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We have seen in previous notes that the operation of forming a Dirichlet series

or twisted Dirichlet series

is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fourier transform, since the functions and are multiplicative characters.

Similarly, it turns out that the operation of forming an *additive* Fourier series

where lies on the (additive) unit circle and is the standard additive character, is an incredibly useful tool for *additive* number theory, particularly when studying additive problems involving three or more variables taking values in sets such as the primes; the deployment of this tool is generally known as the *Hardy-Littlewood circle method*. (In the analytic number theory literature, the minus sign in the phase is traditionally omitted, and what is denoted by here would be referred to instead by , or just .) We list some of the most classical problems in this area:

- (Even Goldbach conjecture) Is it true that every even natural number greater than two can be expressed as the sum of two primes?
- (Odd Goldbach conjecture) Is it true that every odd natural number greater than five can be expressed as the sum of three primes?
- (Waring problem) For each natural number , what is the least natural number such that every natural number can be expressed as the sum of or fewer powers?
- (Asymptotic Waring problem) For each natural number , what is the least natural number such that every
*sufficiently large*natural number can be expressed as the sum of or fewer powers? - (Partition function problem) For any natural number , let denote the number of representations of of the form where and are natural numbers. What is the asymptotic behaviour of as ?

The Waring problem and its asymptotic version will not be discussed further here, save to note that the Vinogradov mean value theorem (Theorem 13 from Notes 5) and its variants are particularly useful for getting good bounds on ; see for instance the ICM article of Wooley for recent progress on these problems. Similarly, the partition function problem was the original motivation of Hardy and Littlewood in introducing the circle method, but we will not discuss it further here; see e.g. Chapter 20 of Iwaniec-Kowalski for a treatment.

Instead, we will focus our attention on the odd Goldbach conjecture as our model problem. (The even Goldbach conjecture, which involves only two variables instead of three, is unfortunately not amenable to a circle method approach for a variety of reasons, unless the statement is replaced with something weaker, such as an averaged statement; see this previous blog post for further discussion. On the other hand, the methods here can obtain weaker versions of the even Goldbach conjecture, such as showing that “almost all” even numbers are the sum of two primes; see Exercise 34 below.) In particular, we will establish the following celebrated theorem of Vinogradov:

Theorem 1 (Vinogradov’s theorem)Every sufficiently large odd number is expressible as the sum of three primes.

Recently, the restriction that be sufficiently large was replaced by Helfgott with , thus establishing the odd Goldbach conjecture in full. This argument followed the same basic approach as Vinogradov (based on the circle method), but with various estimates replaced by “log-free” versions (analogous to the log-free zero-density theorems in Notes 7), combined with careful numerical optimisation of constants and also some numerical work on the even Goldbach problem and on the generalised Riemann hypothesis. We refer the reader to Helfgott’s text for details.

We will in fact show the more precise statement:

Theorem 2 (Quantitative Vinogradov theorem)Let be an natural number. Then

We dropped the hypothesis that is odd in Theorem 2, but note that vanishes when is even. For odd , we have

Unfortunately, due to the ineffectivity of the constants in Theorem 2 (a consequence of the reliance on the Siegel-Walfisz theorem in the proof of that theorem), one cannot quantify explicitly what “sufficiently large” means in Theorem 1 directly from Theorem 2. However, there is a modification of this theorem which gives effective bounds; see Exercise 32 below.

Exercise 4Obtain a heuristic derivation of the main term using the modified Cramér model (Section 1 of Supplement 4).

To prove Theorem 2, we consider the more general problem of estimating sums of the form

for various integers and functions , which we will take to be finitely supported to avoid issues of convergence.

Suppose that are supported on ; for simplicity, let us first assume the pointwise bound for all . (This simple case will not cover the case in Theorem 2, when are truncated versions of the von Mangoldt function , but will serve as a warmup to that case.) Then we have the trivial upper bound

A basic observation is that this upper bound is attainable if all “pretend” to behave like the same additive character for some . For instance, if , then we have when , and then it is not difficult to show that

as .

The key to the success of the circle method lies in the converse of the above statement: the *only* way that the trivial upper bound (2) comes close to being sharp is when all correlate with the same character , or in other words are simultaneously large. This converse is largely captured by the following two identities:

Exercise 5Let be finitely supported functions. Then for any natural number , show that

The traditional approach to using the circle method to compute sums such as proceeds by invoking (3) to express this sum as an integral over the unit circle, then dividing the unit circle into “major arcs” where are large but computable with high precision, and “minor arcs” where one has estimates to ensure that are small in both and senses. For functions of number-theoretic significance, such as truncated von Mangoldt functions, the “major arcs” typically consist of those that are close to a rational number with not too large, and the “minor arcs” consist of the remaining portions of the circle. One then obtains lower bounds on the contributions of the major arcs, and upper bounds on the contribution of the minor arcs, in order to get good lower bounds on .

This traditional approach is covered in many places, such as this text of Vaughan. We will emphasise in this set of notes a slightly different perspective on the circle method, coming from recent developments in additive combinatorics; this approach does not quite give the sharpest quantitative estimates, but it allows for easier generalisation to more combinatorial contexts, for instance when replacing the primes by dense subsets of the primes, or replacing the equation with some other equation or system of equations.

From Exercise 5 and Hölder’s inequality, we immediately obtain

Corollary 6Let be finitely supported functions. Then for any natural number , we haveSimilarly for permutations of the .

In the case when are supported on and bounded by , this corollary tells us that we have is whenever one has uniformly in , and similarly for permutations of . From this and the triangle inequality, we obtain the following conclusion: if is supported on and bounded by , and is *Fourier-approximated* by another function supported on and bounded by in the sense that

Thus, one possible strategy for estimating the sum is, one can effectively replace (or “model”) by a simpler function which Fourier-approximates in the sense that the exponential sums agree up to error . For instance:

Exercise 7Let be a natural number, and let be a random subset of , chosen so that each has an independent probability of of lying in .

- (i) If and , show that with probability as , one has uniformly in . (
Hint:for any fixed , this can be accomplished with quite a good probability (e.g. ) using a concentration of measure inequality, such as Hoeffding’s inequality. To obtain the uniformity in , round to the nearest multiple of (say) and apply the union bound).- (ii) Show that with probability , one has representations of the form with (with treated as an ordered triple, rather than an unordered one).

In the case when is something like the truncated von Mangoldt function , the quantity is of size rather than . This costs us a logarithmic factor in the above analysis, however we can still conclude that we have the approximation (4) whenever is another sequence with such that one has the improved Fourier approximation

uniformly in . (Later on we will obtain a “log-free” version of this implication in which one does not need to gain a factor of in the error term.)

This suggests a strategy for proving Vinogradov’s theorem: find an approximant to some suitable truncation of the von Mangoldt function (e.g. or ) which obeys the Fourier approximation property (5), and such that the expression is easily computable. It turns out that there are a number of good options for such an approximant . One of the quickest ways to obtain such an approximation (which is used in Chapter 19 of Iwaniec and Kowalski) is to start with the standard identity , that is to say

and obtain an approximation by truncating to be less than some threshold (which, in practice, would be a small power of ):

Thus, for instance, if , the approximant would be taken to be

One could also use the slightly smoother approximation

The function is somewhat similar to the continuous Selberg sieve weights studied in Notes 4, with the main difference being that we did not square the divisor sum as we will not need to take to be non-negative. As long as is not too large, one can use some sieve-like computations to compute expressions like quite accurately. The approximation (5) can be justified by using a nice estimate of Davenport that exemplifies the Mobius pseudorandomness heuristic from Supplement 4:

Theorem 8 (Davenport’s estimate)For any and , we haveuniformly for all . The implied constants are ineffective.

This estimate will be proven by splitting into two cases. In the “major arc” case when is close to a rational with small (of size or so), this estimate will be a consequence of the Siegel-Walfisz theorem ( from Notes 2); it is the application of this theorem that is responsible for the ineffective constants. In the remaining “minor arc” case, one proceeds by using a combinatorial identity (such as Vaughan’s identity) to express the sum in terms of bilinear sums of the form , and use the Cauchy-Schwarz inequality and the minor arc nature of to obtain a gain in this case. This will all be done below the fold. We will also use (a rigorous version of) the approximation (6) (or (7)) to establish Vinogradov’s theorem.

A somewhat different looking approximation for the von Mangoldt function that also turns out to be quite useful is

for some that is not too large compared to . The methods used to establish Theorem 8 can also establish a Fourier approximation that makes (8) precise, and which can yield an alternate proof of Vinogradov’s theorem; this will be done below the fold.

The approximation (8) can be written in a way that makes it more similar to (7):

Exercise 9Show that the right-hand side of (8) can be rewritten aswhere

Then, show the inequalities

and conclude that

(

Hint:for the latter estimate, use Theorem 27 of Notes 1.)

The coefficients in the above exercise are quite similar to optimised Selberg sieve coefficients (see Section 2 of Notes 4).

Another approximation to , related to the modified Cramér random model (see Model 10 of Supplement 4) is

where and is a slowly growing function of (e.g. ); a closely related approximation is

for as above and coprime to . These approximations (closely related to a device known as the “-trick”) are not as quantitatively accurate as the previous approximations, but can still suffice to establish Vinogradov’s theorem, and also to count many other linear patterns in the primes or subsets of the primes (particularly if one injects some additional tools from additive combinatorics, and specifically the inverse conjecture for the Gowers uniformity norms); see this paper of Ben Green and myself for more discussion (and this more recent paper of Shao for an analysis of this approach in the context of Vinogradov-type theorems). The following exercise expresses the approximation (9) in a form similar to the previous approximation (8):

Exercise 10With as above, show thatfor all natural numbers .

