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In analytic number theory, there is a well known analogy between the prime factorisation of a large integer, and the cycle decomposition of a large permutation; this analogy is central to the topic of “anatomy of the integers”, as discussed for instance in this survey article of Granville. Consider for instance the following two parallel lists of facts (stated somewhat informally). Firstly, some facts about the prime factorisation of large integers:

- Every positive integer has a prime factorisation
into (not necessarily distinct) primes , which is unique up to rearrangement. Taking logarithms, we obtain a partition

of .

- (Prime number theorem) A randomly selected integer of size will be prime with probability when is large.
- If is a randomly selected large integer of size , and is a randomly selected prime factor of (with each index being chosen with probability ), then is approximately uniformly distributed between and . (See Proposition 9 of this previous blog post.)
- The set of real numbers arising from the prime factorisation of a large random number converges (away from the origin, and in a suitable weak sense) to the Poisson-Dirichlet process in the limit . (See the previously mentioned blog post for a definition of the Poisson-Dirichlet process, and a proof of this claim.)

Now for the facts about the cycle decomposition of large permutations:

- Every permutation has a cycle decomposition
into disjoint cycles , which is unique up to rearrangement, and where we count each fixed point of as a cycle of length . If is the length of the cycle , we obtain a partition

of .

- (Prime number theorem for permutations) A randomly selected permutation of will be an -cycle with probability exactly . (This was noted in this previous blog post.)
- If is a random permutation in , and is a randomly selected cycle of (with each being selected with probability ), then is exactly uniformly distributed on . (See Proposition 8 of this blog post.)
- The set of real numbers arising from the cycle decomposition of a random permutation converges (in a suitable sense) to the Poisson-Dirichlet process in the limit . (Again, see this previous blog post for details.)

See this previous blog post (or the aforementioned article of Granville, or the Notices article of Arratia, Barbour, and Tavaré) for further exploration of the analogy between prime factorisation of integers and cycle decomposition of permutations.

There is however something unsatisfying about the analogy, in that it is not clear *why* there should be such a kinship between integer prime factorisation and permutation cycle decomposition. It turns out that the situation is clarified if one uses another fundamental analogy in number theory, namely the analogy between integers and polynomials over a finite field , discussed for instance in this previous post; this is the simplest case of the more general function field analogy between number fields and function fields. Just as we restrict attention to positive integers when talking about prime factorisation, it will be reasonable to restrict attention to monic polynomials . We then have another analogous list of facts, proven very similarly to the corresponding list of facts for the integers:

- Every monic polynomial has a factorisation
into irreducible monic polynomials , which is unique up to rearrangement. Taking degrees, we obtain a partition

of .

- (Prime number theorem for polynomials) A randomly selected monic polynomial of degree will be irreducible with probability when is fixed and is large.
- If is a random monic polynomial of degree , and is a random irreducible factor of (with each selected with probability ), then is approximately uniformly distributed in when is fixed and is large.
- The set of real numbers arising from the factorisation of a randomly selected polynomial of degree converges (in a suitable sense) to the Poisson-Dirichlet process when is fixed and is large.

The above list of facts addressed the *large limit* of the polynomial ring , where the order of the field is held fixed, but the degrees of the polynomials go to infinity. This is the limit that is most closely analogous to the integers . However, there is another interesting asymptotic limit of polynomial rings to consider, namely the *large limit* where it is now the *degree* that is held fixed, but the order of the field goes to infinity. Actually to simplify the exposition we will use the slightly more restrictive limit where the *characteristic* of the field goes to infinity (again keeping the degree fixed), although all of the results proven below for the large limit turn out to be true as well in the large limit.

The large (or large ) limit is technically a different limit than the large limit, but in practice the asymptotic statistics of the two limits often agree quite closely. For instance, here is the prime number theorem in the large limit:

Theorem 1 (Prime number theorem)The probability that a random monic polynomial of degree is irreducible is in the limit where is fixed and the characteristic goes to infinity.

*Proof:* There are monic polynomials of degree . If is irreducible, then the zeroes of are distinct and lie in the finite field , but do not lie in any proper subfield of that field. Conversely, every element of that does not lie in a proper subfield is the root of a unique monic polynomial in of degree (the minimal polynomial of ). Since the union of all the proper subfields of has size , the total number of irreducible polynomials of degree is thus , and the claim follows.

Remark 2The above argument and inclusion-exclusion in fact gives the well known exact formula for the number of irreducible monic polynomials of degree .

Now we can give a precise connection between the cycle distribution of a random permutation, and (the large limit of) the irreducible factorisation of a polynomial, giving a (somewhat indirect, but still connected) link between permutation cycle decomposition and integer factorisation:

Theorem 3The partition of a random monic polynomial of degree converges in distribution to the partition of a random permutation of length , in the limit where is fixed and the characteristic goes to infinity.

We can quickly prove this theorem as follows. We first need a basic fact:

Lemma 4 (Most polynomials square-free in large limit)A random monic polynomial of degree will be square-free with probability when is fixed and (or ) goes to infinity. In a similar spirit, two randomly selected monic polynomials of degree will be coprime with probability if are fixed and or goes to infinity.

