functor from (or ) to ) comes from exactly one element of the Galois group .

Now we can define the étale fundamental group of a general (connected, Noetherian) scheme with a specified base point . (Actually, one minor advantages of schemes is that they come with a canonical point to pick here, namely the generic point, although the dependence on the base point is not a major issue here in any event.) The analogue of covering spaces or finite extensions are now the finite étale covers of : morphisms from another scheme to that are étale (which, roughly, is like saying is a local diffeomorphism) and finite (roughly, this means that locally looks like the product of with a finite set). Unlike the previous two contexts, in which a universal covering object was available, the category of finite étale covers of a given scheme usually does not have a universal object. (One can already see this in the category of algebraic Riemann surfaces: the universal cover of the punctured plane ought to be the complex plane with covering map given by the exponential map ; this is what happens in the topological setting, but it is not allowed in the algebraic geometry setting because the exponential map is not algebraic.) Nevertheless, one can still define the étale fundamental group without recourse to a universal object, again by using actions on fibres . Namely, the étale fundamental group consists of all objects which act by permutation on the fibres above of every finite cover of , in such a way that this action is natural in the category-theoretic sense. (To actually construct the étale fundamental group as a well-defined set requires a small amount of set-theoretic care, because strictly speaking the class of finite étale covers of is only a class and not a set; but one can start with one representative from each equivalence class of finite étale covers first, with some designated morphisms between them with which to enforce the naturality conditions, and build the fundamental group from there by an inverse limit construction; see e.g. Szamuely’s book for details.) As the étale fundamental group is defined through its actions on finite sets rather than arbitrary discrete sets, it will automatically be a profinite group, and so differs slightly from the topological fundamental group in that regard. For instance, the punctured complex plane has a topological fundamental group of , but has an étale fundamental group of – the profinite integers, rather than the rational integers. More generally, for complex varieties, the étale fundamental group is always the profinite completion of the topological fundamental group (this comes from a deep connection between complex geometry and algebraic geometry known as the Riemann existence theorem), but the situation can be more complicated in finite characteristic or in non-algebraically closed fields, due to the existence of étale finite covers that do not arise from classical topological covers. For instance, the étale fundamental group of a perfect field (which one can view geometrically as a point over ) turns out to be the absolute Galois group of .

The étale fundamental group is functorial: every morphism of schemes gives rise to a homomorphism of fundamental groups. Among other things, this gives rise to a short exact sequence

whenever is a variety defined over a perfect field (and hence also defined over its algebraic closure , via base change) which is geometrically connected (i.e. that is connected). (Again, it is more natural from a scheme-theoretic perspective to not restrict to the perfect case here, but we will do so here for sake of concreteness.) The groups and are known as the *arithmetic fundamental group* and *geometric fundamental group* of respectively. The latter should be viewed as a profinite analogue of the topological fundamental group of (a complex model of) ; this intuition is quite accurate in characteristic zero (due to the Riemann existence theorem mentioned earlier), but only partially accurate in positive characteristic (basically, one has to work with the prime-to- components of these groups in order to see the correspondence). (See for instance these notes of Milne for further discussion.)

The étale fundamental group can also be described “explicitly” in terms of Galois groups as follows. [Caution: I am not 100% confident in the accuracy of the assertions in this paragraph.] Let be the function field on , and let be its separable closure, so that one can form the Galois group . For any closed point of (basically, an orbit of ), we can form a local version of (the Henselization of the discrete valuation ring associated to , which roughly speaking captures the formal power series around ), and a local version of the Galois group, known as the *decomposition group* at . As embeds into , embeds into . Inside is the inertia group , defined (I think) as the elements of which stabilise the residue field of . Informally, this group measures the amount of ramification present at . One can then identify the arithmetic fundamental group with the quotient of by the normal subgroup generated (as a normal subgroup) by all the inertia groups of closed points; informally, the étale fundamental group describes the unramified Galois representations of . The geometric fundamental group has a similar description, but with replaced by the smaller group .

(Note: as pointed out to me by Brian Conrad, one can also identify with where is the maximal Galois extension of which is unramified at the discrete valuations corresponding to all the closed points of . A similar description can be given for higher-dimensional schemes , if one replaces closed points with codimension-1 points, thanks to Abhyankar’s lemma.)

We have seen that fundamental groups (or absolute Galois groups) act on discrete sets, and specifically on the fibres of covering spaces (or field extensions, or étale finite covers). However, fundamental groups also naturally act on other spaces, and in particular on vector spaces over various fields. For instance, suppose one starts with a connected complex manifold and considers the holomorphic functions on this manifold. Typically, there are very few *globally* holomorphic functions on (e.g. if is compact, then Liouville’s theorem will force a globally holomorphic function to be constant), so one usually works instead with *locally* holomorphic functions, defined on some open subset of . These local holomorphic functions then form a sheaf over , with a vector space of holomorphic functions on being attached to each open set , as well as restriction maps from to for every open subset of which obey a small number of axioms which I will not reproduce here (see e.g. the Wikipedia page on sheaves for the list of sheaf axioms).

Now let be a base point in . We can then associate a natural complex vector space to , namely the space of holomorphic germs at (the direct limit of for neighbourhoods of ). Any loop based at then induces a map from to itself, formed by starting with a germ at and performing analytic continuation until one returns to . As analytic continuation is a linear operation, this map from to itself is linear; it is also invertible by reversing the loop, so it lies in the general linear group . From the locally unique nature of analytic continuation, this map is not affected by homotopy of the loop, and this therefore gives a linear representation of the fundamental group. (Actually, to be pedantic, it gives a representation of the opposite group , due to the usual annoyance that composition of functions works in the reverse order from concatenation of paths, but let us ignore this minor technicality.)

Of course, one can perform the same sort of construction for other sheaves over of holomorphic sections of various vector bundles (using the stalk of the sheaf at for the space of germs), giving rise to other complex linear representations of the fundamental group. In analogy with this, we will *define* the notion of a certain type of sheaf over a variety defined over a finite field in terms of such representations. Due to the profinite nature of the étale fundamental group, we do not work with complex representations, but rather with -adic representations, for some prime that is invertible in . However, as the -adics have characteristic zero, we can always choose an embedding into the complex numbers, although it is not unique (and.

We can now give (a special case of) the definition of an -adic sheaf:

Definition 1 (Lisse sheaf)Let be a non-empty affine curve defined over a finite field , and let be a prime not equal to to the characteristic of (i.e. is invertible in ). An-adic lisse sheafover (also known as an-adic local system) is a continuous linear representation of the arithmetic fundamental group of , where is a finite-dimensional vector space over the –adics . (Here, the continuity is with respect to the profinite topology on and the -adic topology on .) We refer to as thefibreof the sheaf. The dimension of is called therankof the sheaf.

One can define more general -adic sheaves (not necessarily lisse) over more general schemes, but the definitions are more complicated, and the lisse case already suffices for many analytic number theory applications. (However, even if one’s applications only involve lisse sheaves, it is natural to generalise to arbitrary sheaves when proving the key theorems about these sheaves, in particular Deligne’s theorems.)

As we see from the above definition, lisse sheaves are essentially just a linear representation of the arithmetic fundamental group (and hence also of the geometric fundamental group). As such, one can directly import many representation-theoretic concepts into the language of -adic lisse sheaves. For instance, one can form the direct sum and direct product of lisse sheaves by using the representation-theoretic direct sum and direct product, and one can take the contragradient sheaf of a sheaf by replacing the representation with its inverse transpose. Morphisms of the underlying curve give rise to a pullback operation from sheaves over to sheaves of . (There is also an important pushforward operation, but it is much more difficult to define and study.) We also have the notion of a trivial sheaf (in which is the identity), an irreducible sheaf (which cannot be decomposed as an extension of a lower rank sheaf), a semisimple sheaf (a sheaf which factors as the direct sum of irreducibles), or an isotypic sheaf (the direct sum of isomorphic sheaves). By replacing the arithmetic fundamental group with its geometric counterpart, one also has geometric versions of many of the above concepts, thus for instance one can talk about geometrically irreducible sheaves, geometrically semisimple sheaves, etc.