A major topic of interest of analytic number theory is the asymptotic behaviour of the Riemann zeta function in the critical strip in the limit . For the purposes of this set of notes, it is a little simpler technically to work with the log-magnitude of the zeta function. (In principle, one can reconstruct a branch of , and hence itself, from using the Cauchy-Riemann equations, or tools such as the Borel-Carathéodory theorem, see Exercise 40 of Supplement 2.)

One has the classical estimate

(See e.g. Exercise 37 from Supplement 3.) In view of this, let us define the normalised log-magnitudes for any by the formula

informally, this is a normalised window into near . One can rephrase several assertions about the zeta function in terms of the asymptotic behaviour of . For instance:

- (i) The bound (1) implies that is asymptotically locally bounded from above in the limit , thus for any compact set we have for and sufficiently large. In fact the implied constant in only depends on the projection of to the real axis.
- (ii) For , we have the bounds
which implies that converges locally uniformly as to zero in the region .

- (iii) The functional equation, together with the symmetry , implies that
which by Exercise 17 of Supplement 3 shows that

as , locally uniformly in . In particular, when combined with the previous item, we see that converges locally uniformly as to in the region .

- (iv) From Jensen’s formula (Theorem 16 of Supplement 2) we see that is a subharmonic function, and thus is subharmonic as well. In particular we have the mean value inequality
for any disk , where the integral is with respect to area measure. From this and (ii) we conclude that

for any disk with and sufficiently large ; combining this with (i) we conclude that is asymptotically locally bounded in in the limit , thus for any compact set we have for sufficiently large .

From (v) and the usual Arzela-Ascoli diagonalisation argument, we see that the are asymptotically compact in the topology of distributions: given any sequence tending to , one can extract a subsequence such that the converge in the sense of distributions. Let us then define a *normalised limit profile* of to be a distributional limit of a sequence of ; they are analogous to limiting profiles in PDE, and also to the more recent introduction of “graphons” in the theory of graph limits. Then by taking limits in (i)-(iv) we can say a lot about such normalised limit profiles (up to almost everywhere equivalence, which is an issue we will address shortly):

- (i) is bounded from above in the critical strip .
- (ii) vanishes on .
- (iii) We have the functional equation for all . In particular for .
- (iv) is subharmonic.

Unfortunately, (i)-(iv) fail to characterise completely. For instance, one could have for any convex function of that equals for , for , and obeys the functional equation , and this would be consistent with (i)-(iv). One can also perturb such examples in a region where is strictly convex to create further examples of functions obeying (i)-(iv). Note from subharmonicity that the function is always going to be convex in ; this can be seen as a limiting case of the Hadamard three-lines theorem (Exercise 41 of Supplement 2).

We pause to address one minor technicality. We have defined as a distributional limit, and as such it is *a priori* only defined up to almost everywhere equivalence. However, due to subharmonicity, there is a unique upper semi-continuous representative of (taking values in ), defined by the formula

for any (note from subharmonicity that the expression in the limit is monotone nonincreasing as , and is also continuous in ). We will now view this upper semi-continuous representative of as *the* canonical representative of , so that is now defined everywhere, rather than up to almost everywhere equivalence.

By a classical theorem of Riesz, a function is subharmonic if and only if the distribution is a non-negative measure, where is the Laplacian in the coordinates. Jensen’s formula (or Greens’ theorem), when interpreted distributionally, tells us that

away from the real axis, where ranges over the non-trivial zeroes of . Thus, if is a normalised limit profile for that is the distributional limit of , then we have

where is a non-negative measure which is the limit in the vague topology of the measures

Thus is a normalised limit profile of the zeroes of the Riemann zeta function.

Using this machinery, we can recover many classical theorems about the Riemann zeta function by “soft” arguments that do not require extensive calculation. Here are some examples:

Theorem 1The Riemann hypothesis implies the Lindelöf hypothesis.

*Proof:* It suffices to show that any limiting profile (arising as the limit of some ) vanishes on the critical line . But if the Riemann hypothesis holds, then the measures are supported on the critical line , so the normalised limit profile is also supported on this line. This implies that is harmonic outside of the critical line. By (ii) and unique continuation for harmonic functions, this implies that vanishes on the half-space (and equals on the complementary half-space, by (iii)), giving the claim.

In fact, we have the following sharper statement:

Theorem 2 (Backlund)The Lindelöf hypothesis is equivalent to the assertion that for any fixed , the number of zeroes in the region is as .

*Proof:* If the latter claim holds, then for any , the measures assign a mass of to any region of the form as for any fixed and . Thus the normalised limiting profile measure is supported on the critical line, and we can repeat the previous argument.

Conversely, suppose the claim fails, then we can find a sequence and such that assigns a mass of to the region . Extracting a normalised limiting profile, we conclude that the normalised limiting profile measure is non-trivial somewhere to the right of the critical line, so the associated subharmonic function is not harmonic everywhere to the right of the critical line. From the maximum principle and (ii) this implies that has to be positive somewhere on the critical line, but this contradicts the Lindelöf hypothesis. (One has to take a bit of care in the last step since only converges to in the sense of distributions, but it turns out that the subharmonicity of all the functions involved gives enough regularity to justify the argument; we omit the details here.)

Theorem 3 (Littlewood)Assume the Lindelöf hypothesis. Then for any fixed , the number of zeroes in the region is as .

*Proof:* By the previous arguments, the only possible normalised limiting profile for is . Taking distributional Laplacians, we see that the only possible normalised limiting profile for the zeroes is Lebesgue measure on the critical line. Thus, can only converge to as , and the claim follows.

Even without the Lindelöf hypothesis, we have the following result:

Theorem 4 (Titchmarsh)For any fixed , there are zeroes in the region for sufficiently large .

Among other things, this theorem recovers a classical result of Littlewood that the gaps between the imaginary parts of the zeroes goes to zero, even without assuming unproven conjectures such as the Riemann or Lindelöf hypotheses.

*Proof:* Suppose for contradiction that this were not the case, then we can find and a sequence such that contains zeroes. Passing to a subsequence to extract a limit profile, we conclude that the normalised limit profile measure assigns no mass to the horizontal strip . Thus the associated subharmonic function is actually harmonic on this strip. But by (ii) and unique continuation this forces to vanish on this strip, contradicting the functional equation (iii).

Exercise 5Use limiting profiles to obtain the matching upper bound of for the number of zeroes in for sufficiently large .

Remark 6One can remove the need to take limiting profiles in the above arguments if one can come up with quantitative (or “hard”) substitutes for qualitative (or “soft”) results such as the unique continuation property for harmonic functions. This would also allow one to replace the qualitative decay rates with more quantitative decay rates such as or . Indeed, the classical proofs of the above theorems come with quantitative bounds that are typically of this form (see e.g. the text of Titchmarsh for details).

Exercise 7Let denote the quantity , where the branch of the argument is taken by using a line segment connecting to (say) , and then to . If we have a sequence producing normalised limit profiles for and the zeroes respectively, show that converges in the sense of distributions to the function , or equivalentlyConclude in particular that if the Lindelöf hypothesis holds, then as .

A little bit more about the normalised limit profiles are known unconditionally, beyond (i)-(iv). For instance, from Exercise 3 of Notes 5 we have as , which implies that any normalised limit profile for is bounded by on the critical line, beating the bound of coming from convexity and (ii), (iii), and then convexity can be used to further bound away from the critical line also. Some further small improvements of this type are known (coming from various methods for estimating exponential sums), though they fall well short of determining completely at our current level of understanding. Of course, given that we believe the Riemann hypothesis (and hence the Lindelöf hypothesis) to be true, the only actual limit profile that should exist is (in fact this assertion is equivalent to the Lindelöf hypothesis, by the arguments above).

Better control on limiting profiles is available if we do not insist on controlling for *all* values of the height parameter , but only for *most* such values, thanks to the existence of several *mean value theorems* for the zeta function, as discussed in Notes 6; we discuss this below the fold.

In analytic number theory, it is a well-known phenomenon that for many arithmetic functions of interest in number theory, it is significazintly easier to estimate logarithmic sums such as

than it is to estimate summatory functions such as

(Here we are normalising to be roughly constant in size, e.g. as .) For instance, when is the von Mangoldt function , the logarithmic sums can be adequately estimated by Mertens’ theorem, which can be easily proven by elementary means (see Notes 1); but a satisfactory estimate on the summatory function requires the prime number theorem, which is substantially harder to prove (see Notes 2). (From a complex-analytic or Fourier-analytic viewpoint, the problem is that the logarithmic sums can usually be controlled just from knowledge of the Dirichlet series for near ; but the summatory functions require control of the Dirichlet series for on or near a large portion of the line . See Notes 2 for further discussion.)

Viewed conversely, whenever one has a difficult estimate on a summatory function such as , one can look to see if there is a “cheaper” version of that estimate that only controls the logarithmic sums , which is easier to prove than the original, more “expensive” estimate. In this post, we shall do this for two theorems, a classical theorem of Halasz on mean values of multiplicative functions on long intervals, and a much more recent result of Matomaki and Radziwiłł on mean values of multiplicative functions in short intervals. The two are related; the former theorem is an ingredient in the latter (though in the special case of the Matomaki-Radziwiłł theorem considered here, we will not need Halasz’s theorem directly, instead using a key tool in the *proof* of that theorem).

