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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 {m} has a prime factorisation

    \displaystyle  m = p_1 p_2 \dots p_r

    into (not necessarily distinct) primes {p_1,\dots,p_r}, which is unique up to rearrangement. Taking logarithms, we obtain a partition

    \displaystyle  \log m = \log p_1 + \log p_2 + \dots + \log p_r

    of {\log m}.

  • (Prime number theorem) A randomly selected integer {m} of size {m \sim N} will be prime with probability {\approx \frac{1}{\log N}} when {N} is large.
  • If {m \sim N} is a randomly selected large integer of size {N}, and {p = p_i} is a randomly selected prime factor of {m = p_1 \dots p_r} (with each index {i} being chosen with probability {\frac{\log p_i}{\log m}}), then {\log p_i} is approximately uniformly distributed between {0} and {\log N}. (See Proposition 9 of this previous blog post.)
  • The set of real numbers {\{ \frac{\log p_i}{\log m}: i=1,\dots,r \}} arising from the prime factorisation {m = p_1 \dots p_r} of a large random number {m \sim N} converges (away from the origin, and in a suitable weak sense) to the Poisson-Dirichlet process in the limit {N \rightarrow \infty}. (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 {\sigma \in S_n} has a cycle decomposition

    \displaystyle  \sigma = C_1 \dots C_r

    into disjoint cycles {C_1,\dots,C_r}, which is unique up to rearrangement, and where we count each fixed point of {\sigma} as a cycle of length {1}. If {|C_i|} is the length of the cycle {C_i}, we obtain a partition

    \displaystyle  n = |C_1| + \dots + |C_r|

    of {n}.

  • (Prime number theorem for permutations) A randomly selected permutation of {S_n} will be an {n}-cycle with probability exactly {1/n}. (This was noted in this previous blog post.)
  • If {\sigma} is a random permutation in {S_n}, and {C_i} is a randomly selected cycle of {\sigma} (with each {i} being selected with probability {|C_i|/n}), then {|C_i|} is exactly uniformly distributed on {\{1,\dots,n\}}. (See Proposition 8 of this blog post.)
  • The set of real numbers {\{ \frac{|C_i|}{n} \}} arising from the cycle decomposition {\sigma = C_1 \dots C_r} of a random permutation {\sigma \in S_n} converges (in a suitable sense) to the Poisson-Dirichlet process in the limit {n \rightarrow \infty}. (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 {P \in {\mathbf F}_q[T]} over a finite field {{\mathbf F}_q}, 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 {P}. We then have another analogous list of facts, proven very similarly to the corresponding list of facts for the integers:

  • Every monic polynomial {f \in {\mathbf F}_q[T]} has a factorisation

    \displaystyle  f = P_1 \dots P_r

    into irreducible monic polynomials {P_1,\dots,P_r \in {\mathbf F}_q[T]}, which is unique up to rearrangement. Taking degrees, we obtain a partition

    \displaystyle  \hbox{deg} f = \hbox{deg} P_1 + \dots + \hbox{deg} P_r

    of {\hbox{deg} f}.

  • (Prime number theorem for polynomials) A randomly selected monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} will be irreducible with probability {\approx \frac{1}{n}} when {q} is fixed and {n} is large.
  • If {f \in {\mathbf F}_q[T]} is a random monic polynomial of degree {n}, and {P_i} is a random irreducible factor of {f = P_1 \dots P_r} (with each {i} selected with probability {\hbox{deg} P_i / n}), then {\hbox{deg} P_i} is approximately uniformly distributed in {\{1,\dots,n\}} when {q} is fixed and {n} is large.
  • The set of real numbers {\{ \hbox{deg} P_i / n \}} arising from the factorisation {f = P_1 \dots P_r} of a randomly selected polynomial {f \in {\mathbf F}_q[T]} of degree {n} converges (in a suitable sense) to the Poisson-Dirichlet process when {q} is fixed and {n} is large.

The above list of facts addressed the large {n} limit of the polynomial ring {{\mathbf F}_q[T]}, where the order {q} 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 {{\bf Z}}. However, there is another interesting asymptotic limit of polynomial rings to consider, namely the large {q} limit where it is now the degree {n} that is held fixed, but the order {q} of the field goes to infinity. Actually to simplify the exposition we will use the slightly more restrictive limit where the characteristic {p} of the field goes to infinity (again keeping the degree {n} fixed), although all of the results proven below for the large {p} limit turn out to be true as well in the large {q} limit.

The large {q} (or large {p}) limit is technically a different limit than the large {n} 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 {q} limit:

Theorem 1 (Prime number theorem) The probability that a random monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} is irreducible is {\frac{1}{n}+o(1)} in the limit where {n} is fixed and the characteristic {p} goes to infinity.

Proof: There are {q^n} monic polynomials {f \in {\mathbf F}_q[T]} of degree {n}. If {f} is irreducible, then the {n} zeroes of {f} are distinct and lie in the finite field {{\mathbf F}_{q^n}}, but do not lie in any proper subfield of that field. Conversely, every element {\alpha} of {{\mathbf F}_{q^n}} that does not lie in a proper subfield is the root of a unique monic polynomial in {{\mathbf F}_q[T]} of degree {f} (the minimal polynomial of {\alpha}). Since the union of all the proper subfields of {{\mathbf F}_{q^n}} has size {o(q^n)}, the total number of irreducible polynomials of degree {n} is thus {\frac{q^n - o(q^n)}{n}}, and the claim follows. \Box

Remark 2 The above argument and inclusion-exclusion in fact gives the well known exact formula {\frac{1}{n} \sum_{d|n} \mu(\frac{n}{d}) q^d} for the number of irreducible monic polynomials of degree {n}.

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

Theorem 3 The partition {\{ \hbox{deg}(P_1), \dots, \hbox{deg}(P_r) \}} of a random monic polynomial {f= P_1 \dots P_r\in {\mathbf F}_q[T]} of degree {n} converges in distribution to the partition {\{ |C_1|, \dots, |C_r|\}} of a random permutation {\sigma = C_1 \dots C_r \in S_n} of length {n}, in the limit where {n} is fixed and the characteristic {p} 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 {q} limit) A random monic polynomial {f \in {\mathbf F}_q[T]} of degree {n} will be square-free with probability {1-o(1)} when {n} is fixed and {q} (or {p}) goes to infinity. In a similar spirit, two randomly selected monic polynomials {f,g} of degree {n,m} will be coprime with probability {1-o(1)} if {n,m} are fixed and {q} or {p} goes to infinity.

Proof: For any polynomial {g} of degree {m}, the probability that {f} is divisible by {g^2} is at most {1/q^{2m}}. Summing over all polynomials of degree {1 \leq m \leq n/2}, and using the union bound, we see that the probability that {f} is not squarefree is at most {\sum_{1 \leq m \leq n/2} \frac{q^m}{q^{2m}} = o(1)}, giving the first claim. For the second, observe from the first claim (and the fact that {fg} has only a bounded number of factors) that {fg} is squarefree with probability {1-o(1)}, giving the claim. \Box

Now we can prove the theorem. Elementary combinatorics tells us that the probability of a random permutation {\sigma \in S_n} consisting of {c_k} cycles of length {k} for {k=1,\dots,r}, where {c_k} are nonnegative integers with {\sum_{k=1}^r k c_k = n}, is precisely

\displaystyle  \frac{1}{\prod_{k=1}^r c_k! k^{c_k}},

since there are {\prod_{k=1}^r c_k! k^{c_k}} ways to write a given tuple of cycles {C_1,\dots,C_r} in cycle notation in nondecreasing order of length, and {n!} 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 {n} that are the product of {c_k} irreducible polynomials of degree {k} is

\displaystyle  \frac{1}{\prod_{k=1}^r c_k!} \prod_{k=1}^r ( (\frac{1}{k}+o(1)) q^k )^{c_k} + o( q^n )

which simplifies to

\displaystyle  \frac{1+o(1)}{\prod_{k=1}^r c_k! k^{c_k}} q^n,

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 {\sigma = C_1 \dots C_r} of permutations and the factorisation {f = P_1 \dots P_r} of a polynomial. One immediate thought might be to try to link the multiplication structure of permutations in {S_n} 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 {S_n} is (extremely) non-abelian but invertible. Also, the multiplication of a degree {n} and a degree {m} polynomial is a degree {n+m} polynomial, whereas the group multiplication law on permutations does not take a permutation in {S_n} and a permutation in {S_m} and return a permutation in {S_{n+m}}.

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.

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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

\displaystyle  {\mathcal D} f(n) := \sum_n \frac{f(n)}{n^s}

or twisted Dirichlet series

\displaystyle  {\mathcal D} \chi f(n) := \sum_n \frac{f(n) \chi(n)}{n^s}

is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fourier transform, since the functions {n \mapsto \frac{1}{n^s}} and {n \mapsto \frac{\chi(n)}{n^s}} are multiplicative characters.