Now let be a closed point in of degree , with its associated decomposition group and inertia group . The quotient can be viewed as the arithmetic fundamental group of , or equivalently the absolute Galois group of the residue field at , and this quotient embeds into the arithmetic fundamental group of since embeds into . On the other hand, as has degree , the residue field is isomorphic to the degree extension of , so is isomorphic to the absolute Galois group of . This latter group is generated (topologically) by the arithmetic Frobenius map , and also by its inverse, known as the geometric Frobenius map. So the geometric Frobenius map is well defined (up to conjugacy) as an element of and hence . By abuse of notation, we will refer to elements of this conjugacy class in as , bearing in mind that this object is only defined up to conjugacy. If we have a lisse -adic sheaf , then is defined up to conjugation, which implies that the trace on is well-defined as an element of . (Actually, a technical point: one should restrict the trace from to the subspace fixed by the inertia group of , but as long as there is no ramification at , this is all of ; we will ignore this technicality, as in practice we will delete the points of in which ramification occurs.) Fixing some embedding , we can then form the trace function

which is a function from to , and in particular a function from to . (A certain subclass of) these trace functions will serve as the “structured” functions mentioned at the start of this section.

Now an important definition. For each unramified closed point , is an invertible linear transformation on the vector space , which has dimension equal to the rank of . In particular, this transformation has eigenvalues (depending on , of course) in , and hence in after selecting an embedding . We say that the sheaf is *pure of weight * for some real number if all of these eigenvalues (and their Galois conjugates) have magnitude exactly , where is the residue field at . There is also the weaker concept of being *mixed of weight *, in which the magnitude of the eigenvalues and their conjugates is merely assumed to be bounded above by . (A technical remark: this upper bound are initially only assumed for unramified closed points, but a result of Deligne allows one to extend this upper bound to ramified points also.) If the sheaf is pure or mixed of weight , then one clearly has the pointwise bound

in particular, for sheaves of weight zero and bounded rank, the trace function is .

The trace functions resemble characters of representations. For instance, taking the direct sum or product of two sheaves results in taking the sum or product of the two trace functions. If two sheaves have weight , then their direct sum has weight as well, while if two sheaves have weights , then their direct product has weight . Taking the contragradient of a sheaf of weight results in a sheaf of weight ; in the weight zero case, the trace function simply gets conjugated. Pulling back a sheaf by some morphism of curves preserves the weight of that sheaf, and pulls back the trace functions accordingly.

It is a convenient fact that pure sheaves automatically have a “geometric semisimplification” which is a pure sheaf of the same weight, and whose trace function is identical to that of the original sheaf. Furthermore, the geometrically irreducible components of the geometrically semisimple sheaf have the same weight as the original sheaf. Because of this, one can often reduce to the study of geometrically irreducible pure sheaves.

In practice, one can normalise the weight of a pure or mixed sheaf to zero by the following simple construction. Given any algebraic integer over the -adics, we can define the *Tate sheaf* associated to to be the unique rank one continuous representation that acts trivially on the geometric fundamental group, and maps the Frobenius map of to . If we set , then this is a pure sheaf of weight , and the associated trace function is equal to ; it is geometrically trivial but arithmetically non-trivial (if ), and conversely all geometrically trivial and geometrically semisimple sheaves come from direct sums of Tate sheaves. Given any other pure or mixed sheaf of some weight , one can then tensor with the Tate sheaf of weight to create a pure or mixed sheaf of weight , which at the level of trace functions amounts to multiplying the trace function by . In particular, setting one can perform a “Tate twist” to normalise these sheaves to be of weight zero.

There is also the notion of the “complexity” of an -adic sheaf, measured by a quantity known as the *conductor* of the sheaf. This quantity is a bit complicated to define here, but basically it incorporates the genus of the underlying curve , the rank of the sheaf, the number of singularities of the sheaf, and something called the Swan conductor at each singularity of that sheaf; bounding the conductor then leads to a bound on all of these quantities. The conductor behaves well with respect to various sheaf-theoretic operations; for instance the direct sum or product of sheaves of bounded conductor will also be a sheaf of bounded conductor. I’ll use “bounded complexity” in place of “bounded conductor” in the text that follows.

The “structured functions” mentioned at the start of this section on a curve are then precisely the trace functions associated to sheaves on open dense subsets of this curve formed by deleting a bounded number of points at most from , which are pure of weight zero and of bounded complexity. As discussed above, this class of functions is closed under pointwise sum, pointwise product, complex conjugation, and pullback, and are also pointwise bounded. It is also a deep fact (essentially due to Deligne, Laumon, and Katz) that this class is closed under other important analytic operations, such as Fourier transform and convolution.

Now we give some basic examples of structured functions, which can be combined with each other using the various closure properties of such functions discussed above. First we show how any additive character can be interpreted as an structured function on the affine line . The affine line is covered by itself via the finite étale covering map defined by . The fibre of this map at is just the field , and so the arithmetic fundamental group acts on . One can show that this action is a translation action (because of the translation symmetry of this covering space), and so we have a map from to , which on composition with (and pulling back to ) gives a rank one sheaf, called the *Artin-Schrier sheaf* associated to . The trace function here is just , and this is clearly a pure sheaf of weight ; it also has bounded complexity (indeed, the genus is zero and the only singularity is at , and the Swan conductor there can be computed to be ).

In a similar vein, any multiplicative character can be viewed as an structured function on the multiplciative group by a similar construction, using instead of (which is still an étale covering map, thanks to a baby case of Hilbert’s Theorem 90), giving rise to a rank one, bounded complexity pure sheaf of weight on known as the *Kummer sheaf*, whose trace function is just .

These two examples, combined with the closure operations defined previously, already give a large number of useful structured functions, such as the function on the affine line with boundedly many points removed, where are on that affine line. But there is a deep and powerful additional closure property due to Deligne, Laumon, and Katz: if is a pure sheaf of weight zero on the affine line (with boundedly many points removed) of bounded complexity that does not contain any Artin-Schrier components, and is a non-trivial additive character, then there is a “Fourier transform” of , which is another pure sheaf of weight zero and bounded complexity with no Artin-Schrier components, with the property that the trace function of is the Fourier transform of the trace function of with respect to on :

See Theorem 8.2.3 of this book of Katz for details. (The construction here is analogous to that used in the Fourier-Mukai transform, although I do not know how tight this analogy is.) This closure under Fourier transforms also implies a closure property with respect to convolutions, by the usual intertwining relationship between convolution and multiplication provided by the Fourier transform. Using these additional closure properties, one can now add many new and interesting examples of structured functions, such as the normalised Kloosterman sums

or more generally the hyper-Kloosterman sums

for some non-principal additive character . These can then be combined with the previous examples of structured functions as before, for instance Kloosterman correlations would now also be examples of structured functions.

We have now demonstrated that the class of structured functions contains many examples of interest to analytic number theory, but we have not yet done anything with this class, in particular we have not obtained any control on “exponential sums” such as

beyond the trivial bound of that comes from the pointwise bound on (and the elementary “Schwarz-Zippel” fact that a bounded complexity curve has at most points over ). However, such control can be obtained through the important Grothendieck-Lefschetz fixed point formula

for any , where are the -adic cohomology groups with compact support in and coefficients in . These groups are defined through the general homological algebra machinery of derived categories (but can also be interpreted using either Galois cohomology or sheaf cohomology), and I do not yet have a sufficiently good understanding of these topics to say much more about these groups, other than that they turn out to be finite-dimensional vector spaces over , and carry an action of the (geometric) Frobenius map of . (In fact they have a richer structure than this, being sheaves over ; this is useful when trying to iterate Deligne’s theorem to control higher-dimensional exponential sums, but we won’t directly use this structure here.) This formula is analogous to the explicit formulae (5), (10), (13), (17), (22), (26), (27) from the introduction, and in fact easily implies the latter four explicit formulae. Specialising to the case , we see that the sum (30) takes the form

To proceed further, we need to understand the eigenvalues of the Frobenius map on these cohomology groups (and we also need some control on the dimensions of these groups, i.e. on Betti numbers). This can be done by the following deep result of Deligne, essentially the main result in his second proof of the Weil conjectures:

Theorem 2 (Deligne’s Weil II)If is a lisse -adic sheaf, pure of weight , and , then any eigenvalue of Frobenius on has magnitude at most , as does any of its Galois conjugates.