We begin with Halasz’s theorem. Here is a version of this theorem, due to Montgomery and to Tenenbaum:

Theorem 1 (Halasz-Montgomery-Tenenbaum)Let be a multiplicative function with for all . Let and , and setThen one has

Informally, this theorem asserts that is small compared with , unless “pretends” to be like the character on primes for some small . (This is the starting point of the “pretentious” approach of Granville and Soundararajan to analytic number theory, as developed for instance here.) We now give a “cheap” version of this theorem which is significantly weaker (both because it settles for controlling logarithmic sums rather than summatory functions, it requires to be completely multiplicative instead of multiplicative, it requires a strong bound on the analogue of the quantity , and because it only gives qualitative decay rather than quantitative estimates), but easier to prove:

Theorem 2 (Cheap Halasz)Let be an asymptotic parameter goingto infinity. Let be a completely multiplicative function (possibly depending on ) such that for all , such that

Note that now that we are content with estimating exponential sums, we no longer need to preclude the possibility that pretends to be like ; see Exercise 11 of Notes 1 for a related observation.

To prove this theorem, we first need a special case of the Turan-Kubilius inequality.

Lemma 3 (Turan-Kubilius)Let be a parameter going to infinity, and let be a quantity depending on such that and as . Then

Informally, this lemma is asserting that

for most large numbers . Another way of writing this heuristically is in terms of Dirichlet convolutions:

This type of estimate was previously discussed as a tool to establish a criterion of Katai and Bourgain-Sarnak-Ziegler for Möbius orthogonality estimates in this previous blog post. See also Section 5 of Notes 1 for some similar computations.

*Proof:* By Cauchy-Schwarz it suffices to show that

Expanding out the square, it suffices to show that

for .

We just show the case, as the cases are similar (and easier). We rearrange the left-hand side as

We can estimate the inner sum as . But a routine application of Mertens’ theorem (handling the diagonal case when separately) shows that

and the claim follows.

Remark 4As an alternative to the Turan-Kubilius inequality, one can use the Ramaré identity(see e.g. Section 17.3 of Friedlander-Iwaniec). This identity turns out to give superior quantitative results than the Turan-Kubilius inequality in applications; see the paper of Matomaki and Radziwiłł for an instance of this.

We now prove Theorem 2. Let denote the left-hand side of (2); by the triangle inequality we have . By Lemma 3 (for some to be chosen later) and the triangle inequality we have

We rearrange the left-hand side as

We now replace the constraint by . The error incurred in doing so is

which by Mertens’ theorem is . Thus we have

But by definition of , we have , thus

From Mertens’ theorem, the expression in brackets can be rewritten as

and so the real part of this expression is

By (1), Mertens’ theorem and the hypothesis on we have

for any . This implies that we can find going to infinity such that

and thus the expression in brackets has real part . The claim follows.

The Turan-Kubilius argument is certainly not the most efficient way to estimate sums such as . In the exercise below we give a significantly more accurate estimate that works when is non-negative.

Exercise 5(Granville-Koukoulopoulos-Matomaki)

- (i) If is a completely multiplicative function with for all primes , show that
as . (

Hint:for the upper bound, expand out the Euler product. For the lower bound, show that , where is the completely multiplicative function with for all primes .)- (ii) If is multiplicative and takes values in , show that
for all .

Now we turn to a very recent result of Matomaki and Radziwiłł on mean values of multiplicative functions in short intervals. For sake of illustration we specialise their results to the simpler case of the Liouville function , although their arguments actually work (with some additional effort) for arbitrary multiplicative functions of magnitude at most that are real-valued (or more generally, stay far from complex characters ). Furthermore, we give a qualitative form of their estimates rather than a quantitative one:

Theorem 6 (Matomaki-Radziwiłł, special case)Let be a parameter going to infinity, and let be a quantity going to infinity as . Then for all but of the integers , one has

A simple sieving argument (see Exercise 18 of Supplement 4) shows that one can replace by the Möbius function and obtain the same conclusion. See this recent note of Matomaki and Radziwiłł for a simple proof of their (quantitative) main theorem in this special case.

Of course, (4) improves upon the trivial bound of . Prior to this paper, such estimates were only known (using arguments similar to those in Section 3 of Notes 6) for unconditionally, or for for some sufficiently large if one assumed the Riemann hypothesis. This theorem also represents some progress towards Chowla’s conjecture (discussed in Supplement 4) that

as for any fixed distinct ; indeed, it implies that this conjecture holds if one performs a small amount of averaging in the .

Below the fold, we give a “cheap” version of the Matomaki-Radziwiłł argument. More precisely, we establish

Theorem 7 (Cheap Matomaki-Radziwiłł)Let be a parameter going to infinity, and let . Then

Note that (5) improves upon the trivial bound of . Again, one can replace with if desired. Due to the cheapness of Theorem 7, the proof will require few ingredients; the deepest input is the improved zero-free region for the Riemann zeta function due to Vinogradov and Korobov. Other than that, the main tools are the Turan-Kubilius result established above, and some Fourier (or complex) analysis.

In the previous set of notes, we saw how zero-density theorems for the Riemann zeta function, when combined with the zero-free region of Vinogradov and Korobov, could be used to obtain prime number theorems in short intervals. It turns out that a more sophisticated version of this type of argument also works to obtain prime number theorems in arithmetic progressions, in particular establishing the celebrated theorem of Linnik:

Theorem 1 (Linnik’s theorem)Let be a primitive residue class. Then contains a prime with .

In fact it is known that one can find a prime with , a result of Xylouris. For sake of comparison, recall from Exercise 65 of Notes 2 that the Siegel-Walfisz theorem gives this theorem with a bound of , and from Exercise 48 of Notes 2 one can obtain a bound of the form if one assumes the generalised Riemann hypothesis. The probabilistic random models from Supplement 4 suggest that one should in fact be able to take .

We will not aim to obtain the optimal exponents for Linnik’s theorem here, and follow the treatment in Chapter 18 of Iwaniec and Kowalski. We will in fact establish the following more quantitative result (a special case of a more powerful theorem of Gallagher), which splits into two cases, depending on whether there is an exceptional zero or not:

Theorem 2 (Quantitative Linnik theorem)Let be a primitive residue class for some . For any , let denote the quantityAssume that for some sufficiently large .

- (i) (No exceptional zero) If all the real zeroes of -functions of real characters of modulus are such that , then
for all and some absolute constant .

- (ii) (Exceptional zero) If there is a zero of an -function of a real character of modulus with for some sufficiently small , then
for all and some absolute constant .

The implied constants here are effective.

Note from the Landau-Page theorem (Exercise 54 from Notes 2) that at most one exceptional zero exists (if is small enough). A key point here is that the error term in the exceptional zero case is an *improvement* over the error term when no exceptional zero is present; this compensates for the potential reduction in the main term coming from the term. The splitting into cases depending on whether an exceptional zero exists or not turns out to be an essential technique in many advanced results in analytic number theory (though presumably such a splitting will one day become unnecessary, once the possibility of exceptional zeroes are finally eliminated for good).

Exercise 3Assuming Theorem 2, and assuming for some sufficiently large absolute constant , establish the lower boundwhen there is no exceptional zero, and

when there is an exceptional zero . Conclude that Theorem 2 implies Theorem 1, regardless of whether an exceptional zero exists or not.

Remark 4The Brun-Titchmarsh theorem (Exercise 33 from Notes 4), in the sharp form of Montgomery and Vaughan, gives thatfor any primitive residue class and any . This is (barely) consistent with the estimate (1). Any lowering of the coefficient in the Brun-Titchmarsh inequality (with reasonable error terms), in the regime when is a large power of , would then lead to at least some elimination of the exceptional zero case. However, this has not led to any progress on the Landau-Siegel zero problem (and may well be just a reformulation of that problem). (When is a relatively small power of , some improvements to Brun-Titchmarsh are possible that are not in contradiction with the presence of an exceptional zero; see this paper of Maynard for more discussion.)

Theorem 2 is deduced in turn from facts about the distribution of zeroes of -functions. Recall from the truncated explicit formula (Exercise 45(iv) of Notes 2) with (say) that

for any non-principal character of modulus , where we assume for some large ; for the principal character one has the same formula with an additional term of on the right-hand side (as is easily deduced from Theorem 21 of Notes 2). Using the Fourier inversion formula

(see Theorem 69 of Notes 1), we thus have

and so it suffices by the triangle inequality (bounding very crudely by , as the contribution of the low-lying zeroes already turns out to be quite dominant) to show that

when no exceptional zero is present, and

when an exceptional zero is present.

To handle the former case (2), one uses two facts about zeroes. The first is the classical zero-free region (Proposition 51 from Notes 2), which we reproduce in our context here:

Proposition 5 (Classical zero-free region)Let . Apart from a potential exceptional zero , all zeroes of -functions with of modulus and are such thatfor some absolute constant .

Using this zero-free region, we have

whenever contributes to the sum in (2), and so the left-hand side of (2) is bounded by

where we recall that is the number of zeroes of any -function of a character of modulus with and (here we use conjugation symmetry to make non-negative, accepting a multiplicative factor of two).

In Exercise 25 of Notes 6, the grand density estimate

is proven. If one inserts this bound into the above expression, one obtains a bound for (2) which is of the form

Unfortunately this is off from what we need by a factor of (and would lead to a weak form of Linnik’s theorem in which was bounded by rather than by ). In the analogous problem for prime number theorems in short intervals, we could use the Vinogradov-Korobov zero-free region to compensate for this loss, but that region does not help here for the contribution of the low-lying zeroes with , which as mentioned before give the dominant contribution. Fortunately, it is possible to remove this logarithmic loss from the zero-density side of things:

Theorem 6 (Log-free grand density estimate)For any and , one hasThe implied constants are effective.

