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As in previous posts, we use the following asymptotic notation: {x} is a parameter going off to infinity, and all quantities may depend on {x} unless explicitly declared to be “fixed”. The asymptotic notation {O(), o(), \ll} is then defined relative to this parameter. A quantity {q} is said to be of polynomial size if one has {q = O(x^{O(1)})}, and said to be bounded if {q=O(1)}. Another convenient notation: we write {X \lessapprox Y} for {X \ll x^{o(1)} Y}. Thus for instance the divisor bound asserts that if {q} has polynomial size, then the number of divisors of {q} is {\lessapprox 1}.

This post is intended to highlight a phenomenon unearthed in the ongoing polymath8 project (and is in fact a key component of Zhang’s proof that there are bounded gaps between primes infinitely often), namely that one can get quite good bounds on relatively short exponential sums when the modulus {q} is smooth, through the basic technique of Weyl differencing (ultimately based on the Cauchy-Schwarz inequality, and also related to the van der Corput lemma in equidistribution theory). Improvements in the case of smooth moduli have appeared before in the literature (e.g. in this paper of Heath-Brown, paper of Graham and Ringrose, this later paper of Heath-Brown, this paper of Chang, or this paper of Goldmakher); the arguments here are particularly close to that of the first paper of Heath-Brown. It now also appears that further optimisation of this Weyl differencing trick could lead to noticeable improvements in the numerology for the polymath8 project, so I am devoting this post to explaining this trick further.

To illustrate the method, let us begin with the classical problem in analytic number theory of estimating an incomplete character sum

\displaystyle  \sum_{M+1 \leq n \leq M+N} \chi(n)

where {\chi} is a primitive Dirichlet character of some conductor {q}, {M} is an integer, and {N} is some quantity between {1} and {q}. Clearly we have the trivial bound

\displaystyle  |\sum_{M+1 \leq n \leq M+N} \chi(n)| \leq N; \ \ \ \ \ (1)

we also have the classical Pólya-Vinogradov inequality

\displaystyle  |\sum_{M+1 \leq n \leq M+N} \chi(n)| \ll q^{1/2} \log q. \ \ \ \ \ (2)

This latter inequality gives improvements over the trivial bound when {N} is much larger than {q^{1/2}}, but not for {N} much smaller than {q^{1/2}}. The Pólya-Vinogradov inequality can be deduced via a little Fourier analysis from the completed exponential sum bound

\displaystyle  | \sum_{n \in {\bf Z}/q{\bf Z}} \chi(n) e_q( an )| \ll q^{1/2}

for any {a \in {\bf Z}/q{\bf Z}}, where {e_q(n) :=e^{2\pi i n/q}}. (In fact, from the classical theory of Gauss sums, this exponential sum is equal to {\tau(\chi) \overline{\chi(a)}} for some complex number {\tau(\chi)} of norm {\sqrt{q}}.)

In the case when {q} is a prime, improving upon the above two inequalities is an important but difficult problem, with only partially satisfactory results so far. To give just one indication of the difficulty, the seemingly modest improvement

\displaystyle |\sum_{M+1 \leq n \leq M+N} \chi(n)| \ll p^{1/2} \log \log p

to the Pólya-Vinogradov inequality when {q=p} was a prime required a 14-page paper in Inventiones by Montgomery and Vaughan to prove, and even then it was only conditional on the generalised Riemann hypothesis! See also this more recent paper of Granville and Soundararajan for an unconditional variant of this result in the case that {\chi} has odd order.

Another important improvement is the Burgess bound, which in our notation asserts that

\displaystyle  |\sum_{M+1 \leq n \leq M+N} \chi(n)| \lessapprox N^{1-1/r} q^{\frac{r+1}{4r^2}} \ \ \ \ \ (3)

for any fixed integer {r \geq 2}, assuming that {q} is square-free (for simplicity) and of polynomial size; see this previous post for a discussion of the Burgess argument. This is non-trivial for {N} as small as {q^{1/4+o(1)}}.

In the case when {q} is prime, there has been very little improvement to the Burgess bound (or its Fourier dual, which can give bounds for {N} as large as {q^{3/4-o(1)}}) in the last fifty years; an improvement to the exponents in (3) in this case (particularly anything that gave a power saving for {N} below {q^{1/4}}) would in fact be rather significant news in analytic number theory.