If we form the zeta function

in analogy with (2), (8), (12), (16), (21), or (25), Deligne’s theorem is equivalent to the Riemann hypothesis for , or at least to the “important” half of that hypothesis, namely that the zeroes have magnitude at most . In many cases, one can use Poincaré duality to derive a functional equation for which shows that the zeroes in fact have magnitude exactly , but for the purposes of upper bounds on exponential sums, it is only the upper bound on the zeroes which is relevant. Interestingly, once one has Deligne’s theorem, the zeta function plays very little role in applications; however, the zeta function is used to some extent in the *proof* of Deligne’s theorem. (In particular, my understanding is that Deligne establishes a preliminary zero-free region for this zeta function analogous to the classical zero-free region for the Riemann zeta function, which he then amplifies using a device similar to the amplification tricks mentioned previously.)

From Deligne’s theorem we conclude an important upper bound for (30) for structured functions:

To use this bound, we need bounds on the dimensions of the the cohomology groups, and we need the cohomology group to be trivial in order to get a non-trivial bound on the exponential sum .

As is usual in cohomology, the extreme cohomology groups are relatively easy to compute; for instance, if is affine can be shown to vanish, and can be shown to vanish when is geometrically irreducible and non-trivial. In any case, when has bounded complexity, both of these groups have bounded dimension. This leads to the bound

when is geometrically irreducible and non-trivial with bounded complexity. Finally, to control the dimension of , one uses a variant of the Grothendieck-Lefschetz formula, namely the *Euler-Poincaré formula*

where is the rank of , is the geometric Euler characteristic of (i.e. , where is the number of points omits from its projective closure ), and are the Swan conductors. (A side note: this identity formally suggests that there is some extension of the Grothendieck-Lefschetz formula to the case, with the right-hand side of (32) being interpretable as some sort of sum over the “field with one element“, whatever that means. I wonder if such an interpretation has been fleshed out further?) All the quantities on the right-hand side of (32) are bounded if has bounded complexity (basically by the definition of complexity), so we conclude that has bounded dimension if has bounded complexity. (Here we are relying on the fact that is one-dimensional, so that there is only one “difficult” Betti number to understand, which can then be recovered through the Euler characteristic; the situation is more complicated, though still reasonably well under control, in higher dimensions.) So we finally recover the square root cancellation bound

whenever is geometrically irreducible and geometrically non-trivial. We thus obtain the more general bound

for any geometrically semisimple lisse sheaf of bounded complexity and pure of weight , where is an algebraic integer reflecting the Tate twists present in the geometrically trivial component of (in particular, if there is no such component). Thus we have a strong “structure vs randomness” dichotomy for this class of functions: either has a geometrically trivial component, or else exhibits square root cancellation. Informally, we are guaranteed square root cancellation for structured functions unless there is a clear geometric reason why such cancellation is not available.

From Schur’s lemma, we then conclude the almost orthogonality relation

for two geometrically irreducible sheaves of bounded complexity and weight , where is when are geometrically isomorphic, and otherwise, and is an algebraic integer measuring the Tate twists in the geometric isomorphism between and (in particular when ). This gives a useful dichotomy: two irreducible structured functions on a curve with boundedly many points removed are either multiples of each other by a root of unity (after restricting to their common domain of definition), or else have an inner product of . (Among other things, this provides polynomial bounds on the number of distinct structured functions of bounded complexity, by using the Kabatjanskii-Levenstein bound mentioned in the previous post.)

A typical application of this dichotomy is to correlations of the form of some structured function (where the asterisk denotes a restriction to those that avoid the singularities of ); if is prime, one can show that this correlation is for all non-zero , unless correlates with a linear phase (i.e. it has an Artin-Schrier component). In a similar fashion, can be shown to be for all non-zero and any , unless correlates with a quadratic phase. These facts are reminiscent of the inverse theorems in the theory of the Gowers uniformity norms, but in the case of structured functions one gets extremely good bounds (either perfect correlation, or square root cancellation). A bit more generally, one can study correlations of the form , where is a fractional linear transformation, leading to an analysis of the automorphy group of with respect to the action; see this note of Fouvry, Kowalski and Michel for details.

## 121 comments

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25 January, 2015 at 12:14 pm

hikliceplehfor

for

You can get a good estimate for the .

– How to evaluate this expression? Will give us a permanent inequality of the real part of non-trivial zeros?

27 March, 2015 at 12:12 am

gninrepoli27 May, 2015 at 5:54 am

taiI solve Riemann Hypothesis.

Please see it.

http://vixra.org/abs/1403.0184

15 August, 2015 at 10:51 pm

A wave equation approach to automorphic forms in analytic number theory | What's new[…] e.g. this previous post for a discussion of this bound) to arrive at the […]

27 September, 2015 at 7:00 am

taiI rewrite the paper.Please see it.

http://vixra.org/abs/1508.0122

17 November, 2015 at 9:18 am

AnonymousRiemann Hypothesis (proven)

Two of the statements of Riemann.

1º- The series for the zeta function will have the particularity of (s= 1).

2º- Is very likely that the roots of a are real.

is to say:

with:

Indicated the beginning of a of the series for

15 November, 2016 at 7:37 am

gninrepoliIs there an equivalent statement of the Riemann hypothesis in Quantum Theory (Quantum Field Theory, Quantum Mechanics, etc.)? If there is a statement in the Quantum Theory, which is logically equivalent to statement. Then this statement is independent of ?

We know that RH is equivalent to a statement. Also we know that: “If a statement is independent of , then it is true”.

19 November, 2016 at 1:33 pm

gninrepoliI think that the sum of prime numbers are the imprint of some sequences +,-,+,-,+,-,+,-,+,- with continious function(‘s) (). If this is true, then an error in the formula must be symmetrical. My assumption is based on Perron’s formula, and the sums of the form .

19 November, 2016 at 4:35 pm

AnonymousOf course, the Riemann Hypothesis is true!

Reference links:

https://www.researchgate.net/publication/300068776_Proof_of_Riemann_Hypothesis;

https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2

27 November, 2016 at 6:54 pm

AnonymousThe Riemann Hypothesis (RH) is true!

It’s a SIN to think otherwise.

Q1. Where are all the primes in the critical strip, [0, 1] ?

Q2. Where are all the powers of primes in the critical strip, [0, 1]?

Q3. How are prime numbers and the nontrivial zeros of the Riemann zeta function connected?

Hints:

Please consider the Fundamental Theorem of Arithmetic, the Generalized Fundamental Theorem of Algebra, the Harmonic Series, and the very important PRIME NUMBER THEOREM since we are counting primes and predicting primes approximately.

If one can answer the above questions, then one understands why the Riemann Hypothesis is true!

Reference link: https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2

27 November, 2016 at 8:04 pm

AnonymousHint for Q1: 1/2 , 1/3, 1/5, 1/7, 1/11, …., 1/p for some prime distinct number, p.

Hint for Q2: We have 1/n where n = ∏(p_i)^k_i over distinct prime numbers, p_i, for some integer, k_i ≥ 1 .

Note: p_1 = 2, p_2 = 3, p_3 = 5, …

Hint for Q3: Prime Number Theorem …

20 February, 2017 at 11:20 am

The Fontaine-Winterberger theorem: going full tilt | Hard Arithmetic[…] For a taste of the veracity of this claim I would highly recommend reading this excellent post of Terry […]

11 March, 2017 at 7:35 am

RemerDear Prof. Tao,

I hope you are doing well.

