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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)


\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


\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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