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

\displaystyle  \hat f(\theta) := \sum_n f(n) e(-n \theta),

where {\theta} lies on the (additive) unit circle {{\bf R}/{\bf Z}} and {e(\theta) := e^{2\pi i \theta}} 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 {e(-n\theta)} is traditionally omitted, and what is denoted by {\hat f(\theta)} here would be referred to instead by {S_f(-\theta)}, {S(f;-\theta)} or just {S(-\theta)}.) We list some of the most classical problems in this area:

  • (Even Goldbach conjecture) Is it true that every even natural number {N} greater than two can be expressed as the sum {p_1+p_2} of two primes?
  • (Odd Goldbach conjecture) Is it true that every odd natural number {N} greater than five can be expressed as the sum {p_1+p_2+p_3} of three primes?
  • (Waring problem) For each natural number {k}, what is the least natural number {g(k)} such that every natural number {N} can be expressed as the sum of {g(k)} or fewer {k^{th}} powers?
  • (Asymptotic Waring problem) For each natural number {k}, what is the least natural number {G(k)} such that every sufficiently large natural number {N} can be expressed as the sum of {G(k)} or fewer {k^{th}} powers?
  • (Partition function problem) For any natural number {N}, let {p(N)} denote the number of representations of {N} of the form {N = n_1 + \dots + n_k} where {k} and {n_1 \geq \dots \geq n_k} are natural numbers. What is the asymptotic behaviour of {p(N)} as {N \rightarrow \infty}?

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 {G(k)}; 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 {N} is expressible as the sum of three primes.

Recently, the restriction that {n} be sufficiently large was replaced by Helfgott with {N > 5}, 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 {N \geq 2} be an natural number. Then

\displaystyle  \sum_{a,b,c: a+b+c=N} \Lambda(a) \Lambda(b) \Lambda(c) = G_3(N) \frac{N^2}{2} + O_A( N^2 \log^{-A} N )

for any {A>0}, where

\displaystyle  G_3(N) = \prod_{p|N} (1-\frac{1}{(p-1)^2}) \times \prod_{p \not | N} (1 + \frac{1}{(p-1)^3}). \ \ \ \ \ (1)

The implied constants are ineffective.

We dropped the hypothesis that {N} is odd in Theorem 2, but note that {G_3(N)} vanishes when {N} is even. For odd {N}, we have

\displaystyle  1 \ll G_3(N) \ll 1.

Exercise 3 Show that Theorem 2 implies Theorem 1.

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 4 Obtain a heuristic derivation of the main term {G_3(N) \frac{N^2}{2}} 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

\displaystyle  \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c)

for various integers {N} and functions {f,g,h: {\bf Z} \rightarrow {\bf C}}, which we will take to be finitely supported to avoid issues of convergence.

Suppose that {f,g,h} are supported on {\{1,\dots,N\}}; for simplicity, let us first assume the pointwise bound {|f(n)|, |g(n)|, |h(n)| \ll 1} for all {n}. (This simple case will not cover the case in Theorem 2, when {f,g,h} are truncated versions of the von Mangoldt function {\Lambda}, but will serve as a warmup to that case.) Then we have the trivial upper bound

\displaystyle  \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c) \ll N^2. \ \ \ \ \ (2)

A basic observation is that this upper bound is attainable if {f,g,h} all “pretend” to behave like the same additive character {n \mapsto e(\theta n)} for some {\theta \in {\bf R}/{\bf Z}}. For instance, if {f(n)=g(n)=h(n) = e(\theta n) 1_{n \leq N}}, then we have {f(a)g(b)h(c) = e(\theta N)} when {a+b+c=N}, and then it is not difficult to show that

\displaystyle  \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c) = (\frac{1}{2}+o(1)) e(\theta N) N^2

as {N \rightarrow \infty}.

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 {f,g,h} all correlate with the same character {n \mapsto e(\theta n)}, or in other words {\hat f(\theta), \hat g(\theta), \hat h(\theta)} are simultaneously large. This converse is largely captured by the following two identities:

Exercise 5 Let {f,g,h: {\bf Z} \rightarrow {\bf C}} be finitely supported functions. Then for any natural number {N}, show that

\displaystyle  \sum_{a,b,c: a+b+c=N} f(a) g(b) h(c) = \int_{{\bf R}/{\bf Z}} \hat f(\theta) \hat g(\theta) \hat h(\theta) e(\theta N)\ d\theta \ \ \ \ \ (3)

and

\displaystyle  \sum_n |f(n)|^2 = \int_{{\bf R}/{\bf Z}} |\hat f(\theta)|^2\ d\theta.

The traditional approach to using the circle method to compute sums such as {\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)} proceeds by invoking (3) to express this sum as an integral over the unit circle, then dividing the unit circle into “major arcs” where {\hat f(\theta), \hat g(\theta),\hat h(\theta)} are large but computable with high precision, and “minor arcs” where one has estimates to ensure that {\hat f(\theta), \hat g(\theta),\hat h(\theta)} are small in both {L^\infty} and {L^2} senses. For functions {f,g,h} of number-theoretic significance, such as truncated von Mangoldt functions, the “major arcs” typically consist of those {\theta} that are close to a rational number {\frac{a}{q}} with {q} 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 {\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}.

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 {a+b+c=N} with some other equation or system of equations.

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

Corollary 6 Let {f,g,h: {\bf Z} \rightarrow {\bf C}} be finitely supported functions. Then for any natural number {N}, we have

\displaystyle  |\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)| \leq (\sum_n |f(n)|^2)^{1/2} (\sum_n |g(n)|^2)^{1/2}

\displaystyle  \times \sup_\theta |\sum_n h(n) e(n\theta)|.

Similarly for permutations of the {f,g,h}.

In the case when {f,g,h} are supported on {[1,N]} and bounded by {O(1)}, this corollary tells us that we have {\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)} is {o(N^2)} whenever one has {\sum_n h(n) e(n\theta) = o(N)} uniformly in {\theta}, and similarly for permutations of {f,g,h}. From this and the triangle inequality, we obtain the following conclusion: if {f} is supported on {[1,N]} and bounded by {O(1)}, and {f} is Fourier-approximated by another function {g} supported on {[1,N]} and bounded by {O(1)} in the sense that

\displaystyle  \sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(N)

uniformly in {\theta}, then we have

\displaystyle  \sum_{a,b,c: a+b+c=N} f(a) f(b) f(c) = \sum_{a,b,c: a+b+c=N} g(a) g(b) g(c) + o(N^2). \ \ \ \ \ (4)

Thus, one possible strategy for estimating the sum {\sum_{a,b,c: a+b+c=N} f(a) f(b) f(c)} is, one can effectively replace (or “model”) {f} by a simpler function {g} which Fourier-approximates {g} in the sense that the exponential sums {\sum_n f(n) e(n\theta), \sum_n g(n) e(n\theta)} agree up to error {o(N)}. For instance:

Exercise 7 Let {N} be a natural number, and let {A} be a random subset of {\{1,\dots,N\}}, chosen so that each {n \in \{1,\dots,N\}} has an independent probability of {1/2} of lying in {A}.

  • (i) If {f := 1_A} and {g := \frac{1}{2} 1_{[1,N]}}, show that with probability {1-o(1)} as {N \rightarrow \infty}, one has {\sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(N)} uniformly in {\theta}. (Hint: for any fixed {\theta}, this can be accomplished with quite a good probability (e.g. {1-o(N^{-2})}) using a concentration of measure inequality, such as Hoeffding’s inequality. To obtain the uniformity in {\theta}, round {\theta} to the nearest multiple of (say) {1/N^2} and apply the union bound).
  • (ii) Show that with probability {1-o(1)}, one has {(\frac{1}{16}+o(1))N^2} representations of the form {N=a+b+c} with {a,b,c \in A} (with {(a,b,c)} treated as an ordered triple, rather than an unordered one).

In the case when {f} is something like the truncated von Mangoldt function {\Lambda(n) 1_{n \leq N}}, the quantity {\sum_n |f(n)|^2} is of size {O( N \log N)} rather than {O( N )}. This costs us a logarithmic factor in the above analysis, however we can still conclude that we have the approximation (4) whenever {g} is another sequence with {\sum_n |g(n)|^2 \ll N \log N} such that one has the improved Fourier approximation

\displaystyle  \sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(\frac{N}{\log N}) \ \ \ \ \ (5)

uniformly in {\theta}. (Later on we will obtain a “log-free” version of this implication in which one does not need to gain a factor of {\frac{1}{\log N}} in the error term.)

This suggests a strategy for proving Vinogradov’s theorem: find an approximant {g} to some suitable truncation {f} of the von Mangoldt function (e.g. {f(n) = \Lambda(n) 1_{n \leq N}} or {f(n) = \Lambda(n) \eta(n/N)}) which obeys the Fourier approximation property (5), and such that the expression {\sum_{a+b+c=N} g(a) g(b) g(c)} is easily computable. It turns out that there are a number of good options for such an approximant {g}. 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 {\Lambda = -\mu L * 1}, that is to say

\displaystyle  \Lambda(n) = - \sum_{d|n} \mu(d) \log d,

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

\displaystyle  \Lambda(n) \approx - \sum_{d \leq R: d|n} \mu(d) \log d. \ \ \ \ \ (6)

Thus, for instance, if {f(n) = \Lambda(n) 1_{n \leq N}}, the approximant {g} would be taken to be

\displaystyle  g(n) := - \sum_{d \leq R: d|n} \mu(d) \log d 1_{n \leq N}.

One could also use the slightly smoother approximation

\displaystyle  \Lambda(n) \approx \sum_{d \leq R: d|n} \mu(d) \log \frac{R}{d} \ \ \ \ \ (7)

in which case we would take

\displaystyle  g(n) := \sum_{d \leq R: d|n} \mu(d) \log \frac{R}{d} 1_{n \leq N}.