We prove this estimate below the fold. The proof follows the methods of the previous section, but one inserts various sieve weights to restrict sums over natural numbers to essentially become sums over “almost primes”, as this turns out to remove the logarithmic losses. (More generally, the trick of restricting to almost primes by inserting suitable sieve weights is quite useful for avoiding any unnecessary losses of logarithmic factors in analytic number theory estimates.)

Now we turn to the case when there is an exceptional zero (3). The argument used to prove (2) applies here also, but does not gain the factor of in the exponent. To achieve this, we need an additional tool, a version of the Deuring-Heilbronn repulsion phenomenon due to Linnik:

Theorem 8 (Deuring-Heilbronn repulsion phenomenon)Suppose is such that there is an exceptional zero with small. Then all other zeroes of -functions of modulus are such thatIn other words, the exceptional zero enlarges the classical zero-free region by a factor of . The implied constants are effective.

Exercise 9Use Theorem 6 and Theorem 8 to complete the proof of (3), and thus Linnik’s theorem.

Exercise 10Use Theorem 8 to give an alternate proof of (Tatuzawa’s version of) Siegel’s theorem (Theorem 62 of Notes 2). (Hint:if two characters have different moduli, then they can be made to have the same modulus by multiplying by suitable principal characters.)

Theorem 8 is proven by similar methods to that of Theorem 6, the basic idea being to insert a further weight of (in addition to the sieve weights), the point being that the exceptional zero causes this weight to be quite small on the average. There is a strengthening of Theorem 8 due to Bombieri that is along the lines of Theorem 6, obtaining the improvement

with effective implied constants for any and in the presence of an exceptional zero, where the prime in means that the exceptional zero is omitted (thus if ). Note that the upper bound on falls below one when for a sufficiently small , thus recovering Theorem 8. Bombieri’s theorem can be established by the methods in this set of notes, and will be given as an exercise to the reader.

Remark 11There are a number of alternate ways to derive the results in this set of notes, for instance using the Turan power sums method which is based on studying derivatives such asfor and large , and performing various sorts of averaging in to attenuate the contribution of many of the zeroes . We will not develop this method here, but see for instance Chapter 9 of Montgomery’s book. See the text of Friedlander and Iwaniec for yet another approach based primarily on sieve-theoretic ideas.

Remark 12When one optimises all the exponents, it turns out that the exponent in Linnik’s theorem isextremelygood in the presence of an exceptional zero – indeed Friedlander and Iwaniec showed can even get a bound of the form for some , which is even stronger than one can obtain from GRH! There are other places in which exceptional zeroes can be used to obtain results stronger than what one can obtain even on the Riemann hypothesis; for instance, Heath-Brown used the hypothesis of an infinite sequence of Siegel zeroes to obtain the twin prime conejcture.

In the previous set of notes, we studied upper bounds on sums such as for that were valid for all in a given range, such as ; this led in turn to upper bounds on the Riemann zeta for in the same range, and for various choices of . While some improvement over the trivial bound of was obtained by these methods, we did not get close to the conjectural bound of that one expects from pseudorandomness heuristics (assuming that is not too large compared with , e.g. .

However, it turns out that one can get much better bounds if one settles for estimating sums such as , or more generally finite Dirichlet series (also known as *Dirichlet polynomials*) such as , for *most* values of in a given range such as . Equivalently, we will be able to get some control on the *large values* of such Dirichlet polynomials, in the sense that we can control the set of for which exceeds a certain threshold, even if we cannot show that this set is empty. These large value theorems are often closely tied with estimates for *mean values* such as of a Dirichlet series; these latter estimates are thus known as *mean value theorems* for Dirichlet series. Our approach to these theorems will follow the same sort of methods used in Notes 3, in particular relying on the generalised Bessel inequality from those notes.

Our main application of the large value theorems for Dirichlet polynomials will be to control the number of zeroes of the Riemann zeta function (or the Dirichlet -functions ) in various rectangles of the form for various and . These rectangles will be larger than the zero-free regions for which we can exclude zeroes completely, but we will often be able to limit the number of zeroes in such rectangles to be quite small. For instance, we will be able to show the following weak form of the Riemann hypothesis: as , a proportion of zeroes of the Riemann zeta function in the critical strip with will have real part . Related to this, the number of zeroes with and can be shown to be bounded by as for any .

In the next set of notes we will use refined versions of these theorems to establish Linnik’s theorem on the least prime in an arithmetic progression.

Our presentation here is broadly based on Chapters 9 and 10 in Iwaniec and Kowalski, who give a number of more sophisticated large value theorems than the ones discussed here.

We return to the study of the Riemann zeta function , focusing now on the task of upper bounding the size of this function within the critical strip; as seen in Exercise 43 of Notes 2, such upper bounds can lead to zero-free regions for , which in turn lead to improved estimates for the error term in the prime number theorem.

In equation (21) of Notes 2 we obtained the somewhat crude estimates

for any and with and . Setting , we obtained the crude estimate

in this region. In particular, if and then we had . Using the functional equation and the Hadamard three lines lemma, we can improve this to ; see Supplement 3.

Now we seek better upper bounds on . We will reduce the problem to that of bounding certain exponential sums, in the spirit of Exercise 33 of Supplement 3:

Proposition 1Let with and . Thenwhere .

*Proof:* We fix a smooth function with for and for , and allow implied constants to depend on . Let with . From Exercise 33 of Supplement 3, we have

for some sufficiently large absolute constant . By dyadic decomposition, we thus have

We can absorb the first term in the second using the case of the supremum. Writing , where

it thus suffices to show that

for each . But from the fundamental theorem of calculus, the left-hand side can be written as

and the claim then follows from the triangle inequality and a routine calculation.

We are thus interested in getting good bounds on the sum . More generally, we consider normalised exponential sums of the form

where is an interval of length at most for some , and is a smooth function. We will assume smoothness estimates of the form

for some , all , and all , where is the -fold derivative of ; in the case , of interest for the Riemann zeta function, we easily verify that these estimates hold with . (One can consider exponential sums under more general hypotheses than (3), but the hypotheses here are adequate for our needs.) We do not bound the zeroth derivative of directly, but it would not be natural to do so in any event, since the magnitude of the sum (2) is unaffected if one adds an arbitrary constant to .

The trivial bound for (2) is

and we will seek to obtain significant improvements to this bound. Pseudorandomness heuristics predict a bound of for (2) for any if ; this assertion (a special case of the *exponent pair hypothesis*) would have many consequences (for instance, inserting it into Proposition 1 soon yields the Lindelöf hypothesis), but is unfortunately quite far from resolution with known methods. However, we can obtain weaker gains of the form when and depends on . We present two such results here, which perform well for small and large values of respectively:

Theorem 2Let , let be an interval of length at most , and let be a smooth function obeying (3) for all and .

The factor of can be removed by a more careful argument, but we will not need to do so here as we are willing to lose powers of . The estimate (6) is superior to (5) when for large, since (after optimising in ) (5) gives a gain of the form over the trivial bound, while (6) gives . We have not attempted to obtain completely optimal estimates here, settling for a relatively simple presentation that still gives good bounds on , and there are a wide variety of additional exponential sum estimates beyond the ones given here; see Chapter 8 of Iwaniec-Kowalski, or Chapters 3-4 of Montgomery, for further discussion.

We now briefly discuss the strategies of proof of Theorem 2. Both parts of the theorem proceed by treating like a polynomial of degree roughly ; in the case of (ii), this is done explicitly via Taylor expansion, whereas for (i) it is only at the level of analogy. Both parts of the theorem then try to “linearise” the phase to make it a linear function of the summands (actually in part (ii), it is necessary to introduce an additional variable and make the phase a *bilinear* function of the summands). The van der Corput estimate achieves this linearisation by squaring the exponential sum about times, which is why the gain is only exponentially small in . The Vinogradov estimate achieves linearisation by raising the exponential sum to a significantly smaller power – on the order of – by using Hölder’s inequality in combination with the fact that the discrete curve becomes roughly equidistributed in the box after taking the sumset of about copies of this curve. This latter fact has a precise formulation, known as the Vinogradov mean value theorem, and its proof is the most difficult part of the argument, relying on using a “-adic” version of this equidistribution to reduce the claim at a given scale to a smaller scale with , and then proceeding by induction.

One can combine Theorem 2 with Proposition 1 to obtain various bounds on the Riemann zeta function:

Exercise 3 (Subconvexity bound)

- (i) Show that for all . (
Hint:use the case of the Van der Corput estimate.)- (ii) For any , show that as .

Exercise 4Let be such that , and let .

- (i) (Littlewood bound) Use the van der Corput estimate to show that whenever .
- (ii) (Vinogradov-Korobov bound) Use the Vinogradov estimate to show that whenever .

As noted in Exercise 43 of Notes 2, the Vinogradov-Korobov bound leads to the zero-free region , which in turn leads to the prime number theorem with error term

for . If one uses the weaker Littlewood bound instead, one obtains the narrower zero-free region

(which is only slightly wider than the classical zero-free region) and an error term

in the prime number theorem.

Exercise 5 (Vinogradov-Korobov in arithmetic progressions)Let be a non-principal character of modulus .

- (i) (Vinogradov-Korobov bound) Use the Vinogradov estimate to show that whenever and
(

Hint:use the Vinogradov estimate and a change of variables to control for various intervals of length at most and residue classes , in the regime (say). For , do not try to capture any cancellation and just use the triangle inequality instead.)- (ii) Obtain a zero-free region
for , for some (effective) absolute constant .