However, in the opposite case when {q} is smooth – that is to say, all of its factors are much smaller than {q} – then one can do better than the Burgess bound in some regimes. This fact has been observed in several places in the literature (in particular, in the papers of Heath-Brown, Graham-Ringrose, Chang, and Goldmakher mentioned previously), but also turns out to (implicitly) be a key insight in Zhang’s paper on bounded prime gaps. In the case of character sums, one such improved estimate (closely related to Theorem 2 of the Heath-Brown paper) is as follows:

Proposition 1 Let {q} be square-free with a factorisation {q = q_1 q_2} and of polynomial size, and let {M,N} be integers with {1 \leq N \leq q}. Then for any primitive character {\chi} with conductor {q}, one has

\displaystyle  | \sum_{M+1 \leq n \leq M+N} \chi(n) | \lessapprox N^{1/2} q_1^{1/2} + N^{1/2} q_2^{1/4}.

This proposition is particularly powerful when {q} is smooth, as this gives many factorisations {q = q_1 q_2} with the ability to specify {q_1,q_2} with a fair amount of accuracy. For instance, if {q} is {y}-smooth (i.e. all prime factors are at most {y}), then by the greedy algorithm one can find a divisor {q_1} of {q} with {y^{-2/3} q^{1/3} \leq q_1 \leq y^{1/3} q^{1/3}}; if we set {q_2 := q/q_1}, then {y^{-1/3} q^{2/3} \leq q_2 \leq y^{2/3} q^{2/3}}, and the above proposition then gives

\displaystyle  | \sum_{M+1 \leq n \leq M+N} \chi(n) | \lessapprox y^{1/6} N^{1/2} q^{1/6}

which can improve upon the Burgess bound when {y} is small. For instance, if {N = q^{1/2}}, then this bound becomes {\lessapprox y^{1/6} q^{5/12}}; in contrast the Burgess bound only gives {\lessapprox q^{7/16}} for this value of {N} (using the optimal choice {r=2} for {r}), which is inferior for {y < q^{1/8}}.

The hypothesis that {q} be squarefree may be relaxed, but for applications to the Polymath8 project, it is only the squarefree moduli that are relevant.

Proof: If {N \ll q_1} then the claim follows from the trivial bound (1), while for {N \gg q_2} the claim follows from (2). Hence we may assume that

\displaystyle  q_1 < N < q_2.

We use the method of Weyl differencing, the key point being to difference in multiples of {q_1}.

Let {K := \lfloor N/q_1 \rfloor}, thus {K \geq 1}. For any {1 \leq k \leq K}, we have

\displaystyle  \sum_{M+1 \leq n \leq M+N} \chi(n) = \sum_n 1_{[M+1,M+N]}(n+kq_1) \chi(n+kq_1)

and thus on averaging

\displaystyle  \sum_{M+1 \leq n \leq M+N} \chi(n) = \frac{1}{K} \sum_n \sum_{k=1}^K 1_{[M+1,M+N]}(n+kq_1) \chi(n+kq_1). \ \ \ \ \ (4)

By the Chinese remainder theorem, we may factor

\displaystyle  \chi(n) = \chi_1(n) \chi_2(n)

where {\chi_1,\chi_2} are primitive characters of conductor {q_1,q_2} respectively. As {\chi_1} is periodic of period {q_1}, we thus have

\displaystyle  \chi(n+kq_1) = \chi_1(n) \chi_2(n+kq_2)

and so we can take {\chi_1} out of the inner summation of the right-hand side of (4) to obtain

\displaystyle  \sum_{M+1 \leq n \leq M+N} \chi(n) = \frac{1}{K} \sum_n \chi_1(n) \sum_{k=1}^K 1_{[M+1,M+N]}(n+kq_1) \chi_2(n+kq_1)

and hence by the triangle inequality

\displaystyle  |\sum_{M+1 \leq n \leq M+N} \chi(n)| \leq \frac{1}{K} \sum_n |\sum_{k=1}^K 1_{[M+1,M+N]}(n+kq_1) \chi_2(n+kq_1)|.

Note how the characters on the right-hand side only have period {q_2} rather than {q=q_1 q_2}. This reduction in the period is ultimately the source of the saving over the Pólya-Vinogradov inequality.

Note that the inner sum vanishes unless {n \in [M+1-Kq_1,M+N]}, which is an interval of length {O(N)} by choice of {K}. Thus by Cauchy-Schwarz one has

\displaystyle  | \sum_{M+1 \leq n \leq M+N} \chi(n) | \ll

\displaystyle  \frac{N^{1/2}}{K} (\sum_n |\sum_{k=1}^K 1_{[M+1,M+N]}(n+kq_1) \chi_2(n+kq_1)|^2)^{1/2}.