The function {g} 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 {g} to be non-negative. As long as {z} is not too large, one can use some sieve-like computations to compute expressions like {\sum_{a+b+c=N} g(a)g(b)g(c)} 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 {A>0} and {x \geq 2}, we have

\displaystyle  \sum_{n \leq x} \mu(n) e(\theta n) \ll_A x \log^{-A} x

uniformly for all {\theta \in {\bf R}/{\bf Z}}. The implied constants are ineffective.

This estimate will be proven by splitting into two cases. In the “major arc” case when {\theta} is close to a rational {a/q} with {q} small (of size {O(\log^{O(1)} x)} 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 {\sum_{n \leq x} \mu(n) e(\theta n)} in terms of bilinear sums of the form {\sum_n \sum_m a_n b_m e(\theta nm)}, and use the Cauchy-Schwarz inequality and the minor arc nature of {\theta} 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

\displaystyle  \Lambda(n) \approx \sum_{q \leq Q} \sum_{a \in ({\bf Z}/q{\bf Z})^\times} \frac{\mu(q)}{\phi(q)} e( \frac{an}{q} ) \ \ \ \ \ (8)

for some {Q} that is not too large compared to {N}. 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 9 Show that the right-hand side of (8) can be rewritten as

\displaystyle  \sum_{d \leq Q: d|n} \mu(d) \rho_d

where

\displaystyle  \rho_d := \frac{d}{\phi(d)} \sum_{m \leq Q/d: (m,d)=1} \frac{\mu^2(m)}{\phi(m)}.

Then, show the inequalities

\displaystyle  \sum_{m \leq Q/d} \frac{\mu^2(m)}{\phi(m)} \leq \rho_d \leq \sum_{m \leq Q} \frac{\mu^2(m)}{\phi(m)}

and conclude that

\displaystyle  \log \frac{Q}{d} - O(1) \leq \rho_d \leq \log Q + O(1).

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

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

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

\displaystyle  \Lambda(n) \approx \frac{W}{\phi(W)} 1_{(n,W)=1} \ \ \ \ \ (9)

where {W := \prod_{p \leq w} p} and {w} is a slowly growing function of {N} (e.g. {w = \log\log N}); a closely related approximation is

\displaystyle  \frac{\phi(W)}{W} \Lambda(Wn+b) \approx 1 \ \ \ \ \ (10)

for {W,w} as above and {1 \leq b \leq W} coprime to {W}. These approximations (closely related to a device known as the “{W}-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 10 With {W} as above, show that

\displaystyle  \frac{W}{\phi(W)} 1_{(n,W)=1} = \sum_{q|W} \sum_{a \in ({\bf Z}/q{\bf Z})^\times} \frac{\mu(q)}{\phi(q)} e( \frac{an}{q} )

for all natural numbers {n}.

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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

\displaystyle  \sum_{n \leq X} \lambda(n+h_1) \dots \lambda(n+h_k) = o(X)

as {X \rightarrow \infty} for any distinct natural numbers {h_1,\dots,h_k}, where {\lambda} denotes the Liouville function. (One could also replace the Liouville function here by the Möbius function {\mu} and obtain a morally equivalent conjecture.) This conjecture remains open for any {k \geq 2}; for instance the assertion

\displaystyle  \sum_{n \leq X} \lambda(n) \lambda(n+2) = o(X)

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 {h_1,\dots,h_k} parameters. More precisely, one has

Theorem 1 (Chowla on the average) Suppose {H = H(X) \leq X} is a quantity that goes to infinity as {X \rightarrow \infty} (but it can go to infinity arbitrarily slowly). Then for any fixed {k \geq 1}, we have

\displaystyle  \sum_{h_1,\dots,h_k \leq H} |\sum_{n \leq X} \lambda(n+h_1) \dots \lambda(n+h_k)| = o( H^k X ).

In fact, we can remove one of the averaging parameters and obtain

\displaystyle  \sum_{h_2,\dots,h_k \leq H} |\sum_{n \leq X} \lambda(n) \lambda(n+h_2) \dots \lambda(n+h_k)| = o( H^{k-1} X ).

Actually we can make the decay rate a bit more quantitative, gaining about {\frac{\log\log H}{\log H}} over the trivial bound. The key case is {k=2}; while the unaveraged Chowla conjecture becomes more difficult as {k} increases, the averaged Chowla conjecture does not increase in difficulty due to the increasing amount of averaging for larger {k}, and we end up deducing the higher {k} case of the conjecture from the {k=2} case by an elementary argument.

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

\displaystyle  \int_{{\mathbf T}} (\int_{\mathbf R} |\sum_{x \leq n \leq x+H} f(n) e(\alpha n)|^2 dx)^2\ d\alpha

\displaystyle = \sum_{|h| \leq H} (H-|h|)^2 |\sum_n f(n) \overline{f}(n+h)|^2

(related to a standard Fourier-analytic identity for the Gowers {U^2} norm) it turns out that the {k=2} case of the above theorem can basically be derived from an estimate of the form

\displaystyle  \int_0^X |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o( H X )

uniformly for all {\alpha \in {\mathbf T}}. For “major arc” {\alpha}, close to a rational {a/q} for small {q}, 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” {\alpha}, 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 {k} copies of {\lambda} in Chowla’s conjecture could be replaced by arbitrary bounded multiplicative functions {g_1,\dots,g_k} as long as these functions were far from a twisted Dirichlet character {n \mapsto \chi(n) n^{it}} in the sense that

\displaystyle  \sum_p \frac{1 - \hbox{Re} g(p) \overline{\chi(p) p^{it}}}{p} = +\infty. \ \ \ \ \ (1)

(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 {g} which behaves like {n^{it_j}} on scales {n \sim N_j} for some {N_j} going to infinity and some slowly varying {t_j}, and such a function will be far from any fixed Dirichlet character whilst still having many large correlations (e.g. the pair correlations {\sum_{n \leq N_j} g(n+1) \overline{g(n)}} 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 {t=0} 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 {\zeta} in the critical strip {\{ \sigma+it: 0 < \sigma < 1; t \in {\bf R} \}} in the limit {t \rightarrow +\infty}. For the purposes of this set of notes, it is a little simpler technically to work with the log-magnitude {\log |\zeta|: {\bf C} \rightarrow [-\infty,+\infty]} of the zeta function. (In principle, one can reconstruct a branch of {\log \zeta}, and hence {\zeta} itself, from {\log |\zeta|} using the Cauchy-Riemann equations, or tools such as the Borel-Carathéodory theorem, see Exercise 40 of Supplement 2.)

One has the classical estimate

\displaystyle  \zeta(\sigma+it) = O( t^{O(1)} )

when {\sigma = O(1)} and {t \geq 10} (say), so that

\displaystyle  \log |\zeta(\sigma+it)| \leq O( \log t ). \ \ \ \ \ (1)

(See e.g. Exercise 37 from Supplement 3.) In view of this, let us define the normalised log-magnitudes {F_T: {\bf C} \rightarrow [-\infty,+\infty]} for any {T \geq 10} by the formula

\displaystyle  F_T( \sigma + it ) := \frac{1}{\log T} \log |\zeta( \sigma + i(T + t) )|;

informally, this is a normalised window into {\log |\zeta|} near {iT}. One can rephrase several assertions about the zeta function in terms of the asymptotic behaviour of {F_T}. For instance:

  • (i) The bound (1) implies that {F_T} is asymptotically locally bounded from above in the limit {T \rightarrow \infty}, thus for any compact set {K \subset {\bf C}} we have {F_T(\sigma+it) \leq O_K(1)} for {\sigma+it \in K} and {T} sufficiently large. In fact the implied constant in {K} only depends on the projection of {K} to the real axis.
  • (ii) For {\sigma > 1}, we have the bounds

    \displaystyle  |\zeta(\sigma+it)|, \frac{1}{|\zeta(\sigma+it)|} \leq \zeta(\sigma)

    which implies that {F_T} converges locally uniformly as {T \rightarrow +\infty} to zero in the region {\{ \sigma+it: \sigma > 1, t \in {\bf R} \}}.

  • (iii) The functional equation, together with the symmetry {\zeta(\sigma-it) = \overline{\zeta(\sigma+it)}}, implies that

    \displaystyle  |\zeta(\sigma+it)| = 2^\sigma \pi^{\sigma-1} |\sin \frac{\pi(\sigma+it)}{2}| |\Gamma(1-\sigma-it)| |\zeta(1-\sigma+it)|

    which by Exercise 17 of Supplement 3 shows that

    \displaystyle  F_T( 1-\sigma+it ) = \frac{1}{2}-\sigma + F_T(\sigma+it) + o(1)

    as {T \rightarrow \infty}, locally uniformly in {\sigma+it}. In particular, when combined with the previous item, we see that {F_T(\sigma+it)} converges locally uniformly as {T \rightarrow +\infty} to {\frac{1}{2}-\sigma} in the region {\{ \sigma+it: \sigma < 0, t \in {\bf R}\}}.

  • (iv) From Jensen’s formula (Theorem 16 of Supplement 2) we see that {\log|\zeta|} is a subharmonic function, and thus {F_T} is subharmonic as well. In particular we have the mean value inequality

    \displaystyle  F_T( z_0 ) \leq \frac{1}{\pi r^2} \int_{z: |z-z_0| \leq r} F_T(z)

    for any disk {\{ z: |z-z_0| \leq r \}}, where the integral is with respect to area measure. From this and (ii) we conclude that

    \displaystyle  \int_{z: |z-z_0| \leq r} F_T(z) \geq O_{z_0,r}(1)

    for any disk with {\hbox{Re}(z_0)>1} and sufficiently large {T}; combining this with (i) we conclude that {F_T} is asymptotically locally bounded in {L^1} in the limit {T \rightarrow \infty}, thus for any compact set {K \subset {\bf C}} we have {\int_K |F_T| \ll_K 1} for sufficiently large {T}.

