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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 ${n}$.

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 (iv) 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.