*Proof:* For any polynomial of degree , the probability that is divisible by is at most . Summing over all polynomials of degree , and using the union bound, we see that the probability that is *not* squarefree is at most , giving the first claim. For the second, observe from the first claim (and the fact that has only a bounded number of factors) that is squarefree with probability , giving the claim.

Now we can prove the theorem. Elementary combinatorics tells us that the probability of a random permutation consisting of cycles of length for , where are nonnegative integers with , is precisely

since there are ways to write a given tuple of cycles in cycle notation in nondecreasing order of length, and ways to select the labels for the cycle notation. On the other hand, by Theorem 1 (and using Lemma 4 to isolate the small number of cases involving repeated factors) the number of monic polynomials of degree that are the product of irreducible polynomials of degree is

which simplifies to

and the claim follows.

This was a fairly short calculation, but it still doesn’t quite explain *why* there is such a link between the cycle decomposition of permutations and the factorisation of a polynomial. One immediate thought might be to try to link the multiplication structure of permutations in with the multiplication structure of polynomials; however, these structures are too dissimilar to set up a convincing analogy. For instance, the multiplication law on polynomials is abelian and non-invertible, whilst the multiplication law on is (extremely) non-abelian but invertible. Also, the multiplication of a degree and a degree polynomial is a degree polynomial, whereas the group multiplication law on permutations does not take a permutation in and a permutation in and return a permutation in .

I recently found (after some discussions with Ben Green) what I feel to be a satisfying conceptual (as opposed to computational) explanation of this link, which I will place below the fold.

Just a short post here to note that the cover story of this month’s Notices of the AMS, by John Friedlander, is about the recent work on bounded gaps between primes by Zhang, Maynard, our own Polymath project, and others.

I may as well take this opportunity to upload some slides of my own talks on this subject: here are my slides on small and large gaps between the primes that I gave at the “Latinos in the Mathematical Sciences” back in April, and here are my slides on the Polymath project for the Schock Prize symposium last October. (I also gave an abridged version of the latter talk at an AAAS Symposium in February, as well as the Breakthrough Symposium from last November.)

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 .

Kaisa Matomaki, Maksym Radziwill, and I have just uploaded to the arXiv our paper “An averaged form of Chowla’s conjecture“. This paper concerns a weaker variant of the famous conjecture of Chowla (discussed for instance in this previous post) that

as for any distinct natural numbers , where denotes the Liouville function. (One could also replace the Liouville function here by the Möbius function and obtain a morally equivalent conjecture.) This conjecture remains open for any ; for instance the assertion

is a variant of the twin prime conjecture (though possibly a tiny bit easier to prove), and is subject to the notorious parity barrier (as discussed in this previous post).

Our main result asserts, roughly speaking, that Chowla’s conjecture can be established unconditionally provided one has non-trivial averaging in the parameters. More precisely, one has

Theorem 1 (Chowla on the average)Suppose is a quantity that goes to infinity as (but it can go to infinity arbitrarily slowly). Then for any fixed , we haveIn fact, we can remove one of the averaging parameters and obtain

Actually we can make the decay rate a bit more quantitative, gaining about over the trivial bound. The key case is ; while the unaveraged Chowla conjecture becomes more difficult as increases, the averaged Chowla conjecture does not increase in difficulty due to the increasing amount of averaging for larger , and we end up deducing the higher case of the conjecture from the case by an elementary argument.

The proof of the theorem proceeds as follows. By exploiting the Fourier-analytic identity

(related to a standard Fourier-analytic identity for the Gowers norm) it turns out that the case of the above theorem can basically be derived from an estimate of the form

uniformly for all . For “major arc” , close to a rational for small , we can establish this bound from a generalisation of a recent result of Matomaki and Radziwill (discussed in this previous post) on averages of multiplicative functions in short intervals. For “minor arc” , we can proceed instead from an argument of Katai and Bourgain-Sarnak-Ziegler (discussed in this previous post).

The argument also extends to other bounded multiplicative functions than the Liouville function. Chowla’s conjecture was generalised by Elliott, who roughly speaking conjectured that the copies of in Chowla’s conjecture could be replaced by arbitrary bounded multiplicative functions as long as these functions were far from a twisted Dirichlet character in the sense that

(This type of distance is incidentally now a fundamental notion in the Granville-Soundararajan “pretentious” approach to multiplicative number theory.) During our work on this project, we found that Elliott’s conjecture is not quite true as stated due to a technicality: one can cook up a bounded multiplicative function which behaves like on scales for some going to infinity and some slowly varying , and such a function will be far from any fixed Dirichlet character whilst still having many large correlations (e.g. the pair correlations will be large). In our paper we propose a technical “fix” to Elliott’s conjecture (replacing (1) by a truncated variant), and show that this repaired version of Elliott’s conjecture is true on the average in much the same way that Chowla’s conjecture is. (If one restricts attention to real-valued multiplicative functions, then this technical issue does not show up, basically because one can assume without loss of generality that in this case; we discuss this fact in an appendix to the paper.)

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 ).

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