From (v) and the usual Arzela-Ascoli diagonalisation argument, we see that the {F_T} are asymptotically compact in the topology of distributions: given any sequence {T_n} tending to {+\infty}, one can extract a subsequence such that the {F_T} converge in the sense of distributions. Let us then define a normalised limit profile of {\log|\zeta|} to be a distributional limit {F} of a sequence of {F_T}; 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 {F} (up to almost everywhere equivalence, which is an issue we will address shortly):

  • (i) {F} is bounded from above in the critical strip {\{ \sigma+it: 0 \leq \sigma \leq 1 \}}.
  • (ii) {F} vanishes on {\{ \sigma+it: \sigma \geq 1\}}.
  • (iii) We have the functional equation {F(1-\sigma+it) = \frac{1}{2}-\sigma + F(\sigma+it)} for all {\sigma+it}. In particular {F(\sigma+it) = \frac{1}{2}-\sigma} for {\sigma<0}.
  • (iv) {F} is subharmonic.

Unfortunately, (i)-(iv) fail to characterise {F} completely. For instance, one could have {F(\sigma+it) = f(\sigma)} for any convex function {f(\sigma)} of {\sigma} that equals {0} for {\sigma \geq 1}, {\frac{1}{2}-\sigma} for {\sigma \leq 1}, and obeys the functional equation {f(1-\sigma) = \frac{1}{2}-\sigma+f(\sigma)}, and this would be consistent with (i)-(iv). One can also perturb such examples in a region where {f} is strictly convex to create further examples of functions obeying (i)-(iv). Note from subharmonicity that the function {\sigma \mapsto \sup_t F(\sigma+it)} is always going to be convex in {\sigma}; 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 {F} 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 {F} (taking values in {[-\infty,+\infty)}), defined by the formula

\displaystyle  F(z_0) = \lim_{r \rightarrow 0^+} \frac{1}{\pi r^2} \int_{B(z_0,r)} F(z)\ dz

for any {z_0 \in {\bf C}} (note from subharmonicity that the expression in the limit is monotone nonincreasing as {r \rightarrow 0}, and is also continuous in {z_0}). We will now view this upper semi-continuous representative of {F} as the canonical representative of {F}, so that {F} is now defined everywhere, rather than up to almost everywhere equivalence.

By a classical theorem of Riesz, a function {F} is subharmonic if and only if the distribution {-\Delta F} is a non-negative measure, where {\Delta := \frac{\partial^2}{\partial \sigma^2} + \frac{\partial^2}{\partial t^2}} is the Laplacian in the {\sigma,t} coordinates. Jensen’s formula (or Greens’ theorem), when interpreted distributionally, tells us that

\displaystyle  -\Delta \log |\zeta| = \frac{1}{2\pi} \sum_\rho \delta_\rho

away from the real axis, where {\rho} ranges over the non-trivial zeroes of {\zeta}. Thus, if {F} is a normalised limit profile for {\log |\zeta|} that is the distributional limit of {F_{T_n}}, then we have

\displaystyle  -\Delta F = \nu

where {\nu} is a non-negative measure which is the limit in the vague topology of the measures

\displaystyle  \nu_{T_n} := \frac{1}{2\pi \log T_n} \sum_\rho \delta_{\rho - T_n}.

Thus {\nu} 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 1 The Riemann hypothesis implies the Lindelöf hypothesis.

Proof: It suffices to show that any limiting profile {F} (arising as the limit of some {F_{T_n}}) vanishes on the critical line {\{1/2+it: t \in {\bf R}\}}. But if the Riemann hypothesis holds, then the measures {\nu_{T_n}} are supported on the critical line {\{1/2+it: t \in {\bf R}\}}, so the normalised limit profile {\nu} is also supported on this line. This implies that {F} is harmonic outside of the critical line. By (ii) and unique continuation for harmonic functions, this implies that {F} vanishes on the half-space {\{ \sigma+it: \sigma \geq \frac{1}{2} \}} (and equals {\frac{1}{2}-\sigma} on the complementary half-space, by (iii)), giving the claim. \Box

In fact, we have the following sharper statement:

Theorem 2 (Backlund) The Lindelöf hypothesis is equivalent to the assertion that for any fixed {\sigma_0 > \frac{1}{2}}, the number of zeroes in the region {\{ \sigma+it: \sigma > \sigma_0, T \leq t \leq T+1 \}} is {o(\log T)} as {T \rightarrow \infty}.

Proof: If the latter claim holds, then for any {T_n \rightarrow \infty}, the measures {\nu_{T_n}} assign a mass of {o(1)} to any region of the form {\{ \sigma+it: \sigma > \sigma_0; t_0 \leq t \leq t_0+1 \}} as {n \rightarrow \infty} for any fixed {\sigma_0>\frac{1}{2}} and {t_0 \in {\bf R}}. Thus the normalised limiting profile measure {\nu} is supported on the critical line, and we can repeat the previous argument.

Conversely, suppose the claim fails, then we can find a sequence {T_n} and {\sigma_0>0} such that {\nu_{T_n}} assigns a mass of {\gg 1} to the region {\{ \sigma+it: \sigma > \sigma_0; 0\leq t \leq 1 \}}. Extracting a normalised limiting profile, we conclude that the normalised limiting profile measure {\nu} is non-trivial somewhere to the right of the critical line, so the associated subharmonic function {F} is not harmonic everywhere to the right of the critical line. From the maximum principle and (ii) this implies that {F} 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 {F_{T_n}} only converges to {F} 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.) \Box

Theorem 3 (Littlewood) Assume the Lindelöf hypothesis. Then for any fixed {\alpha>0}, the number of zeroes in the region {\{ \sigma+it: T \leq t \leq T+\alpha \}} is {(2\pi \alpha+o(1)) \log T} as {T \rightarrow +\infty}.

Proof: By the previous arguments, the only possible normalised limiting profile for {\log |\zeta|} is {\max( 0, \frac{1}{2}-\sigma )}. Taking distributional Laplacians, we see that the only possible normalised limiting profile for the zeroes is Lebesgue measure on the critical line. Thus, {\nu_T( \{\sigma+it: T \leq t \leq T+\alpha \} )} can only converge to {\alpha} as {T \rightarrow +\infty}, and the claim follows. \Box

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

Theorem 4 (Titchmarsh) For any fixed {\alpha>0}, there are {\gg_\alpha \log T} zeroes in the region {\{ \sigma+it: T \leq t \leq T+\alpha \}} for sufficiently large {T}.

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 {\alpha > 0} and a sequence {T_n \rightarrow \infty} such that {\{ \sigma+it: T_n \leq t \leq T_n+\alpha \}} contains {o(\log T)} zeroes. Passing to a subsequence to extract a limit profile, we conclude that the normalised limit profile measure {\nu} assigns no mass to the horizontal strip {\{ \sigma+it: 0 \leq t \leq\alpha \}}. Thus the associated subharmonic function {F} is actually harmonic on this strip. But by (ii) and unique continuation this forces {F} to vanish on this strip, contradicting the functional equation (iii). \Box

Exercise 5 Use limiting profiles to obtain the matching upper bound of {O_\alpha(\log T)} for the number of zeroes in {\{ \sigma+it: T \leq t \leq T+\alpha \}} for sufficiently large {T}.

Remark 6 One 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 {o(1)} with more quantitative decay rates such as {1/\log \log T} or {1/\log\log\log T}. 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 7 Let {S(T)} denote the quantity {S(T) := \frac{1}{\pi} \hbox{arg} \zeta(\frac{1}{2}+iT)}, where the branch of the argument is taken by using a line segment connecting {\frac{1}{2}+iT} to (say) {2+iT}, and then to {2}. If we have a sequence {T_n \rightarrow \infty} producing normalised limit profiles {F, \nu} for {\log|\zeta|} and the zeroes respectively, show that {t \mapsto \frac{1}{\log T_n} S(T_n + t)} converges in the sense of distributions to the function {t \mapsto \frac{1}{\pi} \int_{1/2}^1 \frac{\partial F}{\partial t}(\sigma+it)\ d\sigma}, or equivalently

\displaystyle  t \mapsto \frac{1}{2\pi} \frac{\partial}{\partial t} \int_0^1 F(\sigma+it)\ d\sigma.

Conclude in particular that if the Lindelöf hypothesis holds, then {S(T) = o(\log T)} as {T \rightarrow \infty}.

A little bit more about the normalised limit profiles {F} are known unconditionally, beyond (i)-(iv). For instance, from Exercise 3 of Notes 5 we have {\zeta(1/2 + it ) = O( t^{1/6+o(1)} )} as {t \rightarrow +\infty}, which implies that any normalised limit profile {F} for {\log|\zeta|} is bounded by {1/6} on the critical line, beating the bound of {1/4} coming from convexity and (ii), (iii), and then convexity can be used to further bound {F} 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 {F} 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 {\max(0,\frac{1}{2}-\sigma)} (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 {\zeta} for all values of the height parameter {T}, 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.