I’m interested in this topic (Riemann hypothesis- which I believe it’s true) and since I love finding patterns this made me comment.

I know how busy you are but just in case you have time then I greatly appreciate it.

Actually I have difficulties to how I will start since I’m a novice in this area.

Let me say that I was motivated in writing this comment by the story of James Lovelock, one of the scientist invited by NASA for the search of life in Mars, all of the great minds were gathered for analyzing tools for detecting life/organism, then James Lovelock spoke-out his doubt (What if) this is not the way to detect life… then they realize (I’m not good in English but for those who are interested on the story you can find it in youtube).

I know I don’t have the right to say, but What If we just overlook some simple solution of Reimann. What if we need to look simply at once that the Riemann’s zeta function: enable the separation of composite numbers (to the left plane) and prime numbers (to the critical strip) that extracted from Euler equation at the right of the complex plane. That is the zeros of Riemann’s zeta function give us the properties of real numbers located on the corresponding plane – it just happens there is.

In my works, I named the Complex plane of Riemann by:

1st part where R(s)>1: the Composite and Prime number zone

2nd part where 0<R(s)<1: Prime number zone

3rd part where R(s)<0: Composite number zone

Let me start in 3rd part where the value of is R(s)<0

I have named it since I’ve found connections between the trivial zeros (2,4,6,8..) of the function to the composite numbers.

I define a number as:

1. All natural numbers can be represented by a series of ratio of numbers (which carried its divisibility information), I call it “Spine” (I likely to compare it as their DNA). That is in order for a number to be composite its Spine should contain(s) either or combinations of 2,4,6.. (trivial zeros) – I can show it with formula

2.The Spine of all natural numbers follows the cubic equations – I can prove and illustrate it

By the way I’m not sure if I can show you exactly my works here because I have to put some formulas and it will be supported by tables of numbers and drawings to show the behavior, especially in Prime number zone.

Hopefully there's a link to where I can send my works, that’s if you find it relevant.

Remer

13 March, 2017 at 9:35 am

RemerDear Prof. Tao,

I know how busy you are but just in case you have time then I greatly appreciate it.

I hope I’m not annoying here, for continuing my claimed in my previous post.

(Continuing at Composite number zone : R(s)<0)

I only consider odd numbers for obvious reason

Let me redefined my 1st claimed

1. All natural numbers (odd) can be represented by a series of ratio of numbers carrying its divisibility information; I call it “Spine” (I likely to compare it as their DNA). That is in order for a number to be composite its Spine should contain(s) either or combination of 2, 4, 6… (trivial zeros)

Let’s consider any composite number, P.

(Ex.1) P=9: its Spine can be shown as 5/4

6/3 = 2

7/2

(Ex.2) P=15: its Spine can be shown as 8/7

9/6

10/5 = 2

11/4

12/3 = 4

13/2

(Ex.3) P=21: its Spine can be shown as 11/10

12/9

13/8

14/7 = 2

15/6

16/5

17/4

18/3 = 6

19/2

Anyone may wish to try it to all odd composite numbers, as long as you will maintain the Spine’s Pattern, you will come out with the same result.

• Trivial zeros are the imprint(s) of Composite’s Spine (any pattern lovers may conclude therefore that Non-trivial zeros are the imprint of Prime number’s Spine – yes, but its Spine is so unique that it only follows the rules inside the critical strip- I will illustrate it)

Pattern of Composite’s Spine

o let a= ath place of the odd number (P)

o that is P=2a+1

o sequence = (a+1)/a , (a+2)/(a-1) , (a+3)/(a-2) , (a+4)/(a-3)… (a+n)/2

o the denominator begins at “a” and ends at “2” (there’s a pattern why it should begins and terminate here, which I think unnecessary to explain)

The Composite’s Spine can help us to do

o Primality testing

o Prime sieving (I can illustrate it)

We can say that a number is composite if it satisfies to

[(a+1+n)/(a-n)] = m or to [P/(a-n)]-1 =m

where:

n= 1, 2, 3, 4…. any natural number {(n-1)th place of the number sequence)}

m= 2, 4, 6…. any the trivial zeros,

a= ath place of the odd number (P), That is P=2a+1

This is how beautifully a number can be interpreted (giving hope for non- technical person, like me, to explore) and I hope that there will be someone (mathematician) who will give progress in this insight.

Remer Catacutan

3 April, 2017 at 1:52 am

MatjazGI am curious as to your thoughts on the recent progress in solving the Riemann hypothesis via the Hilbert-Pólya conjecture described in this Physical Review Letters article:

C. M. Bender, Dorje C. Brody, and Markus P. Müller, Hamiltonian for the Zeros of the Riemann Zeta Function, Phys. Rev. Lett. 188, 130201 (2017)

DOI: https://doi.org/10.1103/PhysRevLett.118.130201

Abstract:

A Hamiltonian operator H is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of H is 2xp, which is consistent with the Berry-Keating conjecture. While H is not Hermitian in the conventional sense, iH is PT symmetric with a broken PT symmetry, thus allowing for the possibility that all eigenvalues of H are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that H is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true.

–

Is there hope that this could lead to something substantial?

22 April, 2017 at 2:09 am

int.mathI have found that:

.

Does anyone know equivalent results?

(η(s):dirichlet eta function)

25 April, 2017 at 1:44 am

int.mathI have also found a similar relation with Riemann zeta function:

.

26 April, 2017 at 7:49 am

int.mathPlease, can someone check this:

27 April, 2017 at 4:08 am

AnonymousIt seems that Stirling’s approximation(!) for the Gamma function was used in the Zeta functional equation (anyway, it is not an exact identity.)

25 May, 2017 at 9:45 am

David Cole“If you can’t explain simply, you don’t understand it well enough.” — Albert Einstein.

What does the Riemann Hypothesis (RH) mean?

RH confirms the existence of prime numbers in an optimal way. Or rather, for all positive integers, k > 1, there exists a prime number, p, which divides k such that either p = k or p ≤ sqrt(k) = k^(1/2) where RH states the exponent of k is 1/2.

Please keep that fundamental fact in mind when discussing the truth of RH.

Reference link:

https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2.

.

28 May, 2017 at 1:36 pm

David ColeDoes the nth non-trivial simple zero of the Riemann zeta function indicate the nth prime, p_n, occurs as a prime factor in all multiples of p_n?

30 May, 2017 at 7:55 am

David ColeHmm. The short answer is a resounding yes! Wow! The great Riemann saw it all.

Reference link:

‘On the Number of Prime Numbers less than a

Given Quantity.(Ueber die Anzahl der Primzahlen unter einer

gegebenen Grosse.) ‘,

31 May, 2017 at 8:38 am

David ColeFYI:

Please an answer at link:

https://www.researchgate.net/post/Does_the_nth_nontrivial_simple_zero_of_the_Riemann_zeta_function_indicate_the_nth_prime_p_n_occurs_as_a_prime_factor_in_all_multiples_of_p_n

7 June, 2018 at 3:15 am

Heat flow and zeroes of polynomials II: zeroes on a circle | What's new[…] , then, as famously proven by Weil, the associated local zeta function (as defined for instance in this previous blog post) is known to take the […]

1 July, 2018 at 2:57 am

Pallav GoyalThe summand in identity (2) isn’t defined when , so maybe the sum should start from ?

1 July, 2018 at 6:52 am

AnonymousSince , it seems that the first summand is .

2 July, 2018 at 1:35 am

Pallav GoyalYeah. I guess the meaning of the expression is clear as such, this is just a pedantic correction.

14 January, 2019 at 1:50 am

AnonymousI have always liked this image,http://en.wikipedia.org/wiki/Image:Zeta_polar.svg, but could someone explain it? Is important or useful? Hides the secret of RH? Has some modular arithmetica?