We expand the right-hand side as

\displaystyle  \frac{N^{1/2}}{K} |\sum_{1 \leq k,k' \leq K} \sum_n

\displaystyle  1_{[M+1,M+N]}(n+kq_1) 1_{[M+1,M+N]}(n+k'q_1) \chi_2(n+kq_1) \overline{\chi_2(n+k'q_1)}|^{1/2}.

We first consider the diagonal contribution {k=k'}. In this case we use the trivial bound {O(N)} for the inner summation, and we soon see that the total contribution here is {O( K^{-1/2} N ) = O( N^{1/2}q_1^{1/2} )}.

Now we consider the off-diagonal case; by symmetry we can take {k < k'}. Then the indicator functions {1_{[M+1,M+N]}(n+kq_1) 1_{[M+1,M+N]}(n+k'q_1)} restrict {n} to the interval {[M+1-kq_1, M+N-k'q_1]}. On the other hand, as a consequence of the Weil conjectures for curves one can show that

\displaystyle  |\sum_{n \in {\bf Z}/q_2{\bf Z}} \chi_2(n+kq_1) \overline{\chi_2(n+k'q_1)} e_{q_2}(an)| \lessapprox q_2^{1/2} (k-k',q_2)^{1/2}

for any {a \in {\bf Z}/q_2{\bf Z}}; indeed one can use the Chinese remainder theorem and the square-free nature of {q_2} to reduce to the case when {q_2} is prime, in which case one can apply (for instance) the original paper of Weil to establish this bound, noting also that {q_1} and {q_2} are coprime since {q} is squarefree. Applying the method of completion of sums (or the Parseval formula), this shows that

\displaystyle  |\sum_n 1_{[M+1,M+N]}(n+kq_1) 1_{[M+1,M+N]}(n+k'q_1) \chi_2(n+kq_1) \overline{\chi_2(n+k'q_1)}|

\displaystyle  \lessapprox q_2^{1/2} (k-k',q_2)^{1/2}.

Summing in {k,k'} (using Lemma 5 from this previous post) we see that the total contribution to the off-diagonal case is

\displaystyle  \lessapprox \frac{N^{1/2}}{K} ( K^2 q_2^{1/2} )^{1/2}

which simplifies to {\lessapprox N^{1/2} q_2^{1/4}}. The claim follows. \Box

A modification of the above argument (using more complicated versions of the Weil conjectures) allows one to replace the summand {\chi(n)} by more complicated summands such as {\chi(f(n)) e_q(g(n))} for some polynomials or rational functions {f,g} of bounded degree and obeying a suitable non-degeneracy condition (after restricting of course to those {n} for which the arguments {f(n),g(n)} are well-defined). We will not detail this here, but instead turn to the question of estimating slightly longer exponential sums, such as

\displaystyle  \sum_{1 \leq n \leq N} e_{d_1}( \frac{c_1}{n} ) e_{d_2}( \frac{c_2}{n+l} )

where {N} should be thought of as a little bit larger than {(d_1d_2)^{1/2}}.

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This is one of the continuations of the online reading seminar of Zhang’s paper for the polymath8 project. (There are two other continuations; this previous post, which deals with the combinatorial aspects of the second part of Zhang’s paper, and a post to come that covers the Type III sums.) The main purpose of this post is to present (and hopefully, to improve upon) the treatment of two of the three key estimates in Zhang’s paper, namely the Type I and Type II estimates.

The main estimate was already stated as Theorem 16 in the previous post, but we quickly recall the relevant definitions here. As in other posts, we always take {x} to be a parameter going off to infinity, with the usual asymptotic notation {O(), o(), \ll} associated to this parameter.

Definition 1 (Coefficient sequences) A coefficient sequence is a finitely supported sequence {\alpha: {\bf N} \rightarrow {\bf R}} that obeys the bounds

\displaystyle  |\alpha(n)| \ll \tau^{O(1)}(n) \log^{O(1)}(x) \ \ \ \ \ (1)

for all {n}, where {\tau} is the divisor function.

  • (i) If {\alpha} is a coefficient sequence and {a\ (q) = a \hbox{ mod } q} is a primitive residue class, the (signed) discrepancy {\Delta(\alpha; a\ (q))} of {\alpha} in the sequence is defined to be the quantity

    \displaystyle  \Delta(\alpha; a \ (q)) := \sum_{n: n = a\ (q)} \alpha(n) - \frac{1}{\phi(q)} \sum_{n: (n,q)=1} \alpha(n). \ \ \ \ \ (2)

  • (ii) A coefficient sequence {\alpha} is said to be at scale {N} for some {N \geq 1} if it is supported on an interval of the form {[(1-O(\log^{-A_0} x)) N, (1+O(\log^{-A_0} x)) N]}.
  • (iii) A coefficient sequence {\alpha} at scale {N} is said to obey the Siegel-Walfisz theorem if one has

    \displaystyle  | \Delta(\alpha 1_{(\cdot,q)=1}; a\ (r)) | \ll \tau(qr)^{O(1)} N \log^{-A} x \ \ \ \ \ (3)

    for any {q,r \geq 1}, any fixed {A}, and any primitive residue class {a\ (r)}.