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In analytic number theory, it is a well-known phenomenon that for many arithmetic functions {f: {\bf N} \rightarrow {\bf C}} of interest in number theory, it is significazintly easier to estimate logarithmic sums such as

\displaystyle \sum_{n \leq x} \frac{f(n)}{n}

than it is to estimate summatory functions such as

\displaystyle \sum_{n \leq x} f(n).

(Here we are normalising {f} to be roughly constant in size, e.g. {f(n) = O( n^{o(1)} )} as {n \rightarrow \infty}.) For instance, when {f} is the von Mangoldt function {\Lambda}, the logarithmic sums {\sum_{n \leq x} \frac{\Lambda(n)}{n}} 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 {\sum_{n \leq x} \Lambda(n)} 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 {\sum_{n \leq x} \frac{f(n)}{n}} can usually be controlled just from knowledge of the Dirichlet series {\sum_n \frac{f(n)}{n^s}} for {s} near {1}; but the summatory functions require control of the Dirichlet series {\sum_n \frac{f(n)}{n^s}} for {s} on or near a large portion of the line {\{ 1+it: t \in {\bf R} \}}. See Notes 2 for further discussion.)

Viewed conversely, whenever one has a difficult estimate on a summatory function such as {\sum_{n \leq x} f(n)}, one can look to see if there is a “cheaper” version of that estimate that only controls the logarithmic sums {\sum_{n \leq x} \frac{f(n)}{n}}, 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 {f: {\bf N} \rightarrow {\bf C}} be a multiplicative function with {|f(n)| \leq 1} for all {n}. Let {x \geq 3} and {T \geq 1}, and set

\displaystyle M := \min_{|t| \leq T} \sum_{p \leq x} \frac{1 - \hbox{Re}( f(p) p^{-it} )}{p}.

Then one has

\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M} + \frac{1}{\sqrt{T}}.

Informally, this theorem asserts that {\sum_{n \leq x} f(n)} is small compared with {x}, unless {f} “pretends” to be like the character {p \mapsto p^{it}} on primes for some small {y}. (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 {f} to be completely multiplicative instead of multiplicative, it requires a strong bound on the analogue of the quantity {M}, and because it only gives qualitative decay rather than quantitative estimates), but easier to prove:

Theorem 2 (Cheap Halasz) Let {x} be an asymptotic parameter goingto infinity. Let {f: {\bf N} \rightarrow {\bf C}} be a completely multiplicative function (possibly depending on {x}) such that {|f(n)| \leq 1} for all {n}, such that

\displaystyle \sum_{p \leq x} \frac{1 - \hbox{Re}( f(p) )}{p} \gg \log\log x. \ \ \ \ \ (1)

 

Then

\displaystyle \frac{1}{\log x} \sum_{n \leq x} \frac{f(n)}{n} = o(1). \ \ \ \ \ (2)

 

Note that now that we are content with estimating exponential sums, we no longer need to preclude the possibility that {f(p)} pretends to be like {p^{it}}; 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 {x} be a parameter going to infinity, and let {1 < P < x} be a quantity depending on {x} such that {P = x^{o(1)}} and {P \rightarrow \infty} as {x \rightarrow \infty}. Then

\displaystyle \sum_{n \leq x} \frac{ | \frac{1}{\log \log P} \sum_{p \leq P: p|n} 1 - 1 |}{n} = o( \log x ).

Informally, this lemma is asserting that

\displaystyle \sum_{p \leq P: p|n} 1 \approx \log \log P

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

\displaystyle 1 \approx 1 * \frac{1}{\log\log P} 1_{{\mathcal P} \cap [1,P]}.

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

\displaystyle \sum_{n \leq x} \frac{ | \frac{1}{\log \log P} \sum_{p \leq P: p|n} 1 - 1 |^2}{n} = o( \log x ).

Expanding out the square, it suffices to show that

\displaystyle \sum_{n \leq x} \frac{ (\frac{1}{\log \log P} \sum_{p \leq P: p|n} 1)^j}{n} = \log x + o( \log x )

for {j=0,1,2}.

We just show the {j=2} case, as the {j=0,1} cases are similar (and easier). We rearrange the left-hand side as

\displaystyle \frac{1}{(\log\log P)^2} \sum_{p_1, p_2 \leq P} \sum_{n \leq x: p_1,p_2|n} \frac{1}{n}.

We can estimate the inner sum as {(1+o(1)) \frac{1}{[p_1,p_2]} \log x}. But a routine application of Mertens’ theorem (handling the diagonal case when {p_1=p_2} separately) shows that

\displaystyle \sum_{p_1, p_2 \leq P} \frac{1}{[p_1,p_2]} = (1+o(1)) (\log\log P)^2

and the claim follows. \Box

Remark 4 As an alternative to the Turan-Kubilius inequality, one can use the Ramaré identity

\displaystyle \sum_{p \leq P: p|n} \frac{1}{\# \{ p' \leq P: p'|n\} + 1} - 1 = 1_{(p,n)=1 \hbox{ for all } p \leq P}

(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 {Q} denote the left-hand side of (2); by the triangle inequality we have {Q=O(1)}. By Lemma 3 (for some {P = x^{o(1)}} to be chosen later) and the triangle inequality we have

\displaystyle \sum_{n \leq x} \frac{\frac{1}{\log \log P} \sum_{p \leq P: p|n} f(n)}{n} = Q \log x + o( \log x ).

We rearrange the left-hand side as

\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p} \sum_{m \leq x/p} \frac{f(m)}{m}.

We now replace the constraint {m \leq x/p} by {m \leq x}. The error incurred in doing so is

\displaystyle O( \frac{1}{\log\log P} \sum_{p \leq P} \frac{1}{p} \sum_{x/P \leq m \leq x} \frac{1}{m} )

which by Mertens’ theorem is {O(\log P) = o( \log x )}. Thus we have

\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p} \sum_{m \leq x} \frac{f(m)}{m} = Q \log x + o( \log x ).

But by definition of {Q}, we have {\sum_{m \leq x} \frac{f(m)}{m} = Q \log x}, thus

\displaystyle [1 - \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p}] Q = o(1). \ \ \ \ \ (3)

 

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

\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{1 - f(p)}{p} + o(1)

and so the real part of this expression is

\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{1 - \hbox{Re} f(p)}{p} + o(1).

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

\displaystyle \sum_{p \leq x^\varepsilon} \frac{(1 - \hbox{Re} f(p)) \log p}{p} \gg \log\log x^\varepsilon - O_\varepsilon(1)

for any {\varepsilon > 0}. This implies that we can find {P = x^{o(1)}} going to infinity such that

\displaystyle \sum_{p \leq P} \frac{(1 - \hbox{Re} f(p)) \log p}{p} \gg (1-o(1))\log\log P

and thus the expression in brackets has real part {\gg 1-o(1)}. The claim follows.

The Turan-Kubilius argument is certainly not the most efficient way to estimate sums such as {\frac{1}{n} \sum_{n \leq x} f(n)}. In the exercise below we give a significantly more accurate estimate that works when {f} is non-negative.

Exercise 5 (Granville-Koukoulopoulos-Matomaki)

  • (i) If {g} is a completely multiplicative function with {g(p) \in \{0,1\}} for all primes {p}, show that

    \displaystyle (e^{-\gamma}-o(1)) \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1} \leq \sum_{n \leq x} \frac{g(n)}{n} \leq \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1}.

    as {x \rightarrow \infty}. (Hint: for the upper bound, expand out the Euler product. For the lower bound, show that {\sum_{n \leq x} \frac{g(n)}{n} \times \sum_{n \leq x} \frac{h(n)}{n} \ge \sum_{n \leq x} \frac{1}{n}}, where {h} is the completely multiplicative function with {h(p) = 1-g(p)} for all primes {p}.)

  • (ii) If {g} is multiplicative and takes values in {[0,1]}, show that

    \displaystyle \sum_{n \leq x} \frac{g(n)}{n} \asymp \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1}

    \displaystyle \asymp \exp( \sum_{p \leq x} \frac{g(p)}{p} )

    for all {x \geq 1}.

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 {\lambda}, although their arguments actually work (with some additional effort) for arbitrary multiplicative functions of magnitude at most {1} that are real-valued (or more generally, stay far from complex characters {p \mapsto p^{it}}). Furthermore, we give a qualitative form of their estimates rather than a quantitative one:

Theorem 6 (Matomaki-Radziwiłł, special case) Let {X} be a parameter going to infinity, and let {2 \leq h \leq X} be a quantity going to infinity as {X \rightarrow \infty}. Then for all but {o(X)} of the integers {x \in [X,2X]}, one has

\displaystyle \sum_{x \leq n \leq x+h} \lambda(n) = o( h ).

Equivalently, one has

\displaystyle \sum_{X \leq x \leq 2X} |\sum_{x \leq n \leq x+h} \lambda(n)|^2 = o( h^2 X ). \ \ \ \ \ (4)

 

A simple sieving argument (see Exercise 18 of Supplement 4) shows that one can replace {\lambda} by the Möbius function {\mu} 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 {O( h^2 X )}. Prior to this paper, such estimates were only known (using arguments similar to those in Section 3 of Notes 6) for {h \geq X^{1/6+\varepsilon}} unconditionally, or for {h \geq \log^A X} for some sufficiently large {A} if one assumed the Riemann hypothesis. This theorem also represents some progress towards Chowla’s conjecture (discussed in Supplement 4) that

\displaystyle \sum_{n \leq x} \lambda(n+h_1) \dots \lambda(n+h_k) = o( x )

as {x \rightarrow \infty} for any fixed distinct {h_1,\dots,h_k}; indeed, it implies that this conjecture holds if one performs a small amount of averaging in the {h_1,\dots,h_k}.