5 February, 2019 at 9:06 am

Richard HarrisThis is summary of the concluding part of Section 3 that will come as the last of small installments.

Within the critical region, if is root, the necessary and sufficient condition for the existence of another root with the same imaginary part is the special relation between some constant entity and the difference of two others over a certain operation *. Formally, the necessary and sufficient condition is: . When is shown to be for , it is obvious that the condition can not be satisfied, implying RH is true.

***

, , or something may not display properly due to limitations of formatting or LaTex rendering. My apologies for any error or imperfection.

However, a PDF version of the expos\'e in its entirety can be requested.

6 February, 2019 at 8:10 am

Richard HarrisThis is introduction to a demonstration that none of the nontrivial zeros of can be off the critical line.

The section to shortly follow ought to be a pretty colloquial piece, laying out quite precisely the proof strategy adopted, paving way with preparatory material.

The section after the detailed proof roadmap will, in a formal manner, reiterate the major notions involved and then make the final connection of the (literally).

Such two-tiered style offers the option for people to go directly to the concise exposition but leaves available the elaboration for people to refer back to if, at points, details deem desired.

In an effort to bring Riemann Hypothesis (RH) to general readership, no complex analysis may loom into view. Instead, taking an equivalence, the proof can be carried out in a purely real-valued fashion. All results and conclusions, intermediate or final, tend to be drawn from what are well-known and established, largely folklore mathematics and not arisen from specialized investigation of RH or of the function.

We assume an understanding of the reader of the following (i.e. RH):

6 February, 2019 at 8:43 am

Terence TaoThe comment section is not intended for extremely lengthy texts. If you have such a text, you can host it on another site and post a link to it here.

6 February, 2019 at 9:13 am

Richard HarrisNot extremely lengthy. Just a bit more following the next one.

2 July, 2019 at 4:13 am

Richard HarrisNobody might give a damn if not for this wonderful piece of advice. It is no surprise that the whole thing then became vibrant with life through an -pixel area (for some compendious ).

I apologize for the excessive length which came to a fraction of what you had written on the topic. But there was, more by design than by accident perhaps, the humble attempt at limiting the size to no more than 8 pages and, with the 2-page-worth Section 4 subdued, it would only come to 0.5 dozen when put on A4. Even though a mere three-word title, “Zero Zero Off” was left out to not waste any more of your precious blog real estate.

In the first sentence of the very first installment, I had, I believe, the explicit word which, in the blind greed for something looking , should be conveniently overlooked. Is it now habitual mathematical breakthrough, following the other elite prophetic polymath, that before capturing a glimpse we already know everything, even its length? Actual suppression of truth of any size, employing any cute means (not excluding camouflaged instigation or vociferous silence), would hopefully not produce volumes but could in effect result in endless inconclusive tomes. I do not know for sure if, for such a problem as Riemann Hypothesis, a resolution of the modest length of Section 3 of PROOF is permissible, but I can surely not prevent one from playing innocent to artfully turn a blind eye to the plain facts, including the obvious that the proof can be easily compressed — actually been done — down to only one simple expression:

If you can further shorten it to better fit in any margin of any page in any book, there is for you, as reality or illusion, the entire genuine community that has been known to be eager to show interest. Besides, any URL I can incompetently come up with seems lengthier than that single morphemic unit.

The world, a perfect , is wonderful enough to embrace all genres and scenes that we can fancy. Emperors, adulators, onlookers, bare wisdom, clad nudity, etc. etc. It is admirable to be able to show the ability to at the same time be on Goldbach, Twin Primes, Navier, … But reality is that not all can be all over the place at once. If one can only prove Grand Riemann Hypothesis, could the poor soul be allowed a minor negligible chance?

Who came up with the brilliant idea of the block(head-shaking clickable) to be so brilliantly promoted that incomprehensible math insight can bravely gather behind? My low intelligence never figured out any crap of mathematical sense from those addictive mousy mouths. Care to share what you can get?

It is also quite a spectacle that the click-count HERE isn’t monotonously increasing! Is it law of nature of “what goes up also comes down”? Or is it some kind of mechanism manipulated to properly guide public opinion? May the perfect be one day told the secret of how to CLICK the count DOWN?

***

Also wondering if the 20 bucks can count toward the $32 (since she recognized him) for that 40+ minutes of Bach flowed from the $MM+ red Strad. Does that ring a Bell?

At L’Enfant Plaza Station, unfortunately, there was no button to click, though not entirely sure if law enforcement came with the polite suggestion to not impede purposeful traffic.

Memory is also of a young European who did a tour across the pond, fiddling Schoenberg of course. When asked to describe his thundering applauding audience, he came up with the word “enthusiastic”.

I am sorry to have sounded like piling praise on some brilliantly useless button. I can cowardly worry about thumb-downs 24/7 without anybody else’s effort. Pathetically, I am able to stop worrying on only 2 occasions throughout the year:

6 February, 2019 at 9:10 am

Richard HarrisThis proof is rather of a geometric flavor and is to show whether a certain property can be satisfied.

If the satisfaction, or absence of it, brings about the resolution, then it must be shown equivalent to the existence/non-existence of distinct roots symmetric to the critical line. Therefore, the proof is to first show the equivalence, followed by the demonstration of the lack of the spatial property for suspected roots.

We make it explicit that in our discussion, unless stated otherwise, , the value range of is , and the complex plane (or points on the complex plane) we talk about, in the context of RH, will always exclude 1, the singularity. In the process of the proof, there will be spatial entities, such as planes and infinite-dimensional space, within which there can be points or vectors irrelevant to RH. For example, a set of vectors may possess a certain property and when we refer to vectors having such property, we implicitly exclude those having the property but irrelevant to RH, unless explicitly stated otherwise. When we say a point or a vector is on a plane, we mean that the end point of the vector, or the point being the end point of a vector, is on that plane. In that sense, a point (as the end point of a vector) is interchangeable with the vector.

Pertaining to any two planes (in -dimensional space for ), independent of our topic, their spatial relation will be exactly one of the following scenarios:

By “intersect” we mean cutting into/through each other within certain region(s) of the planes.

In Scenario A, the two planes share no common points.

In Scenario B, a line (segment) will form where the two planes intersect, and all points, which will be infinite on that line (segment), are commonly shared by the two planes.

In Scenario C, the planes must meet but only at isolated, discrete points.

Observe that the spatial relations introduced are independent of the size (finite or infinite) or curvature (Euclidean or not) of the planes and independent of the number of dimensions of the containing space. Two-dimensional space is of course special where “planes” are not planes but lines/curves. Otherwise, in -dimensional space, (hyper)planes will always obey exactly one of the spatial relations, regardless if the context is RH.

Now imagine that the demonstration of the validity of RH involves two planes, referred to as and , satisfying the following (still assumptions at this point):

It is obvious that if , as assumed, contains all uniquely -root-related points, then only Scenario C is relevant to RH and is only on one side of , since Scenario A means “no root” and Scenario B means “infinite number of roots” and “infinite number of root-mapped points” within a finite region.

Let denote the unique mapping to points on . More formally: iff where and and are points on .

Assume that has some unique property, called the or simply , and that any point on corresponds uniquely to a root of . More formally, on is iff there is a unique such that . (, not , is unique with regard to )

Since said () property is UNIQUE to , only -points (i.e. points on ) can have that property. Therefore, there is only ONE WAY for , a -point, to be : is also a -point.

In other words, and share , a meeting point of the two planes. Geometrically, the meeting points of and are all and the only -root-mapped points. This suggests that if any distinct and are shared by and where , then RH is false. Otherwise, RH is true.

Therefore, the question of whether and can both be roots is reduced to (equivalent to) whether and can both be -points, the geometric version, the equivalence we desire.

7 February, 2019 at 2:25 am

Richard HarrisThe resolution of RH, therefore, depends on whether we can establish what are assumed, and on whether we can find such planes and . To be concrete, we have to achieve the following tasks:

We accomplish by finding a unique mapping from complex numbers to points on .