- (iii) Obtain the prime number theorem in arithmetic progressions with error term
whenever , , is primitive, and depends (ineffectively) on .

We continue the discussion of sieve theory from Notes 4, but now specialise to the case of the *linear sieve* in which the sieve dimension is equal to , which is one of the best understood sieving situations, and one of the rare cases in which the precise limits of the sieve method are known. A bit more specifically, let be quantities with for some fixed , and let be a multiplicative function with

for all primes and some fixed (we allow all constants below to depend on ). Let , and for each prime , let be a set of integers, with for . We consider finitely supported sequences of non-negative reals for which we have bounds of the form

for all square-free and some , and some remainder terms . One is then interested in upper and lower bounds on the quantity

The fundamental lemma of sieve theory (Corollary 19 of Notes 4) gives us the bound

This bound is strong when is large, but is not as useful for smaller values of . We now give a sharp bound in this regime. We introduce the functions by

where we adopt the convention . Note that for each one has only finitely many non-zero summands in (6), (7). These functions are closely related to the Buchstab function from Exercise 28 of Supplement 4; indeed from comparing the definitions one has

for all .

Exercise 1 (Alternate definition of )Show that is continuously differentiable except at , and is continuously differentiable except at where it is continuous, obeying the delay-differential equationsfor , with the initial conditions

for and

for . Show that these properties of determine completely.

For future reference, we record the following explicit values of :

We will show

Theorem 2 (Linear sieve)Let the notation and hypotheses be as above, with . Then, for any , one has the upper boundif is sufficiently large depending on . Furthermore, this claim is sharp in the sense that the quantity cannot be replaced by any smaller quantity, and similarly cannot be replaced by any larger quantity.

Comparing the linear sieve with the fundamental lemma (and also testing using the sequence for some extremely large ), we conclude that we necessarily have the asymptotics

for all ; this can also be proven directly from the definitions of , or from Exercise 1, but is somewhat challenging to do so; see e.g. Chapter 11 of Friedlander-Iwaniec for details.

Exercise 3Establish the integral identitiesand

for . Argue heuristically that these identities are consistent with the bounds in Theorem 2 and the Buchstab identity (Equation (16) from Notes 4).

Exercise 4Use the Selberg sieve (Theorem 30 from Notes 4) to obtain a slightly weaker version of (12) in the range in which the error term is worsened to , but the main term is unchanged.

We will prove Theorem 2 below the fold. The optimality of is closely related to the parity problem obstruction discussed in Section 5 of Notes 4; a naive application of the parity arguments there only give the weak bounds and for , but this can be sharpened by a more careful counting of various sums involving the Liouville function .

As an application of the linear sieve (specialised to the ranges in (10), (11)), we will establish a famous theorem of Chen, giving (in some sense) the closest approach to the twin prime conjecture that one can hope to achieve by sieve-theoretic methods:

Theorem 5 (Chen’s theorem)There are infinitely many primes such that is the product of at most two primes.

The same argument gives the version of Chen’s theorem for the even Goldbach conjecture, namely that for all sufficiently large even , there exists a prime between and such that is the product of at most two primes.

The discussion in these notes loosely follows that of Friedlander-Iwaniec (who study sieving problems in more general dimension than ).

Many problems in non-multiplicative prime number theory can be recast as *sieving* problems. Consider for instance the problem of counting the number of pairs of twin primes contained in for some large ; note that the claim that for arbitrarily large is equivalent to the twin prime conjecture. One can obtain this count by any of the following variants of the sieve of Eratosthenes:

- Let be the set of natural numbers in . For each prime , let be the union of the residue classes and . Then is the cardinality of the
*sifted set*. - Let be the set of primes in . For each prime , let be the residue class . Then is the cardinality of the
*sifted set*. - Let be the set of primes in . For each prime , let be the residue class . Then is the cardinality of the
*sifted set*. - Let be the set . For each prime , let be the residue class Then is the cardinality of the
*sifted set*.

Exercise 1Develop similar sifting formulations of the other three Landau problems.

In view of these sieving interpretations of number-theoretic problems, it becomes natural to try to estimate the size of sifted sets for various finite sets of integers, and subsets of integers indexed by primes dividing some squarefree natural number (which, in the above examples, would be the product of all primes up to ). As we see in the above examples, the sets in applications are typically the union of one or more residue classes modulo , but we will work at a more abstract level of generality here by treating as more or less arbitrary sets of integers, without caring too much about the arithmetic structure of such sets.

It turns out to be conceptually more natural to replace sets by functions, and to consider the more general the task of estimating sifted *sums*

for some finitely supported sequence of non-negative numbers; the previous combinatorial sifting problem then corresponds to the indicator function case . (One could also use other index sets here than the integers if desired; for much of sieve theory the index set and its subsets are treated as abstract sets, so the exact arithmetic structure of these sets is not of primary importance.)

Continuing with twin primes as a running example, we thus have the following sample sieving problem:

Problem 2 (Sieving problem for twin primes)Let , and let denote the number of natural numbers which avoid the residue classes for all primes . In other words, we havewhere , is the product of all the primes strictly less than (we omit itself for minor technical reasons), and is the union of the residue classes . Obtain upper and lower bounds on which are as strong as possible in the asymptotic regime where goes to infinity and the

sifting levelgrows with (ideally we would like to grow as fast as ).

From the preceding discussion we know that the number of twin prime pairs in is equal to , if is not a perfect square; one also easily sees that the number of twin prime pairs in is at least , again if is not a perfect square. Thus we see that a sufficiently good answer to Problem 2 would resolve the twin prime conjecture, particularly if we can get the sifting level to be as large as .

We return now to the general problem of estimating (1). We may expand

where (with the convention that ). We thus arrive at the Legendre sieve identity

Specialising to the case of an indicator function , we recover the inclusion-exclusion formula

Such exact sieving formulae are already satisfactory for controlling sifted sets or sifted sums when the amount of sieving is relatively small compared to the size of . For instance, let us return to the running example in Problem 2 for some . Observe that each in this example consists of residue classes modulo , where is defined to equal when and when is odd. By the Chinese remainder theorem, this implies that for each , consists of residue classes modulo . Using the basic bound

for any and any residue class , we conclude that

for any , where is the multiplicative function

Since and there are at most primes dividing , we may crudely bound , thus

Also, the number of divisors of is at most . From the Legendre sieve (3), we thus conclude that

We can factorise the main term to obtain

This is compatible with the heuristic

coming from the equidistribution of residues principle (Section 3 of Supplement 4), bearing in mind (from the modified Cramér model, see Section 1 of Supplement 4) that we expect this heuristic to become inaccurate when becomes very large. We can simplify the right-hand side of (7) by recalling the twin prime constant

(see equation (7) from Supplement 4); note that

so from Mertens’ third theorem (Theorem 42 from Notes 1) one has

as . Bounding crudely by , we conclude in particular that

when with . This is somewhat encouraging for the purposes of getting a sufficiently good answer to Problem 2 to resolve the twin prime conjecture, but note that is currently far too small: one needs to get as large as before one is counting twin primes, and currently can only get as large as .

The problem is that the number of terms in the Legendre sieve (3) basically grows exponentially in , and so the error terms in (4) accumulate to an unacceptable extent once is significantly larger than . An alternative way to phrase this problem is that the estimate (4) is only expected to be truly useful in the regime ; on the other hand, the moduli appearing in (3) can be as large as , which grows exponentially in by the prime number theorem.

To resolve this problem, it is thus natural to try to *truncate* the Legendre sieve, in such a way that one only uses information about the sums for a relatively small number of divisors of , such as those which are below a certain threshold . This leads to the following general sieving problem:

Problem 3 (General sieving problem)Let be a squarefree natural number, and let be a set of divisors of . For each prime dividing , let be a set of integers, and define for all (with the convention that ). Suppose that is an (unknown) finitely supported sequence of non-negative reals, whose sumsare known for all . What are the best upper and lower bounds one can conclude on the quantity (1)?

Here is a simple example of this type of problem (corresponding to the case , , , , and ):

Exercise 4Let be a finitely supported sequence of non-negative reals such that , , and . Show thatand give counterexamples to show that these bounds cannot be improved in general, even when is an indicator function sequence.

Problem 3 is an example of a linear programming problem. By using linear programming duality (as encapsulated by results such as the Hahn-Banach theorem, the separating hyperplane theorem, or the Farkas lemma), we can rephrase the above problem in terms of *upper and lower bound sieves*:

Theorem 5 (Dual sieve problem)Let be as in Problem 3. We assume that Problem 3 isfeasible, in the sense that there exists at least one finitely supported sequence of non-negative reals obeying the constraints in that problem. Define an (normalised)upper bound sieveto be a function of the formfor some coefficients , and obeying the pointwise lower bound

for all (in particular is non-negative). Similarly, define a (normalised)

lower bound sieveto be a function of the formfor some coefficients , and obeying the pointwise upper bound

for all . Thus for instance and are (trivially) upper bound sieves and lower bound sieves respectively.

- (i) The supremal value of the quantity (1), subject to the constraints in Problem 3, is equal to the infimal value of the quantity , as ranges over all upper bound sieves.
- (ii) The infimal value of the quantity (1), subject to the constraints in Problem 3, is equal to the supremal value of the quantity , as ranges over all lower bound sieves.

*Proof:* We prove part (i) only, and leave part (ii) as an exercise. Let be the supremal value of the quantity (1) given the constraints in Problem 3, and let be the infimal value of . We need to show that .