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

Theorem 7 (Cheap Matomaki-Radziwiłł) Let {X} be a parameter going to infinity, and let {1 \leq T \leq X}. Then

\displaystyle \int_X^{X^A} \left|\sum_{x \leq n \leq e^{1/T} x} \frac{\lambda(n)}{n}\right|^2\frac{dx}{x} = o\left( \frac{\log X}{T^2} \right), \ \ \ \ \ (5)

 

for any fixed {A>1}.

Note that (5) improves upon the trivial bound of {O( \frac{\log X}{T^2} )}. Again, one can replace {\lambda} with {\mu} 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.

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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 {a\ (q)} be a primitive residue class. Then {a\ (q)} contains a prime {p} with {p \ll q^{O(1)}}.

In fact it is known that one can find a prime {p} with {p \ll q^{5}}, 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 {p \ll \exp( q^{o(1)} )}, and from Exercise 48 of Notes 2 one can obtain a bound of the form {p \ll \phi(q)^2 \log^2 q} if one assumes the generalised Riemann hypothesis. The probabilistic random models from Supplement 4 suggest that one should in fact be able to take {p \ll q^{1+o(1)}}.

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 {a\ (q)} be a primitive residue class for some {q \geq 2}. For any {x > 1}, let {\psi(x;q,a)} denote the quantity

\displaystyle  \psi(x;q,a) := \sum_{n \leq x: n=a\ (q)} \Lambda(n).

Assume that {x \geq q^C} for some sufficiently large {C}.

  • (i) (No exceptional zero) If all the real zeroes {\beta} of {L}-functions {L(\cdot,\chi)} of real characters {\chi} of modulus {q} are such that {1-\beta \gg \frac{1}{\log q}}, then

    \displaystyle  \psi(x;q,a) = \frac{x}{\phi(q)} ( 1 + O( \exp( - c \frac{\log x}{\log q} ) ) + O( \frac{\log^2 q}{q} ) )

    for all {x \geq 1} and some absolute constant {c>0}.

  • (ii) (Exceptional zero) If there is a zero {\beta} of an {L}-function {L(\cdot,\chi_1)} of a real character {\chi_1} of modulus {q} with {\beta = 1 - \frac{\varepsilon}{\log q}} for some sufficiently small {\varepsilon>0}, then

    \displaystyle  \psi(x;q,a) = \frac{x}{\phi(q)} ( 1 - \chi_1(a) \frac{x^{\beta-1}}{\beta} \ \ \ \ \ (1)

    \displaystyle + O( \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) )

    \displaystyle  + O( \frac{\log^2 q}{q} ) )

    for all {x \geq 1} and some absolute constant {c>0}.

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 {\varepsilon} is small enough). A key point here is that the error term {O( \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) )} 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 {\chi_1(a) \frac{x^{\beta-1}}{\beta}} 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 3 Assuming Theorem 2, and assuming {x \geq q^C} for some sufficiently large absolute constant {C}, establish the lower bound

\displaystyle  \psi(x;a,q) \gg \frac{x}{\phi(q)}

when there is no exceptional zero, and

\displaystyle  \psi(x;a,q) \gg \varepsilon \frac{x}{\phi(q)}

when there is an exceptional zero {\beta = 1 - \frac{\varepsilon}{\log q}}. Conclude that Theorem 2 implies Theorem 1, regardless of whether an exceptional zero exists or not.

Remark 4 The Brun-Titchmarsh theorem (Exercise 33 from Notes 4), in the sharp form of Montgomery and Vaughan, gives that

\displaystyle  \pi(x; q, a) \leq 2 \frac{x}{\phi(q) \log (x/q)}

for any primitive residue class {a\ (q)} and any {x \geq q}. This is (barely) consistent with the estimate (1). Any lowering of the coefficient {2} in the Brun-Titchmarsh inequality (with reasonable error terms), in the regime when {x} is a large power of {q}, 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 {x} is a relatively small power of {q}, 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 {L}-functions. Recall from the truncated explicit formula (Exercise 45(iv) of Notes 2) with (say) {T := q^2} that

\displaystyle  \sum_{n \leq x} \Lambda(n) \chi(n) = - \sum_{\hbox{Re}(\rho) > 3/4; |\hbox{Im}(\rho)| \leq q^2; L(\rho,\chi)=0} \frac{x^\rho}{\rho} + O( \frac{x}{q^2} \log^2 q)

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

\displaystyle  1_{n = a\ (q)} = \frac{1}{\phi(q)} \sum_{\chi\ (q)} \overline{\chi(a)} \chi(n)

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

\displaystyle  \psi(x;a,q) = \frac{x}{\phi(q)} ( 1 - \sum_{\chi\ (q)} \overline{\chi(a)} \sum_{\hbox{Re}(\rho) > 3/4; |\hbox{Im}(\rho)| \leq q^2; L(\rho,\chi)=0} \frac{x^{\rho-1}}{\rho}

\displaystyle  + O( \frac{\log^2 q}{q} ) )

and so it suffices by the triangle inequality (bounding {1/\rho} very crudely by {O(1)}, as the contribution of the low-lying zeroes already turns out to be quite dominant) to show that

\displaystyle  \sum_{\chi\ (q)} \sum_{\sigma > 3/4; |t| \leq q^2; L(\sigma+it,\chi)=0} x^{\sigma-1} \ll \exp( - c \frac{\log x}{\log q} ) \ \ \ \ \ (2)

when no exceptional zero is present, and

\displaystyle  \sum_{\chi\ (q)} \sum_{\sigma > 3/4; |t| \leq q^2; L(\sigma+it,\chi)=0; \sigma+it \neq \beta} x^{\sigma-1} \ll \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) \ \ \ \ \ (3)

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 {q, T \geq 2}. Apart from a potential exceptional zero {\beta}, all zeroes {\sigma+it} of {L}-functions {L(\cdot,\chi)} with {\chi} of modulus {q} and {|t| \leq T} are such that

\displaystyle  \sigma \leq 1 - \frac{c}{\log qT}

for some absolute constant {c>0}.

Using this zero-free region, we have

\displaystyle  x^{\sigma-1} \ll \log x \int_{1/2}^{1-c/\log q} 1_{\alpha < \sigma} x^{\alpha-1}\ d\alpha

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

\displaystyle  \ll \log x \int_{1/2}^{1 - c/\log q} N( \alpha, q, q^2 ) x^{\alpha-1}\ d\alpha

where we recall that {N(\alpha,q,T)} is the number of zeroes {\sigma+it} of any {L}-function of a character {\chi} of modulus {q} with {\sigma \geq \alpha} and {0 \leq t \leq T} (here we use conjugation symmetry to make {t} non-negative, accepting a multiplicative factor of two).

In Exercise 25 of Notes 6, the grand density estimate

\displaystyle  N(\alpha,q,T) \ll (qT)^{4(1-\alpha)} \log^{O(1)}(qT) \ \ \ \ \ (4)

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

\displaystyle  \ll (\log^{O(1)} q) \exp( - c \frac{\log x}{\log q} ).

Unfortunately this is off from what we need by a factor of {\log^{O(1)} q} (and would lead to a weak form of Linnik’s theorem in which {p} was bounded by {O( \exp( \log^{O(1)} q ) )} rather than by {q^{O(1)}}). 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 {t = O(1)}, 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 {q, T > 1} and {1/2 \leq \alpha \leq 1}, one has

\displaystyle  N(\alpha,q,T) \ll (qT)^{O(1-\alpha)}.

The 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.)

Exercise 7 Use Theorem 6 to complete the proof of (2).

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 {\log \frac{1}{\varepsilon}} 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 {q \geq 2} is such that there is an exceptional zero {\beta = 1 - \frac{\varepsilon}{\log q}} with {\varepsilon} small. Then all other zeroes {\sigma+it} of {L}-functions of modulus {q} are such that

\displaystyle  \sigma \leq 1 - c \frac{\log \frac{1}{\varepsilon}}{\log(q(2+|t|))}.

In other words, the exceptional zero enlarges the classical zero-free region by a factor of {\log \frac{1}{\varepsilon}}. The implied constants are effective.

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

Exercise 10 Use 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 {1 * \chi_1} (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

\displaystyle  N'(\alpha,q,T) \ll \varepsilon (1 + \frac{\log T}{\log q}) (qT)^{O(1-\alpha)} \ \ \ \ \ (5)

with effective implied constants for any {1/2 \leq \alpha \leq 1} and {T \geq 1} in the presence of an exceptional zero, where the prime in {N'(\alpha,q,T)} means that the exceptional zero {\beta} is omitted (thus {N'(\alpha,q,T) = N(\alpha,q,T)-1} if {\alpha \leq \beta}). Note that the upper bound on {N'(\alpha,q,T)} falls below one when {\alpha > 1 - c \frac{\log \frac{1}{\varepsilon}}{\log(qT)}} for a sufficiently small {c>0}, 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 11 There 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 as

\displaystyle \frac{L'}{L}(s,\chi)^{(k)} = (-1)^k \sum_n \frac{\Lambda(n) \chi(n) \log^k n}{n^s}

\displaystyle  \approx (-1)^{k+1} k! \sum_\rho \frac{1}{(s-\rho)^{k+1}}

for {\hbox{Re}(s)>1} and large {k}, and performing various sorts of averaging in {k} to attenuate the contribution of many of the zeroes {\rho}. 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 12 When one optimises all the exponents, it turns out that the exponent in Linnik’s theorem is extremely good in the presence of an exceptional zero – indeed Friedlander and Iwaniec showed can even get a bound of the form {p \ll q^{2-c}} for some {c>0}, 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.