Representing Vector

The representing vector of is:

In other words, the th dimension value of the representing vector is just the th term in the summation.

Now, for any on the complex plane, there is a unique point on , and for any on , there is obviously a unique . This defines the unique mapping between complex number and vector whose end point is on .

Next, we define to be the (flat, Euclidean) plane to (which is the unique identity vector) and introduce Hadamard Product and Dot Product as follows:

() of vectors and is a vector whose th entry value is the product of the th entry value of with the th of . E.g. assuming and , then their Hadamard Product is:

() of vectors and is a scalar (number) resulting from taking the summation of all the entries of the Hadamard Product of the two vectors. E.g. assuming the same for and , then the Dot Product of and is just the summation of all entries of (the Hadamard Product of and ):

We observe that. by the definition of representing vector, has an equivalent vector form:

and by the well-known relation (that two vectors are perpendicular if and only if their Dot Product is zero), we see that we have found , the plane perpendicular to , having the to contain all vectors, including for any root , orthogonal to $V(0)$. is completed.

In particular, the following are equivalent, stating the same thing:

is root

is perpendicular to

is on unique plane orthogonal to

The last thing to show (i.e. ) is whether there can be more than one distinct root. Without loss of generality, we may assume to be root and try to find out if there can be another root , where and .

Since is (assumed) root, is a vector on , orthogonal to . Therefore, if is also root (i.e. is also on ), then must also be a vector on (but then obviously not on ). Therefore, to prove that there is no other distinct root, we only have to show that no such orthogonal-to- exists.

***

Hadamard Product may be seen as redundant in this version, but alternative proofs may use it, like other variations of this proof. Besides, can always be expressed as to make the constant identity vector prominent and explicit.

7 February, 2019 at 2:25 am

Richard HarrisRepresenting Vector (of ):

It is immediate, from , that is one-one correspondence and that has the following equivalent vector form (where is Hadamard Product and is Dot Product):

By the well-established Cauchy-Schwartz Inequality ( and as vectors, and as the angle in between):

we have the equivalence: is root iff is on , the plane orthogonal to .

It follows that distinct and being roots is equivalent to and being on . To show that if is on , can not be on , we only need to demonstrate that can not be a vector on .

Recall that RH concerns complex numbers whose real parts, i.e. and , are confined within the range (0, 1) and observe that if and are distinct. Let , , and . If is on , then must be zero. But by :

and the validity of Riemann Hypothesis is established.

7 February, 2019 at 5:01 am

Richard Harris$V(\alpha-\beta)$ is $\mathscr{Q}$-point, but not $r$.

Just to be sure the obvious is not distorted (again).

29 March, 2019 at 10:48 pm

Richard HarrisMathematics is a solemn enterprise in which comedy seems foreign. But feigned seriousness can at times overshadow the acting-out of farce, and is no more foreign than it is frequent. Pessoptimistic drama in mathematics often unfolds with an attempted departure of Keynesian difficulty from stale ideas. And in the situation with Riemann Hypothesis, it may be harder than Sartrean Exit.

RH is so amazing that a person can be rumored to have gone berserk over it, and another to have wanted to trade life-supporting organs for a settlement of it, and still another to have wished to make its resolution the highest curiosity in second life (or maybe secondary life). Mathematics never ceases to amuse and always seems willing to stop to. To bring a proof down to an undesirably wordy level isn’t quite complimentary but, for a solution that hundreds of others rely on, repetitive verbiage may be given the special privilege to be allowed to recycle.

Honestly, nothing is not well-established or is beyond elementary in PROOF (i.e. the proof made public here around Feb. 5 through 7, 2019) which even a reader with little mathematical training can establish (without proving anything himself) by confirming each of the individual proof steps with honest people having the basics in mathematics.

There can be more than one way to state RH and, quite possibly, more than one way to prove. A simplest point capturing RH for it to be proved is: the Hypothesis is true if is strictly decreasing in the range of (0, 1). In other words, RH is false only if there exist different real values and between 0 and 1 such that .

1. We do not have to worry about how high up in the complex plane there may suddenly appear distinct roots symmetric to the critical line if we just from current frame of mind for a bit. Neither do we need to look too far to the left or to the right. We know that, excluding the real-axis, only the region between 0 and 1 may have roots. In fact, we can (temporarily) ignore the symmetry, and forget about 0.5, the value associated with the critical line. In that sense, we only need to prove:

2. If RH is true in the above sense (of unique ), then another assumed root , where but , will have to induce absurdity, reaching a conclusion contradictory to some fact. The contradiction will firmly establish the validity of RH beyond any doubt.

3. To show the contradiction, we want to show that all roots have some unique property (orthogonality) and operations on and (the assumed roots) will have to result in some other entity having the same orthogonal property. However, the resulting entity having that property is in direct contradiction to simple, elementary truth, namely, is strictly decreasing in the real value range of (0, 1).

4. The unique property referred to in 3 comes from the vector representation of , or more precisely from , for which the reader may refer to Section 3 (or Section 2) of PROOF concerning Representing Vector. In essence, for any complex number , is root if and only if its representing vector is orthogonal to . The orthogonal property is of course consequence from the well-established Cauchy-Schwarz Inequality, which any reader can prove himself or confirm with help from others. We can redundantly clarify that Dot Product () of with is just “each of the terms in multiplied by each corresponding term of (i.e. by 1) and then summed up”, which is no more or less than .

5. By the vector representation of and the orthogonal property, any representing vector of must be perpendicular to . Therefore, any corresponding vector of root must be on the (unique) plane (we refer to it as ) that is orthogonal to . If is perpendicular to , then all vectors (completely) on , including all vectors where is root, must be perpendicular to .

6. We know the basic fact that two (non-zero) vectors that are not multiples of each other uniquely determine a Euclidean plane that they are completely on. (A vector is completely on a Euclidean plane iff multiples of the vector are on the plane). Furthermore, the sum and difference of the two are also vectors completely on that plane. In particular, and being vectors completely on results in their difference also being completely on , implying that must also be orthogonal to , i.e. .

7. If we show that can not be zero, then either or (or both) can not be root(s). The concluding part of Section 3 of PROOF shows exactly that, via the elementary fact that is strictly decreasing for real . The demonstration of takes advantage of the well-established elementary fact that is isomorphic to . A reader does not necessarily have to establish this himself. Such simple and elementary theorem can be confirmed with any person having basic math knowledge and integrity. Using this fact means that we need no complex number calculation. We can carry out the computation, which is just symbolic manipulation, in purely real-valued sense. It is so elementary and trivial that an explanation may be taken as an insult. To be complete, however, we do it once more.

By Cauchy-Schwarz, if and are real vectors, then , where is the angle between and . And for complex vectors and , , where is the angle between and .

Since is isomorphic to , and , where and are complex vectors, and and . In particular:

By the way, people not sure of the isomorphism can, as an exercise, verify first with one-dimensional and so that is just , and then use induction on the number of dimensions (but remember the conjugation, though). If the reader regards the one-dimensional (or ) just as a vector with two real dimensions orthogonal to each other (and operations on them inter-affecting the magnitude of each other), as if there is never the convolution of ‘imaginary’, things may be simpler.

With the equivalence (), we can try to determine if (which equals ) can be zero:

(for , etc. above, refer to concluding part of Section 3 of PROOF)

We clearly see that, assuming two roots and within the critical region, with the same non-zero imaginary part, we will have to reach the conclusion for , a direct contradiction to being strictly decreasing for real value range of (0, 1). Since can not be zero, can not be orthogonal to and can not be on . If is not on , then at least one of and is not on (because if they both are, their difference must be on ). Not both being on means that either or (or both) can not be root(s). In other words, there can never be two roots with the same non-zero imaginary part, or implies .