We first establish the easy inequality . If the sequence obeys the constraints in Problem 3, and is an upper bound sieve, then

and hence (by the non-negativity of and )

taking suprema in and infima in we conclude that .

Now suppose for contradiction that , thus for some real number . We will argue using the hyperplane separation theorem; one can also proceed using one of the other duality results mentioned above. (See this previous blog post for some discussion of the connections between these various forms of linear duality.) Consider the affine functional

on the vector space of finitely supported sequences of reals. On the one hand, since , this functional is positive for every sequence obeying the constraints in Problem 3. Next, let be the space of affine functionals of the form

for some real numbers , some non-negative function which is a finite linear combination of the for , and some non-negative . This is a closed convex cone in a finite-dimensional vector space ; note also that lies in . Suppose first that , thus we have a representation of the form

for any finitely supported sequence . Comparing coefficients, we conclude that

for any (i.e., is an upper bound sieve), and also

and thus , a contradiction. Thus lies outside of . But then by the hyperplane separation theorem, we can find an affine functional on that is non-negative on and negative on . By duality, such an affine functional takes the form for some finitely supported sequence and (indeed, can be supported on a finite set consisting of a single representative for each atom of the finite -algebra generated by the ). Since is non-negative on the cone , we see (on testing against multiples of the functionals or ) that the and are non-negative, and that for all ; thus is feasible for Problem 3. Since is negative on , we see that

and thus , giving the desired contradiction.

Exercise 6Prove part (ii) of the above theorem.

Exercise 7Show that the infima and suprema in the above theorem are actually attained (so one can replace “infimal” and “supremal” by “minimal” and “maximal” if desired).

Exercise 8What are the optimal upper and lower bound sieves for Exercise 4?

In the case when consists of *all* the divisors of , we see that the Legendre sieve is both the optimal upper bound sieve and the optimal lower bound sieve, regardless of what the quantities are. However, in most cases of interest, will only be some strict subset of the divisors of , and there will be a gap between the optimal upper and lower bounds.

Observe that a sequence of real numbers will form an upper bound sieve if one has the inequalities

and

for all ; we will refer to such sequences as *upper bound sieve coefficients*. (Conversely, if the sets are in “general position” in the sense that every set of the form for is non-empty, we see that every upper bound sieve arises from a sequence of upper bound sieve coefficients.) Similarly, a sequence of real numbers will form a lower bound sieve if one has the inequalities

and

for all with ; we will refer to such sequences as *lower bound sieve coefficients*.

Exercise 9 (Brun pure sieve)Let be a squarefree number, and a non-negative integer. Show that the sequence defined bywhere is the number of prime factors of , is a sequence of upper bound sieve coefficients for even , and a sequence of lower bound sieve coefficients for odd . Deduce the Bonferroni inequalities

when is odd, whenever one is in the situation of Problem 3 (and contains all with ). The resulting upper and lower bound sieves are sometimes known as

Brun pure sieves. The Legendre sieve can be viewed as the limiting case when .

In many applications the sums in (9) take the form

for some quantity independent of , some multiplicative function with , and some remainder term whose effect is expected to be negligible on average if is restricted to be small, e.g. less than a threshold ; note for instance that (5) is of this form if for some fixed (note from the divisor bound, Lemma 23 of Notes 1, that if ). We are thus led to the following idealisation of the sieving problem, in which the remainder terms are ignored:

Problem 10 (Idealised sieving)Let (we refer to as thesifting leveland as thelevel of distribution), let be a multiplicative function with , and let . How small can one make the quantityfor a sequence of upper bound sieve coefficients, and how large can one make the quantity

Thus, for instance, the trivial upper bound sieve and the trivial lower bound sieve show that (14) can equal and (15) can equal . Of course, one hopes to do better than these trivial bounds in many situations; usually one can improve the upper bound quite substantially, but improving the lower bound is significantly more difficult, particularly when is large compared with .

If the remainder terms in (13) are indeed negligible on average for , then one expects the upper and lower bounds in Problem 3 to essentially be the optimal bounds in (14) and (15) respectively, multiplied by the normalisation factor . Thus Problem 10 serves as a good model problem for Problem 3, in which all the arithmetic content of the original sieving problem has been abstracted into two parameters and a multiplicative function . In many applications, will be approximately on the average for some fixed , known as the *sieve dimension*; for instance, in the twin prime sieving problem discussed above, the sieve dimension is . The larger one makes the level of distribution compared to , the more choices one has for the upper and lower bound sieves; it is thus of interest to obtain equidistribution estimates such as (13) for as large as possible. When the sequence is of arithmetic origin (for instance, if it is the von Mangoldt function ), then estimates such as the Bombieri-Vinogradov theorem, Theorem 17 from Notes 3, turn out to be particularly useful in this regard; in other contexts, the required equidistribution estimates might come from other sources, such as homogeneous dynamics, or the theory of expander graphs (the latter arises in the recent theory of the *affine sieve*, discussed in this previous blog post). However, the sieve-theoretic tools developed in this post are not particularly sensitive to *how* a certain level of distribution is attained, and are generally content to use sieve axioms such as (13) as “black boxes”.

In some applications one needs to modify Problem 10 in various technical ways (e.g. in altering the product , the set , or the definition of an upper or lower sieve coefficient sequence), but to simplify the exposition we will focus on the above problem without such alterations.

As the exercise below (or the heuristic (7)) suggests, the “natural” size of (14) and (15) is given by the quantity (so that the natural size for Problem 3 is ):

Exercise 11Let be as in Problem 10, and set .

- (i) Show that the quantity (14) is always at least when is a sequence of upper bound sieve coefficients. Similarly, show that the quantity (15) is always at most when is a sequence of lower bound sieve coefficients. (
Hint:compute the expected value of when is a random factor of chosen according to a certain probability distribution depending on .)- (ii) Show that (14) and (15) can both attain the value of when . (
Hint:translate the Legendre sieve to this setting.)

The problem of finding good sequences of upper and lower bound sieve coefficients in order to solve problems such as Problem 10 is one of the core objectives of sieve theory, and has been intensively studied. This is more of an optimisation problem rather than a genuinely number theoretic problem; however, the optimisation problem is sufficiently complicated that it has not been solved exactly or even asymptotically, except in a few special cases. (It can be reduced to a optimisation problem involving multilinear integrals of certain unknown functions of several variables, but this problem is rather difficult to analyse further; see these lecture notes of Selberg for further discussion.) But while we do not yet have a definitive solution to this problem in general, we do have a number of good general-purpose upper and lower bound sieve coefficients that give fairly good values for (14), (15), often coming within a constant factor of the idealised value , and which work well for sifting levels as large as a small power of the level of distribution . Unfortunately, we also know of an important limitation to the sieve, known as the *parity problem*, that prevents one from taking as large as while still obtaining non-trivial lower bounds; as a consequence, sieve theory is not able, on its own, to sift out primes for such purposes as establishing the twin prime conjecture. However, it is still possible to use these sieves, in conjunction with additional tools, to produce various types of primes or prime patterns in some cases; examples of this include the theorem of Ben Green and myself in which an upper bound sieve is used to demonstrate the existence of primes in arbitrarily long arithmetic progressions, or the more recent theorem of Zhang in which (among other things) used an upper bound sieve was used to demonstrate the existence of infinitely many pairs of primes whose difference was bounded. In such arguments, the upper bound sieve was used not so much to count the primes or prime patterns directly, but to serve instead as a sort of “container” to efficiently envelop such prime patterns; when used in such a manner, the upper bound sieves are sometimes known as *enveloping sieves*. If the original sequence was supported on primes, then the enveloping sieve can be viewed as a “smoothed out indicator function” that is concentrated on *almost primes*, which in this context refers to numbers with no small prime factors.

In a somewhat different direction, it can be possible in some cases to break the parity barrier by assuming additional equidistribution axioms on the sequence than just (13), in particular controlling certain *bilinear* sums involving rather than just *linear* sums of the . This approach was in particular pursued by Friedlander and Iwaniec, leading to their theorem that there are infinitely many primes of the form .

The study of sieves is an immense topic; see for instance the recent 527-page text by Friedlander and Iwaniec. We will limit attention to two sieves which give good general-purpose results, if not necessarily the most optimal ones:

- (i) The
*beta sieve*(or*Rosser-Iwaniec sieve*), which is a modification of the classical combinatorial sieve of Brun. (A collection of sieve coefficients is called*combinatorial*if its coefficients lie in .) The beta sieve is a family of upper and lower bound combinatorial sieves, and are particularly useful for efficiently sieving out all primes up to a parameter from a set of integers of size , in the regime where is moderately large, leading to what is sometimes known as the fundamental lemma of sieve theory. - (ii) The
*Selberg upper bound sieve*, which is a general-purpose sieve that can serve both as an upper bound sieve for classical sieving problems, as well as an enveloping sieve for sets such as the primes. (One can also convert the Selberg upper bound sieve into a lower bound sieve in a number of ways, but we will only touch upon this briefly.) A key advantage of the Selberg sieve is that, due to the “quadratic” nature of the sieve, the difficult optimisation problem in Problem 10 is replaced with a much more tractable quadratic optimisation problem, which can often be solved for exactly.

Remark 12It is possible to compose two sieves together, for instance by using the observation that the product of two upper bound sieves is again an upper bound sieve, or that the product of an upper bound sieve and a lower bound sieve is a lower bound sieve. Such a composition of sieves is useful in some applications, for instance if one wants to apply the fundamental lemma as a “preliminary sieve” to sieve out small primes, but then use a more precise sieve like the Selberg sieve to sieve out medium primes. We will see an example of this in later notes, when we discuss the linear beta-sieve.