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In the previous set of notes, we studied upper bounds on sums such as {|\sum_{N \leq n \leq N+M} n^{-it}|} for {1 \leq M \leq N} that were valid for all {t} in a given range, such as {[T,2T]}; this led in turn to upper bounds on the Riemann zeta {\zeta(\sigma+it)} for {t} in the same range, and for various choices of {\sigma}. While some improvement over the trivial bound of {O(N)} was obtained by these methods, we did not get close to the conjectural bound of {O( N^{1/2+o(1)})} that one expects from pseudorandomness heuristics (assuming that {T} is not too large compared with {N}, e.g. {T = O(N^{O(1)})}.

However, it turns out that one can get much better bounds if one settles for estimating sums such as {|\sum_{N \leq n \leq N+M} n^{-it}|}, or more generally finite Dirichlet series (also known as Dirichlet polynomials) such as {|\sum_n a_n n^{-it}|}, for most values of {t} in a given range such as {[T,2T]}. 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 {t} for which {|\sum_n a_n n^{-it}|} 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 {\frac{1}{T}\int_T^{2T} |\sum_n a_n n^{-it}|^{2k}\ dt} 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 {\zeta(s)} (or the Dirichlet {L}-functions {L(s,\chi)}) in various rectangles of the form {\{ \sigma+it: \sigma \geq \alpha, |t| \leq T \}} for various {T > 1} and {1/2 < \alpha < 1}. 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 {T \rightarrow \infty}, a proportion {1-o(1)} of zeroes of the Riemann zeta function in the critical strip with {|\hbox{Im}(s)| \leq T} will have real part {1/2+o(1)}. Related to this, the number of zeroes with {|\hbox{Im}(s)| \leq T} and {|\hbox{Re}(s)| \geq \alpha} can be shown to be bounded by {O( T^{O(1-\alpha)+o(1)} )} as {T \rightarrow \infty} for any {1/2 < \alpha < 1}.

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.

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We return to the study of the Riemann zeta function {\zeta(s)}, 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 {\zeta}, 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

\displaystyle  \zeta(s) = \sum_{n \leq x} \frac{1}{n^s} - \frac{x^{1-s}}{1-s} + O( \frac{|s|}{\sigma} \frac{1}{x^\sigma} ) \ \ \ \ \ (1)

for any {x > 0} and {s = \sigma+it} with {\sigma>0} and {s \neq 1}. Setting {x=1}, we obtained the crude estimate

\displaystyle  \zeta(s) = \frac{1}{s-1} + O( \frac{|s|}{\sigma} )

in this region. In particular, if {0 < \varepsilon \leq \sigma \ll 1} and {|t| \gg 1} then we had {\zeta(s) = O_\varepsilon( |t| )}. Using the functional equation and the Hadamard three lines lemma, we can improve this to {\zeta(s) \ll_\varepsilon |t|^{\frac{1-\sigma}{2}+\varepsilon}}; see Supplement 3.

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

Proposition 1 Let {s = \sigma+it} with {0 < \varepsilon \leq \sigma \ll 1} and {|t| \gg 1}. Then

\displaystyle  \zeta(s) \ll_\varepsilon \log(2+|t|) \sup_{1 \leq M \leq N \ll |t|} N^{1-\sigma} |\frac{1}{N} \sum_{N \leq n < N+M} e( -\frac{t}{2\pi} \log n)|

where {e(x) := e^{2\pi i x}}.

Proof: We fix a smooth function {\eta: {\bf R} \rightarrow {\bf C}} with {\eta(t)=1} for {t \leq -1} and {\eta(t)=0} for {t \geq 1}, and allow implied constants to depend on {\eta}. Let {s=\sigma+it} with {\varepsilon \leq \sigma \ll 1}. From Exercise 33 of Supplement 3, we have

\displaystyle  \zeta(s) = \sum_n \frac{1}{n^s} \eta( \log n - \log C|t| ) + O_\varepsilon( 1 )

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

\displaystyle  \zeta(s) \ll_{\varepsilon} 1 + \log(2+|t|) \sup_{1 \leq N \ll |t|} |\sum_{N \leq n < 2N} \frac{1}{n^s} \eta( \log n - \log C|t| )|.

We can absorb the first term in the second using the {N=1} case of the supremum. Writing {\frac{1}{n^s} \eta( \log n - \log|C| t ) = N^{-\sigma} e( - \frac{t}{2\pi} \log n ) F_N(n)}, where

\displaystyle  F_N(n) := (N/n)^\sigma \eta(\log n - \log C|t| ),

it thus suffices to show that

\displaystyle  \sum_{N \leq n < 2N} e(-\frac{t}{2\pi} \log N) F_N(n) \ll \sup_{1 \leq M \leq N} |\sum_{N \leq n < N+M} e(-\frac{t}{2\pi} \log n)|

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

\displaystyle  F_N(2N) \sum_{N \leq n < 2N} e(-\frac{t}{2\pi} \log n)

\displaystyle - \int_0^{N} (\sum_{N \leq n < N+M} e(-\frac{t}{2\pi} \log n)) F'_N(M)\ dM

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

We are thus interested in getting good bounds on the sum {\sum_{N \leq n < N+M} e( -\frac{t}{2\pi} \log n )}. More generally, we consider normalised exponential sums of the form

\displaystyle  \frac{1}{N} \sum_{n \in I} e( f(n) ) \ \ \ \ \ (2)

where {I \subset {\bf R}} is an interval of length at most {N} for some {N \geq 1}, and {f: {\bf R} \rightarrow {\bf R}} is a smooth function. We will assume smoothness estimates of the form

\displaystyle  |f^{(j)}(x)| = \exp( O(j^2) ) \frac{T}{N^j} \ \ \ \ \ (3)

for some {T>0}, all {x \in I}, and all {j \geq 1}, where {f^{(j)}} is the {j}-fold derivative of {f}; in the case {f(x) := -\frac{t}{2\pi} \log x}, {I \subset [N,2N]} of interest for the Riemann zeta function, we easily verify that these estimates hold with {T := |t|}. (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 {f^{(0)}=f} of {f} 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 {f(n)}.

The trivial bound for (2) is

\displaystyle  \frac{1}{N} \sum_{n \in I} e(f(n)) \ll 1 \ \ \ \ \ (4)

and we will seek to obtain significant improvements to this bound. Pseudorandomness heuristics predict a bound of {O_\varepsilon(N^{-1/2+\varepsilon})} for (2) for any {\varepsilon>0} if {T = O(N^{O(1)})}; 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 {O(N^{1-c_K})} when {T \ll N^K} and {c_K > 0} depends on {K}. We present two such results here, which perform well for small and large values of {K} respectively:

Theorem 2 Let {2 \leq N \ll T}, let {I} be an interval of length at most {N}, and let {f: I \rightarrow {\bf R}} be a smooth function obeying (3) for all {j \geq 1} and {x \in I}.

  • (i) (van der Corput estimate) For any natural number {k \geq 2}, one has

    \displaystyle  \frac{1}{N} \sum_{n \in I} e( f(n) ) \ll (\frac{T}{N^k})^{\frac{1}{2^k-2}} \log^{1/2} (2+T). \ \ \ \ \ (5)

  • (ii) (Vinogradov estimate) If {k} is a natural number and {T \leq N^{k}}, then

    \displaystyle  \frac{1}{N} \sum_{n \in I} e( f(n) ) \ll N^{-c/k^2} \ \ \ \ \ (6)

    for some absolute constant {c>0}.

The factor of {\log^{1/2} (2+T)} 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 {\log T}. The estimate (6) is superior to (5) when {T \sim N^K} for {K} large, since (after optimising in {k}) (5) gives a gain of the form {N^{-c/2^{cK}}} over the trivial bound, while (6) gives {N^{-c/K^2}}. We have not attempted to obtain completely optimal estimates here, settling for a relatively simple presentation that still gives good bounds on {\zeta}, 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 {f} like a polynomial of degree roughly {k}; 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 {k} times, which is why the gain is only exponentially small in {k}. The Vinogradov estimate achieves linearisation by raising the exponential sum to a significantly smaller power – on the order of {k^2} – by using Hölder’s inequality in combination with the fact that the discrete curve {\{ (n,n^2,\dots,n^k): n \in \{1,\dots,M\}\}} becomes roughly equidistributed in the box {\{ (a_1,\dots,a_k): a_j = O( M^j ) \}} after taking the sumset of about {k^2} 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 “{p}-adic” version of this equidistribution to reduce the claim at a given scale {M} to a smaller scale {M/p} with {p \sim M^{1/k}}, 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 {\zeta(\frac{1}{2}+it) \ll (1+|t|)^{1/6} \log^{O(1)}(1+|t|)} for all {t \in {\bf R}}. (Hint: use the {k=3} case of the Van der Corput estimate.)
  • (ii) For any {0 < \sigma < 1}, show that {\zeta(\sigma+it) \ll (1+|t|)^{\min( \frac{1-\sigma}{3}, \frac{1}{2} - \frac{\sigma}{3}) + o(1)}} as {|t| \rightarrow \infty}.