***

Loosely, the simple perspective is, for ANY two complex numbers and , regardless if they are inside the critical region, the difference of their mapped points/vectors and , i.e. , has to be on for both to be roots. If not, either or (or both) can not be root(s) of . Of course, such is a slightly more general view than the focus on the critical region only. Needless to say, either, that being on does not mean that is also a representing vector, and that and are not necessarily the same thing.

The vector view of may offer a sense of direction and orientation. With as reference, for example, the ‘pulling’ of on by Hamadard may exhibit as being in opposite directions with regard to (not necessarily at ). Viewing RH in geometric terms, there are characteristics we can expect. For example, of all straight lines tangent to the line segment , for fixed and , that contains a root-mapped point, only one can be parallel to with the tangent point being . In that sense, there may be interpretations to the derivative of , Lehmer Phenomenon, maxima/minima, etc.

25 May, 2019 at 3:16 pm

VladThis like one correct proof. Seven proof steps all good and enough easy. But I can wrong and can not see problem a little place. Anybody can say exact error where?

18 August, 2019 at 12:21 am

AnonymousThis may seem like a stupid question, but what exactly does the Riemann hypothesis mean for the prime numbers? I mean the twin prime conjecture just means that there are infinitely many pairs of primes with difference 2, is there such a straightforward formulation for the consequences of the Riemann hypothesis?

9 November, 2019 at 2:17 pm

RexDear Terry,

Your definition of mixed sheaf seems to be a priori weaker than the usual one (there exists a finite filtration of subsheaves $\mathscr{F}_1 \subset \cdots \subset \mathscr{F}_n = \mathscr{F}$ such that the successive quotients are pure). Does it turn out to be equivalent? If so, is this easy to see?

5 July, 2021 at 3:02 am

Richard HarrisOnly for a modest purpose.[1]The discussion is via

SZCQ(i.e.SimpleZeroConjectureQuestion asked here).It may be obvious that we can not answer SZCQ without a proof (of either RH or SZC). However, to include the proof(s) will make this discussion too lengthy[4] (and I do not necessarily mean 160+ years or any period shorter than that). Therefore, the proof(s) will be running separately, parallel to this in space but not necessarily in time.

To unfold our discussion smoothly (or with wrinkles), and for the sake of our discussion, let us just assume SZC. In particular, we assume that SZC and RH have been proved “in one breath”.

From the answers/comments concerning SZCQ over at MO, it is obvious that virtually all the opinions are leaning horizontal toward “stronger” (and ??equivalently?? “harder”), and the notation used is “RH + SZC”. Obvious too is the context where “stronger” means that SZC has RH as a premise, in other words, SZC with the assumption that RH is true. Therefore, there is such claim as:

RH + SZC is stronger than RH (alone)being “formally trivial”.Obviously, by the

currentstate of the art (with effective denials of any RH proof), there is NO showing that RH implies SZC or vice versa; and there is NO showing that one will have to have the other as prerequisite, either. The leaning opinion (or the leaning angle) can hardly be right[5], but it is perhaps not even wrong, and there is no such thing as “formally trivial” from thin air. Religion seems to have been promoted as if it were rigorously established fact.So whatever the analogy[6], as long as it carries this “stronger” notion — this cooked-up ‘reality’, it may only lend verisimilitude to that assertion at best.

If we have to have one, the more appropriate analogy should be:

allowing the possibility for (a sense of “proving both in one breath”).

It is also possible that confident minds got confused with the best can be drawn (in the absence of a proof):

But that may be flagged by people, me to say the least, as “smart truism”[7].

Without going into the details of proofs, we can contemplate the following possibility:

Therefore, “stronger/harder” is moot. We (may) have been viewing a problem in a weaker form that simply does not exist!

And now, consider the possibility of “RH = SZC”[8] and the question behind SZCQ: Is there the possibility

reasonthat we can not prove?[1] Had been sure that ephemeral pixel life isn’t worth much, but then enlightened to believe that some blog real estate could be worth a lot. Now, this ‘piece of land’ having been laid largely barren and abandoned got me thinking again. Since 2019, this corner of the world has been awfully quiet. Various settings seem to neither vary much nor vary into new stuff much, yet some old settings may still need be properly considered or re-considered. So am thinking of pondering on ‘RH vs SZC’. Please be absolutely clear that our purpose is modest. We make sure that no obstacle is created. If anyone wants to or absolutely needs to devote his/her lifetime to the settlement of Riemann Hypothesis after its settlement, there is absolutely no hindrance.

[2] SZC (

SimpleZeroConjecture) is the conjecture that all non-trivial roots of aresimple.[3] Nobody can because it is false. (Replace ‘because’ with ‘if’ if the religion is not atheism.)

[4] Full proof may be too much (even if it is a single word). It nevertheless does not hurt to take a look at the gist of one proof (in elaboration):

[5] Simply, the existence of non-trivial root with does not necessarily contradict RH.

[6]

[7] Deriving X with the assumption of Y so that X implies Y. Pretty neat sense of “stronger”.

[8] If uncomfortable with (or even hostile to) “RH = SZC” as fact, try having it as a conjecture.

5 July, 2021 at 7:16 am

AnonymousA surprisingly short comment on such important question.

5 July, 2021 at 2:43 pm

How do you work in midage?How are you able to do outstanding research in late 40s? Do you have any learning disabilities because of midage? Is there immunity to midage debilitation?

20 July, 2021 at 3:20 am

Richard HarrisSZC = RH?A proof is better than a world of beliefs.

Therefore, we are going to actually do with an actual proof, one of those, among all the versions, that were made public.

It is inconceivable that some one in this universe would think so highly of his time and space as to want others to explain again and again that a proof is a proof when it already is, in order that he can persuade the world to turn a deaf ear to. So what is going to be said, concerning the proof re-displayed beneath, is not to re-elaborate its entirety for RH but to highlight its relevance to SZC.

For those who are unfamiliar with the notations used in the proof, may I please suggest the whole expos — made available here earlier around Feb. 5 through Feb. 7, 2019?

A look at the proof first, with an overview to start with:

Now we consider the following:

But who can guarantee that line (6) is zero?

Mathematics is not a belief system. But it alters nothing by asking: How much do you now believe that RH is right?

(

. . . to be continued)———

[1] This question can be seen as prompted by SZCQ, which was seeking the reason for “SZC > RH” when unwittingly provoked the “SZC + RH > RH” assertion, generating the many clicks and comments and analogies and more.

A thousand clicks plus comments plus maneuverings won’t trump a proof, and a formally trivial one showing “stronger” would carry more strength. So we only need to have a proof presented, and there is no need, for example, to quibble over the bearings of almost verbatim words from another version.

[2] And a truth-seeking question, such as SZCQ, is better than a thousand wiggles for awkward limbo. Such a question points to the very heart of one of the most important questions of our times. It can mark a rare moment of courage in our Brave New World, not intimidated by great advances in mathematics or by “formally trivial” or by being looked at as weak in intelligence.

For weeks, TPTW’s factual revelation of the awkward state of the art received only one (net) up-vote, while the more opinionated one got twenty. Welcome to the sane world of smoke and mirrors!

[3] Different versions of RH proof will be assembled and made public, but only not at this moment. (“parallel in space but not necessarily in time.”)

[4] Please note that it is just

a look at the proof. Our modest goal is not to win over any believers. If it did not convince then, it may do little to remove any doubt now.20 July, 2021 at 3:24 am

Richard HarrisSorry, footnote [3] is misplaced. Please ignore.

— RH

22 July, 2021 at 3:01 am

Richard HarrisSZC = RH?(continued)That brief walk-through of the RH proof may not constitute a definitive conclusion and may not be accepted as such (before, or even with, a direct or indirect demonstration of the equality between line (5) and line (6)). But I am deliberately having this as discussion for all rather than as lecture by myself. Enough freedom is proffered so that readers can fill in whatever details they deem needed.

However, the walk-through does, IMHO, indicate one thing: Riemann Hypothesis can never have a Planck of hope to be something not provable. And it definitely says more than just that.