We will also briefly present the (arithmetic) *large sieve*, which gives a rather different approach to Problem 3 in the case that each consists of some number (typically a large number) of residue classes modulo , and is powered by the (analytic) large sieve inequality of the preceding section. As an application of these methods, we will utilise the Selberg upper bound sieve as an enveloping sieve to establish Zhang’s theorem on bounded gaps between primes. Finally, we give an informal discussion of the *parity barrier* which gives some heuristic limitations on what sieve theory is able to accomplish with regards to counting prime patters such as twin primes.

These notes are only an introduction to the vast topic of sieve theory; more detailed discussion can be found in the Friedlander-Iwaniec text, in these lecture notes of Selberg, and in many further texts.

A fundamental and recurring problem in analytic number theory is to demonstrate the presence of *cancellation* in an oscillating sum, a typical example of which might be a correlation

between two arithmetic functions and , which to avoid technicalities we will assume to be finitely supported (or that the variable is localised to a finite range, such as ). A key example to keep in mind for the purposes of this set of notes is the twisted von Mangoldt summatory function

that measures the correlation between the primes and a Dirichlet character . One can get a “trivial” bound on such sums from the triangle inequality

for instance, from the triangle inequality and the prime number theorem we have

as . But the triangle inequality is insensitive to the phase oscillations of the summands, and often we expect (e.g. from the probabilistic heuristics from Supplement 4) to be able to improve upon the trivial triangle inequality bound by a substantial amount; in the best case scenario, one typically expects a “square root cancellation” that gains a factor that is roughly the square root of the number of summands. (For instance, for Dirichlet characters of conductor , it is expected from probabilistic heuristics that the left-hand side of (3) should in fact be for any .)

It has proven surprisingly difficult, however, to establish significant cancellation in many of the sums of interest in analytic number theory, particularly if the sums do not have a strong amount of algebraic structure (e.g. multiplicative structure) which allow for the deployment of specialised techniques (such as multiplicative number theory techniques). In fact, we are forced to rely (to an embarrassingly large extent) on (many variations of) a single basic tool to capture at least some cancellation, namely the Cauchy-Schwarz inequality. In fact, in many cases the classical case

considered by Cauchy, where at least one of is finitely supported, suffices for applications. Roughly speaking, the Cauchy-Schwarz inequality replaces the task of estimating a *cross-correlation* between two different functions , to that of measuring *self-correlations* between and itself, or and itself, which are usually easier to compute (albeit at the cost of capturing less cancellation). Note that the Cauchy-Schwarz inequality requires almost no hypotheses on the functions or , making it a very widely applicable tool.

There is however some skill required to decide exactly how to deploy the Cauchy-Schwarz inequality (and in particular, how to select and ); if applied blindly, one loses all cancellation and can even end up with a worse estimate than the trivial bound. For instance, if one tries to bound (2) directly by applying Cauchy-Schwarz with the functions and , one obtains the bound

The right-hand side may be bounded by , but this is worse than the trivial bound (3) by a logarithmic factor. This can be “blamed” on the fact that and are concentrated on rather different sets ( is concentrated on primes, while is more or less uniformly distributed amongst the natural numbers); but even if one corrects for this (e.g. by weighting Cauchy-Schwarz with some suitable “sieve weight” that is more concentrated on primes), one still does not do any better than (3). Indeed, the Cauchy-Schwarz inequality suffers from the same key weakness as the triangle inequality: it is insensitive to the phase oscillation of the factors .

While the Cauchy-Schwarz inequality can be poor at estimating a single correlation such as (1), its power improves when considering an average (or sum, or square sum) of *multiple* correlations. In this set of notes, we will focus on one such situation of this type, namely that of trying to estimate a square sum

that measures the correlations of a single function with multiple other functions . One should think of the situation in which is a “complicated” function, such as the von Mangoldt function , but the are relatively “simple” functions, such as Dirichlet characters. In the case when the are orthonormal functions, we of course have the classical Bessel inequality:

Lemma 1 (Bessel inequality)Let be finitely supported functions obeying the orthonormality relationshipfor all . Then for any function , we have

For sake of comparison, if one were to apply the Cauchy-Schwarz inequality (4) separately to each summand in (5), one would obtain the bound of , which is significantly inferior to the Bessel bound when is large. Geometrically, what is going on is this: the Cauchy-Schwarz inequality (4) is only close to sharp when and are close to parallel in the Hilbert space . But if are orthonormal, then it is not possible for any other vector to be simultaneously close to parallel to too many of these orthonormal vectors, and so the inner products of with most of the should be small. (See this previous blog post for more discussion of this principle.) One can view the Bessel inequality as formalising a repulsion principle: if correlates too much with some of the , then it does not have enough “energy” to have large correlation with the rest of the .

In analytic number theory applications, it is useful to generalise the Bessel inequality to the situation in which the are not necessarily orthonormal. This can be accomplished via the Cauchy-Schwarz inequality:

Proposition 2 (Generalised Bessel inequality)Let be finitely supported functions, and let be a non-negative function. Let be such that vanishes whenever vanishes, we havefor some sequence of complex numbers with , with the convention that vanishes whenever both vanish.

Note by relabeling that we may replace the domain here by any other at most countable set, such as the integers . (Indeed, one can give an analogue of this lemma on arbitrary measure spaces, but we will not do so here.) This result first appears in this paper of Boas.

*Proof:* We use the *method of duality* to replace the role of the function by a dual sequence . By the converse to Cauchy-Schwarz, we may write the left-hand side of (6) as

for some complex numbers with . Indeed, if all of the vanish, we can set the arbitrarily, otherwise we set to be the unit vector formed by dividing by its length. We can then rearrange this expression as

Applying Cauchy-Schwarz (dividing the first factor by and multiplying the second by , after first removing those for which vanish), this is bounded by

and the claim follows by expanding out the second factor.

Observe that Lemma 1 is a special case of Proposition 2 when and the are orthonormal. In general, one can expect Proposition 2 to be useful when the are *almost orthogonal* relative to , in that the correlations tend to be small when are distinct. In that case, one can hope for the diagonal term in the right-hand side of (6) to dominate, in which case one can obtain estimates of comparable strength to the classical Bessel inequality. The flexibility to choose different weights in the above proposition has some technical advantages; for instance, if is concentrated in a sparse set (such as the primes), it is sometimes useful to tailor to a comparable set (e.g. the almost primes) in order not to lose too much in the first factor . Also, it can be useful to choose a fairly “smooth” weight , in order to make the weighted correlations small.

Remark 3In harmonic analysis, the use of tools such as Proposition 2 is known as themethod of almost orthogonality, or themethod. The explanation for the latter name is as follows. For sake of exposition, suppose that is never zero (or we remove all from the domain for which vanishes). Given a family of finitely supported functions , consider the linear operator defined by the formulaThis is a bounded linear operator, and the left-hand side of (6) is nothing other than the norm of . Without any further information on the function other than its norm , the best estimate one can obtain on (6) here is clearly

where denotes the operator norm of .

The adjoint is easily computed to be

The composition of and its adjoint is then given by

From the spectral theorem (or singular value decomposition), one sees that the operator norms of and are related by the identity

and as is a self-adjoint, positive semi-definite operator, the operator norm is also the supremum of the quantity

where ranges over unit vectors in . Putting these facts together, we obtain Proposition 2; furthermore, we see from this analysis that the bound here is essentially optimal if the only information one is allowed to use about is its norm.

For further discussion of almost orthogonality methods from a harmonic analysis perspective, see Chapter VII of this text of Stein.

Exercise 4Under the same hypotheses as Proposition 2, show thatas well as the variant inequality

Proposition 2 has many applications in analytic number theory; for instance, we will use it in later notes to control the large value of Dirichlet series such as the Riemann zeta function. One of the key benefits is that it largely eliminates the need to consider further correlations of the function (other than its self-correlation relative to , which is usually fairly easy to compute or estimate as is usually chosen to be relatively simple); this is particularly useful if is a function which is significantly more complicated to analyse than the functions . Of course, the tradeoff for this is that one now has to deal with the coefficients , which if anything are even less understood than , since literally the only thing we know about these coefficients is their square sum . However, as long as there is enough almost orthogonality between the , one can estimate the by fairly crude estimates (e.g. triangle inequality or Cauchy-Schwarz) and still get reasonably good estimates.

In this set of notes, we will use Proposition 2 to prove some versions of the *large sieve inequality*, which controls a square-sum of correlations

of an arbitrary finitely supported function with various additive characters (where ), or alternatively a square-sum of correlations

of with various primitive Dirichlet characters ; it turns out that one can prove a (slightly sub-optimal) version of this inequality quite quickly from Proposition 2 if one first prepares the sum by inserting a smooth cutoff with well-behaved Fourier transform. The large sieve inequality has many applications (as the name suggests, it has particular utility within *sieve theory*). For the purposes of this set of notes, though, the main application we will need it for is the Bombieri-Vinogradov theorem, which in a very rough sense gives a prime number theorem in arithmetic progressions, which, “on average”, is of strength comparable to the results provided by the Generalised Riemann Hypothesis (GRH), but has the great advantage of being unconditional (it does not require any unproven hypotheses such as GRH); it can be viewed as a significant extension of the Siegel-Walfisz theorem from Notes 2. As we shall see in later notes, the Bombieri-Vinogradov theorem is a very useful ingredient in sieve-theoretic problems involving the primes.