  • (iv) A coefficient sequence {\alpha} at scale {N} is said to be smooth if it takes the form {\alpha(n) = \psi(n/N)} for some smooth function {\psi: {\bf R} \rightarrow {\bf C}} supported on {[1-O(\log^{-A_0} x), 1+O(\log^{-A_0} x)]} obeying the derivative bounds

    \displaystyle  \psi^{(j)}(t) = O( \log^{j A_0} x ) \ \ \ \ \ (4)

    for all fixed {j \geq 0} (note that the implied constant in the {O()} notation may depend on {j}).

In Lemma 8 of this previous post we established a collection of “crude estimates” which assert, roughly speaking, that for the purposes of averaged estimates one may ignore the {\tau^{O(1)}(n)} factor in (1) and pretend that {\alpha(n)} was in fact {O( \log^{O(1)} n)}. We shall rely frequently on these “crude estimates” without further citation to that precise lemma.

For any {I \subset {\bf R}}, let {{\mathcal S}_I} denote the square-free numbers whose prime factors lie in {I}.

Definition 2 (Singleton congruence class system) Let {I \subset {\bf R}}. A singleton congruence class system on {I} is a collection {{\mathcal C} = (\{a_q\})_{q \in {\mathcal S}_I}} of primitive residue classes {a_q \in ({\bf Z}/q{\bf Z})^\times} for each {q \in {\mathcal S}_I}, obeying the Chinese remainder theorem property

\displaystyle  a_{qr}\ (qr) = (a_q\ (q)) \cap (a_r\ (r)) \ \ \ \ \ (5)

whenever {q,r \in {\mathcal S}_I} are coprime. We say that such a system {{\mathcal C}} has controlled multiplicity if the

\displaystyle  \tau_{\mathcal C}(n) := |\{ q \in {\mathcal S}_I: n = a_q\ (q) \}|

obeys the estimate

\displaystyle  \sum_{C^{-1} x \leq n \leq Cx: n = a\ (r)} \tau_{\mathcal C}(n)^2 \ll \frac{x}{r} \tau(r)^{O(1)} \log^{O(1)} x + x^{o(1)}. \ \ \ \ \ (6)

for any fixed {C>1} and any congruence class {a\ (r)} with {r \in {\mathcal S}_I}.

The main result of this post is then the following:

Theorem 3 (Type I/II estimate) Let {\varpi, \delta, \sigma > 0} be fixed quantities such that

\displaystyle  11 \varpi + 3\delta + 2 \sigma < \frac{1}{4} \ \ \ \ \ (7)

and

\displaystyle  29\varpi + 5 \delta < \frac{1}{4} \ \ \ \ \ (8)

and let {\alpha,\beta} be coefficient sequences at scales {M,N} respectively with

\displaystyle  x \ll MN \ll x \ \ \ \ \ (9)

and

\displaystyle  x^{\frac{1}{2}-\sigma} \ll N \ll M \ll x^{\frac{1}{2}+\sigma} \ \ \ \ \ (10)

with {\beta} obeying a Siegel-Walfisz theorem. Then for any {I \subset [1,x^\delta]} and any singleton congruence class system {(\{a_q\})_{q \in {\mathcal S}_I}} with controlled multiplicity we have

\displaystyle  \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}} |\Delta(\alpha \ast \beta; a_q)| \ll x \log^{-A} x.

The proof of this theorem relies on five basic tools:

  • (i) the Bombieri-Vinogradov theorem;
  • (ii) completion of sums;
  • (iii) the Weil conjectures;
  • (iv) factorisation of smooth moduli {q \in {\mathcal S}_I}; and
  • (v) the Cauchy-Schwarz and triangle inequalities (Weyl differencing and the dispersion method).

For the purposes of numerics, it is the interplay between (ii), (iii), and (v) that drives the final conditions (7), (8). The Weil conjectures are the primary source of power savings ({x^{-c}} for some fixed {c>0}) in the argument, but they need to overcome power losses coming from completion of sums, and also each use of Cauchy-Schwarz tends to halve any power savings present in one’s estimates. Naively, one could thus expect to get better estimates by relying more on the Weil conjectures, and less on completion of sums and on Cauchy-Schwarz.

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