Taking the assumption that RH is valid, (5) has to equal (6). Be that the case, the terms related to the imaginary can obviously cancel out.

We then, instead of merely accepting (that they do cancel out), should be wondering why they can. And this little inquisitive moment may sooner or later reveal the property of . And that property should at once settle SZC (as well as the other I brushed upon concerning simple zeros).

As such property is not just speculated, but real and concrete, it will show off itself. Suppose that our exploration also reveals the property in a different branch of mathematics, even if we are quite cautious about reaching a conclusion, we are likely not hesitant to declare Riemann Hypothesis as resolved.

When RH is verified for up to, say, non-trivial zeros, some would like to take that as evidence. However in the mathematics world, even checking up to can only be coincidence but evidence. On the other hand, if different branches of mathematics are saying some same thing (concerning the property), it can hardly be coincidence. It can be the source for confidence.

So to put “SZC = RH” in the opposite direction, suppose that we do not have anything to do with that proof step when trying to prove RH in algebra, by accident we prove SZC, with the discovery of said property, in another branch of mathematics. We can suddenly (also inevitably) find that RH is then trivially true in light of that discovery.

If we look at the entertaining aspect of it, it is like proving a conjecture in order to prove one of its proof steps.

Findings that suggest some inner working or structure ought to catch attention, while discovering a numerical bound, or making progress toward tightening of such a bound may be different. It depends on the problem. Not all instances of that sort are for sure a breakthrough.

Let us take the situation that nobody knows what that property is and nobody can even be sure that such property actually exists. But does it hurt to investigate and explore? Should we prevent new ideas? Can we be allowed to ask a simple, harmless question of “SZC vs RH” and try to provide a proof? Is the math community so scared of being tolerant that the suppression of ideas has to be tolerated?

At this point, I think I have said all I need to say for the moment about SZC and SZC in relation to RH. But since we are already here and we already re-examined the proof, I would like to move a bit further, as further clarification of issues related to the RH proof.

(

. . . to be continued)22 July, 2021 at 11:36 am

Anonymous“… to be continued” is a promise or a threat?

23 July, 2021 at 2:28 am

Richard HarrisThanks for the question through human words (and believing that mathematically non-zero chance of a Shakespeare Sonnet by chance may just be myth). :)

We live in a wonderful time when FLP (P as in Promise or mistakenly as in Problem) is always spoken of as FLT (T as in Threat).

23 July, 2021 at 12:40 am

Richard HarrisSZC = RH(continued and completed)If RH is true, then (5) = (6).

But that is so bizarre! It looks so wrong! It seems that is equated with .

It can not be right as they obviously are not equal!

Yet again, if RH is true, (5) = (6) has to be correct.

To calm our disbelief of such an outlandish thing, which seems to be rammed down our throat and seems to have to be swallowed even it is horizontal going down, we can give it a name: post-doc’s dream.

It would be crazy, impossibly crazy, to have come up with what seems to be equating with .

But if we start from (4) ending in (5) and then, instead of proceeding from there, we start afresh from (8) going backward to (6) and stop (getting stuck) there as well, we will suddenly realize that if RH is true, (5) and (6) must equal (and equal zero)!

. . . Then things could look much less crazy.

The proof was not just casually thrown out there. A bit of thought went into it. It is so designed that if anybody has any problem, it will be with that one single proof step, since all others are too obvious and elementary. Even if that proof step is not proven, we have another equivalent statement of RH that is different from all other known formulations. So there is only one proof step, one little thing people need to focus on.

No matter how many ways there are to prove that step, it is obviously not mandatory to have it proved directly. As long as RH is true, that proof step must be valid. This means, if RH is proved in any other way, this proof, or rather this proof step, is made to stand.

We see that the proof has two parts, the first part (first three lines) is geometric, indicating a way (to prove) different from that of the second part (which starts from line (4)). In fact, the proof is specially designed as an actual merge of two (simplified) proofs whose originals purely involve two separate branches of mathematics. Even for the second part, that proof step may be bypassed and the conjecture proved in a number of slightly different ways with the same general approach.

Therefore, a proof of RH may no longer be important. The important thing is actually and factually the question behind: Why can we not settle Riemann Hypothesis? And such a question, or any falling into its category, will always have two inquisitive parts to it:

The former is interesting, allowing us to muse over what has been less desirable and adversely entrapping in our view of the problem itself. The latter, however, is boring (though obvious and extremely easy) to answer (material for answer here). No amount of manipulation or wriggling can change the history that we all have seen and experienced, history that we actually wrote ourselves without any ink from historians.

Dear readers, you now have a chance to answer the ultimate question for both of its inquisitive parts.

Thanks for your kind patience that has carried you thus far till this point. Please do not forget to express your support, or your high-value emotion, by clicking …

———

[5] (

Too long a footnote. Omitted. But different versions of RH proof will be assembled and made public, only not at this moment — “parallel to this in space but not necessarily in time”.)[6] Damn! Just as I have found the use of buttons, they are gone! But we should not panic. :) Math, the unique democracy via the freedom of speech by the <SOAM>mouth</SOAM> mouse, can still be exercised at places where life seems to have difficulty catching on. :) (where

SOAMstands for Strike Out As Mispronounced)Besides, all this stuff, no matter how rigorous, may just be fairy tale (if we keep on telling).

23 July, 2021 at 2:25 am

Richard HarrisOnce upon a time . . .

Even before a single thread was ever seen by anybody, the beauty weaved in was already extolled in heated debates that had hardly any disagreements in the great details about patterns and colors displayed on the never-witnessed fabric. When the procession came into sight, however, glorification turned into hushed awe. Suddenly, in the quasi-silence filled with didactic vocalizations of worship, outside

fairy talewhile in, a crisp voice was heard.“He’s got nothing on!”

A hustle then came. It came like a shudder from the crowd shocked at the prospect that blatant vulgarity could cover up naked truth, yet fail to at the same time.

“Your Majesty, the fashion show is doing great!” Fearing that the regal steps might become less confident thuds, the tongue of the exceptionally competent eunuch wasted no time in dropping toward the boot. His Royal Highness, meanwhile, felt the soothing and encouraging caress on the bare toe.

“Yeah! You are doing just fabulous!” Another, believing to be holding membership to some court coterie, also cheered him on. “What we just heard about the expository coverage may sound like something legible but is mere inexplicable noise. Your admirers are obviously insulted, not by the obvious but what it tries to conceal, and you can rest assured that the annoying noise maker will be promptly chased off. Benevolence, nevertheless, can be cleverly shown, benevolence of allowing him a YouAreEl, a modern arrow, pointing in the direction of the remote place he may be driven to, where he hopefully won’t be heard, ever! The visibility of this high style is the greatest symbolism in the universe, synonymous to intelligence and superiority. All who fail to see or fail to fall over one another to declare the ability to see are simply admitting stupidity. Besides, you can pin ultimate hope on your subjects, solidly on the semi-liquid brains of moron, of nothingon, …, of lepton and of anyon that shall never develop the aptitude of growing guts to repeat the noise. As we are making history and making up history all at once, a chance is a terrible thing to miss. If you just hold your head high and carry on the parade with more resolve and, better, with more fanfare on anything that can distract and redirect attention away from the noise to wisdom au naturel, it will be first time ever in human history that sizable twofold half-wits make an all-out effort to loom intimidatingly even bigger.”

———

[1] Carrying around a heavy overdose of excessive IQ, or the love of it, reality would be doomed to easily take an abrupt turn to become more incredible than fiction.

[2] Only when fallen in

Love & Mathis one able to resort to exhibitionistic tactic to get attention. We just can not make this up:reality hiddenwhen you strip.[3] In b(h)f-based existence, all matter, laughing or otherwise, is made of

fun, the mental particles.[4] Quaylian for

anyone, and the only correct spelling for condensate mass in bozonick systems with superimposed spins.[5] Mathematical identity (of complete wholes): .