There is however one additional important trick, beyond the large sieve, which we will need in order to establish the Bombieri-Vinogradov theorem. As it turns out, after some basic manipulations (and the deployment of some multiplicative number theory, and specifically the Siegel-Walfisz theorem), the task of proving the Bombieri-Vinogradov theorem is reduced to that of getting a good estimate on sums that are roughly of the form

for some primitive Dirichlet characters . This looks like the type of sum that can be controlled by the large sieve (or by Proposition 2), except that this is an ordinary sum rather than a square sum (i.e., an norm rather than an norm). One could of course try to control such a sum in terms of the associated square-sum through the Cauchy-Schwarz inequality, but this turns out to be very wasteful (it loses a factor of about ). Instead, one should try to exploit the special structure of the von Mangoldt function , in particular the fact that it can be expressible as a Dirichlet convolution of two further arithmetic sequences (or as a finite linear combination of such Dirichlet convolutions). The reason for introducing this convolution structure is through the basic identity

for any finitely supported sequences , as can be easily seen by multiplying everything out and using the completely multiplicative nature of . (This is the multiplicative analogue of the well-known relationship between ordinary convolution and Fourier coefficients.) This factorisation, together with yet another application of the Cauchy-Schwarz inequality, lets one control (7) by square-sums of the sort that can be handled by the large sieve inequality.

As we have seen in Notes 1, the von Mangoldt function does indeed admit several factorisations into Dirichlet convolution type, such as the factorisation . One can try directly inserting this factorisation into the above strategy; it almost works, however there turns out to be a problem when considering the contribution of the portion of or that is supported at very small natural numbers, as the large sieve loses any gain over the trivial bound in such settings. Because of this, there is a need for a more sophisticated decomposition of into Dirichlet convolutions which are non-degenerate in the sense that are supported away from small values. (As a non-example, the trivial factorisation would be a totally inappropriate factorisation for this purpose.) Fortunately, it turns out that through some elementary combinatorial manipulations, some satisfactory decompositions of this type are available, such as the Vaughan identity and the Heath-Brown identity. By using one of these identities we will be able to complete the proof of the Bombieri-Vinogradov theorem. (These identities are also useful for other applications in which one wishes to control correlations between the von Mangoldt function and some other sequence; we will see some examples of this in later notes.)

For further reading on these topics, including a significantly larger number of examples of the large sieve inequality, see Chapters 7 and 17 of Iwaniec and Kowalski.

Remark 5We caution that the presentation given in this set of notes is highly ahistorical; we are using modern streamlined proofs of results that were first obtained by more complicated arguments.

We now move away from the world of *multiplicative* prime number theory covered in Notes 1 and Notes 2, and enter the wider, and complementary, world of *non-multiplicative* prime number theory, in which one studies statistics related to non-multiplicative patterns, such as twins . This creates a major jump in difficulty; for instance, even the most basic multiplicative result about the primes, namely Euclid’s theorem that there are infinitely many of them, remains unproven for twin primes. Of course, the situation is even worse for stronger results, such as Euler’s theorem, Dirichlet’s theorem, or the prime number theorem. Finally, even many multiplicative questions about the primes remain open. The most famous of these is the Riemann hypothesis, which gives the asymptotic (see Proposition 24 from Notes 2). But even if one assumes the Riemann hypothesis, the precise distribution of the error term in the above asymptotic (or in related asymptotics, such as for the sum that measures the distribution of primes in short intervals) is not entirely clear.

Despite this, we do have a number of extremely convincing and well supported *models* for the primes (and related objects) that let us *predict* what the answer to many prime number theory questions (both multiplicative and non-multiplicative) should be, particularly in asymptotic regimes where one can work with aggregate statistics about the primes, rather than with a small number of individual primes. These models are based on taking some statistical distribution related to the primes (e.g. the primality properties of a randomly selected -tuple), and replacing that distribution by a *model* distribution that is easy to compute with (e.g. a distribution with strong joint independence properties). One can then predict the asymptotic value of various (normalised) statistics about the primes by replacing the relevant statistical distributions of the primes with their simplified models. In this non-rigorous setting, many difficult conjectures on the primes reduce to relatively simple calculations; for instance, all four of the (still unsolved) Landau problems may now be justified in the affirmative by one or more of these models. Indeed, the models are so effective at this task that analytic number theory is in the curious position of being able to confidently predict the answer to a large proportion of the open problems in the subject, whilst not possessing a clear way forward to rigorously confirm these answers!

As it turns out, the models for primes that have turned out to be the most accurate in practice are *random* models, which involve (either explicitly or implicitly) one or more random variables. This is despite the prime numbers being obviously deterministic in nature; no coins are flipped or dice rolled to create the set of primes. The point is that while the primes have a lot of obvious *multiplicative* structure (for instance, the product of two primes is never another prime), they do not appear to exhibit much discernible *non-multiplicative* structure asymptotically, in the sense that they rarely exhibit statistical anomalies in the asymptotic limit that cannot be easily explained in terms of the multiplicative properties of the primes. As such, when considering non-multiplicative statistics of the primes, the primes appear to behave pseudorandomly, and can thus be modeled with reasonable accuracy by a random model. And even for multiplicative problems, which are in principle controlled by the zeroes of the Riemann zeta function, one can obtain good predictions by positing various pseudorandomness properties of these zeroes, so that the distribution of these zeroes can be modeled by a random model.

Of course, one cannot expect *perfect* accuracy when replicating a deterministic set such as the primes by a probabilistic model of that set, and each of the heuristic models we discuss below have some limitations to the range of statistics about the primes that they can expect to track with reasonable accuracy. For instance, many of the models about the primes do not fully take into account the multiplicative structure of primes, such as the connection with a zeta function with a meromorphic continuation to the entire complex plane; at the opposite extreme, we have the GUE hypothesis which appears to accurately model the zeta function, but does not capture such basic properties of the primes as the fact that the primes are all natural numbers. Nevertheless, each of the models described below, when deployed within their sphere of reasonable application, has (possibly after some fine-tuning) given predictions that are in remarkable agreement with numerical computation and with known rigorous theoretical results, as well as with other models in overlapping spheres of application; they are also broadly compatible with the general heuristic (discussed in this previous post) that in the absence of any exploitable structure, asymptotic statistics should default to the most “uniform”, “pseudorandom”, or “independent” distribution allowable.

As hinted at above, we do not have a single unified model for the prime numbers (other than the primes themselves, of course), but instead have an overlapping family of useful models that each appear to accurately describe some, but not all, aspects of the prime numbers. In this set of notes, we will discuss four such models:

- The Cramér random model and its refinements, which model the set of prime numbers by a random set.
- The
*Möbius pseudorandomness principle*, which predicts that the Möbius function does not correlate with any genuinely different arithmetic sequence of reasonable “complexity”. - The
*equidistribution of residues principle*, which predicts that the residue classes of a large number modulo a small or medium-sized prime behave as if they are independently and uniformly distributed as varies. - The
*GUE hypothesis*, which asserts that the zeroes of the Riemann zeta function are distributed (at microscopic and mesoscopic scales) like the zeroes of a GUE random matrix, and which generalises the*pair correlation conjecture*regarding pairs of such zeroes.

This is not an exhaustive list of models for the primes and related objects; for instance, there is also the model in which the major arc contribution in the Hardy-Littlewood circle method is predicted to always dominate, and with regards to various finite groups of number-theoretic importance, such as the class groups discussed in Supplement 1, there are also heuristics of Cohen-Lenstra type. Historically, the first heuristic discussion of the primes along these lines was by Sylvester, who worked informally with a model somewhat related to the equidistribution of residues principle. However, we will not discuss any of these models here.

A word of warning: the discussion of the above four models will inevitably be largely informal, and “fuzzy” in nature. While one can certainly make precise formalisations of at least some aspects of these models, one should not be inflexibly wedded to a specific such formalisation as being “the” correct way to pin down the model rigorously. (To quote the statistician George Box: “all models are wrong, but some are useful”.) Indeed, we will see some examples below the fold in which some finer structure in the prime numbers leads to a correction term being added to a “naive” implementation of one of the above models to make it more accurate, and it is perfectly conceivable that some further such fine-tuning will be applied to one or more of these models in the future. These sorts of mathematical models are in some ways closer in nature to the scientific theories used to model the physical world, than they are to the axiomatic theories one is accustomed to in rigorous mathematics, and one should approach the discussion below accordingly. In particular, and in contrast to the other notes in this course, the material here is not *directly* used for proving further theorems, which is why we have marked it as “optional” material. Nevertheless, the heuristics and models here are still used *indirectly* for such purposes, for instance by

- giving a clearer indication of what results one
*expects*to be true, thus guiding one to fruitful conjectures; - providing a quick way to scan for possible errors in a mathematical claim (e.g. by finding that the main term is off from what a model predicts, or an error term is too small);
- gauging the relative strength of various assertions (e.g. classifying some results as “unsurprising”, others as “potential breakthroughs” or “powerful new estimates”, others as “unexpected new phenomena”, and yet others as “way too good to be true”); or
- setting up heuristic barriers (such as the parity barrier) that one has to resolve before resolving certain key problems (e.g. the twin prime conjecture).

See also my previous essay on the distinction between “rigorous” and “post-rigorous” mathematics, or Thurston’s essay discussing, among other things, the “definition-theorem-proof” model of mathematics and its limitations.

Remark 1The material in this set of notes presumes some prior exposure to probability theory. See for instance this previous post for a quick review of the relevant concepts.

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