Exercise 4 Let {t} be such that {|t| \geq 100}, and let {\sigma \geq 1/2}.

  • (i) (Littlewood bound) Use the van der Corput estimate to show that {\zeta(\sigma+it) \ll \log^{O(1)} |t|} whenever {\sigma \geq 1 - O( \frac{(\log\log |t|)^2}{\log |t|} ))}.
  • (ii) (Vinogradov-Korobov bound) Use the Vinogradov estimate to show that {\zeta(\sigma+it) \ll \log^{O(1)} |t|} whenever {\sigma \geq 1 - O( \frac{(\log\log |t|)^{2/3}}{\log^{2/3} |t|} )}.

As noted in Exercise 43 of Notes 2, the Vinogradov-Korobov bound leads to the zero-free region {\{ \sigma+it: \sigma > 1 - c \frac{1}{(\log |t|)^{2/3} (\log\log |t|)^{1/3}}; |t| \geq 100 \}}, which in turn leads to the prime number theorem with error term

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + O\left( x \exp\left( - c \frac{\log^{3/5} x}{(\log\log x)^{1/5}} \right) \right)

for {x > 100}. If one uses the weaker Littlewood bound instead, one obtains the narrower zero-free region

\displaystyle  \{ \sigma+it: \sigma > 1 - c \frac{\log\log|t|}{\log |t|}; |t| \geq 100 \}

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

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + O( x \exp( - c \sqrt{\log x \log\log x} ) )

in the prime number theorem.

Exercise 5 (Vinogradov-Korobov in arithmetic progressions) Let {\chi} be a non-principal character of modulus {q}.

  • (i) (Vinogradov-Korobov bound) Use the Vinogradov estimate to show that {L(\sigma+it,\chi) \ll \log^{O(1)}(q|t|)} whenever {|t| \geq 100} and

    \displaystyle  \sigma \geq 1 - O( \min( \frac{\log\log(q|t|)}{\log q}, \frac{(\log\log(q|t|))^{2/3}}{\log^{2/3} |t|} ) ).

    (Hint: use the Vinogradov estimate and a change of variables to control {\sum_{n \in I: n = a\ (q)} \exp( -it \log n)} for various intervals {I} of length at most {N} and residue classes {a\ (q)}, in the regime {N \geq q^2} (say). For {N < q^2}, do not try to capture any cancellation and just use the triangle inequality instead.)

  • (ii) Obtain a zero-free region

    \displaystyle  \{ \sigma+it: \sigma > 1 - c \min( \frac{1}{(\log |t|)^{2/3} (\log\log |t|)^{1/3}}, \frac{1}{\log q} );

    \displaystyle  |t| \geq 100 \}

    for {L(s,\chi)}, for some (effective) absolute constant {c>0}.

  • (iii) Obtain the prime number theorem in arithmetic progressions with error term

    \displaystyle  \sum_{n \leq x: n = a\ (q)} \Lambda(n) = x + O\left( x \exp\left( - c_A \frac{\log^{3/5} x}{(\log\log x)^{1/5}} \right) \right)

    whenever {x > 100}, {q \leq \log^A x}, {a\ (q)} is primitive, and {c_A>0} depends (ineffectively) on {A}.

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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 {\kappa} is equal to {1}, 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 {z, D \geq 1} be quantities with {z = D^{1/s}} for some fixed {s>1}, and let {g} be a multiplicative function with

\displaystyle  g(p) = \frac{1}{p} + O(\frac{1}{p^2}) \ \ \ \ \ (1)

and

\displaystyle  0 \leq g(p) \leq 1-c \ \ \ \ \ (2)

for all primes {p} and some fixed {c>0} (we allow all constants below to depend on {c}). Let {P(z) := \prod_{p<z} p}, and for each prime {p < z}, let {E_p} be a set of integers, with {E_d := \bigcap_{p|d} E_p} for {d|P(z)}. We consider finitely supported sequences {(a_n)_{n \in {\bf Z}}} of non-negative reals for which we have bounds of the form

\displaystyle  \sum_{n \in E_d} a_n = g(d) X + r_d. \ \ \ \ \ (3)

for all square-free {d \leq D} and some {X>0}, and some remainder terms {r_d}. One is then interested in upper and lower bounds on the quantity

\displaystyle  \sum_{n\not \in\bigcup_{p <z} E_p} a_n.

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

\displaystyle  \sum_{n\not \in\bigcup_{p <z} E_p} a_n = (1 + O(e^{-s})) X V(z) + O( \sum_{d \leq D: \mu^2(d)=1} |r_d| ) \ \ \ \ \ (4)

where {V(z)} is the quantity

\displaystyle  V(z) := \prod_{p<z} (1-g(p)). \ \ \ \ \ (5)

This bound is strong when {s} is large, but is not as useful for smaller values of {s}. We now give a sharp bound in this regime. We introduce the functions {F, f: (0,+\infty) \rightarrow {\bf R}^+} by

\displaystyle  F(s) := 2e^\gamma ( \frac{1_{s>1}}{s} \ \ \ \ \ (6)

\displaystyle  + \sum_{j \geq 3, \hbox{ odd}} \frac{1}{j!} \int_{[1,+\infty)^{j-1}} 1_{t_1+\dots+t_{j-1}\leq s-1} \frac{dt_1 \dots dt_{j-1}}{t_1 \dots t_j} )

and

\displaystyle  f(s) := 2e^\gamma \sum_{j \geq 2, \hbox{ even}} \frac{1}{j!} \int_{[1,+\infty)^{j-1}} 1_{t_1+\dots+t_{j-1}\leq s-1} \frac{dt_1 \dots dt_{j-1}}{t_1 \dots t_j} \ \ \ \ \ (7)

where we adopt the convention {t_j := s - t_1 - \dots - t_{j-1}}. Note that for each {s} one has only finitely many non-zero summands in (6), (7). These functions are closely related to the Buchstab function {\omega} from Exercise 28 of Supplement 4; indeed from comparing the definitions one has

\displaystyle  F(s) + f(s) = 2 e^\gamma \omega(s)

for all {s>0}.

Exercise 1 (Alternate definition of {F, f}) Show that {F(s)} is continuously differentiable except at {s=1}, and {f(s)} is continuously differentiable except at {s=2} where it is continuous, obeying the delay-differential equations

\displaystyle  \frac{d}{ds}( s F(s) ) = f(s-1) \ \ \ \ \ (8)

for {s > 1} and

\displaystyle  \frac{d}{ds}( s f(s) ) = F(s-1) \ \ \ \ \ (9)

for {s>2}, with the initial conditions

\displaystyle  F(s) = \frac{2e^\gamma}{s} 1_{s>1}

for {s \leq 3} and

\displaystyle  f(s) = 0

for {s \leq 2}. Show that these properties of {F, f} determine {F, f} completely.

For future reference, we record the following explicit values of {F, f}:

\displaystyle  F(s) = \frac{2e^\gamma}{s} \ \ \ \ \ (10)

for {1 < s \leq 3}, and

\displaystyle  f(s) = \frac{2e^\gamma}{s} \log(s-1) \ \ \ \ \ (11)

for {2 \leq s \leq 4}.

We will show

Theorem 2 (Linear sieve) Let the notation and hypotheses be as above, with {s > 1}. Then, for any {\varepsilon > 0}, one has the upper bound

\displaystyle  \sum_{n\not \in\bigcup_{p <z} E_p} a_n \leq (F(s) + O(\varepsilon)) X V(z) + O( \sum_{d \leq D: \mu^2(d)=1} |r_d| ) \ \ \ \ \ (12)

and the lower bound

\displaystyle  \sum_{n\not \in\bigcup_{p <z} E_p} a_n \geq (f(s) - O(\varepsilon)) X V(z) + O( \sum_{d \leq D: \mu^2(d)=1} |r_d| ) \ \ \ \ \ (13)

if {D} is sufficiently large depending on {\varepsilon, s, c}. Furthermore, this claim is sharp in the sense that the quantity {F(s)} cannot be replaced by any smaller quantity, and similarly {f(s)} cannot be replaced by any larger quantity.

Comparing the linear sieve with the fundamental lemma (and also testing using the sequence {a_n = 1_{1 \leq n \leq N}} for some extremely large {N}), we conclude that we necessarily have the asymptotics

\displaystyle  1 - O(e^{-s}) \leq f(s) \leq 1 \leq F(s) \leq 1 + O( e^{-s} )

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

Exercise 3 Establish the integral identities

\displaystyle  F(s) = 1 + \frac{1}{s} \int_s^\infty (1 - f(t-1))\ dt

and

\displaystyle  f(s) = 1 + \frac{1}{s} \int_s^\infty (1 - F(t-1))\ dt

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

Exercise 4 Use the Selberg sieve (Theorem 30 from Notes 4) to obtain a slightly weaker version of (12) in the range {1 < s < 3} in which the error term {|r_d|} is worsened to {\tau_3(d) |r_d|}, but the main term is unchanged.

We will prove Theorem 2 below the fold. The optimality of {F, f} 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 {F(s) \geq \frac{2 e^\gamma}{s}} and {f(s)=0} for {s \leq 2}, but this can be sharpened by a more careful counting of various sums involving the Liouville function {\lambda}.

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 {p} such that {p+2} 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 {N}, there exists a prime {p} between {2} and {N} such that {N-p} 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 {\kappa=1}).

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