As in previous posts, we use the following asymptotic notation: is a parameter going off to infinity, and all quantities may depend on unless explicitly declared to be “fixed”. The asymptotic notation is then defined relative to this parameter. A quantity is said to be of polynomial size if one has , and said to be bounded if . Another convenient notation: we write for . Thus for instance the divisor bound asserts that if has polynomial size, then the number of divisors of is .
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 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
where is a primitive Dirichlet character of some conductor , is an integer, and is some quantity between and . Clearly we have the trivial bound
we also have the classical Pólya-Vinogradov inequality
This latter inequality gives improvements over the trivial bound when is much larger than , but not for much smaller than . The Pólya-Vinogradov inequality can be deduced via a little Fourier analysis from the completed exponential sum bound
for any , where . (In fact, from the classical theory of Gauss sums, this exponential sum is equal to for some complex number of norm .)
In the case when 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
to the Pólya-Vinogradov inequality when 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 has odd order.
Another important improvement is the Burgess bound, which in our notation asserts that
for any fixed integer , assuming that 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 as small as .
In the case when is prime, there has been very little improvement to the Burgess bound (or its Fourier dual, which can give bounds for as large as ) in the last fifty years; an improvement to the exponents in (3) in this case (particularly anything that gave a power saving for below ) would in fact be rather significant news in analytic number theory.
However, in the opposite case when is smooth – that is to say, all of its factors are much smaller than – 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:
This proposition is particularly powerful when is smooth, as this gives many factorisations with the ability to specify with a fair amount of accuracy. For instance, if is -smooth (i.e. all prime factors are at most ), then by the greedy algorithm one can find a divisor of with ; if we set , then , and the above proposition then gives
which can improve upon the Burgess bound when is small. For instance, if , then this bound becomes ; in contrast the Burgess bound only gives for this value of (using the optimal choice for ), which is inferior for .
The hypothesis that be squarefree may be relaxed, but for applications to the Polymath8 project, it is only the squarefree moduli that are relevant.
We use the method of Weyl differencing, the key point being to difference in multiples of .
Let , thus . For any , we have
By the Chinese remainder theorem, we may factor
where are primitive characters of conductor respectively. As is periodic of period , we thus have
and so we can take out of the inner summation of the right-hand side of (4) to obtain
and hence by the triangle inequality
Note how the characters on the right-hand side only have period rather than . This reduction in the period is ultimately the source of the saving over the Pólya-Vinogradov inequality.
Note that the inner sum vanishes unless , which is an interval of length by choice of . Thus by Cauchy-Schwarz one has
We expand the right-hand side as
We first consider the diagonal contribution . In this case we use the trivial bound for the inner summation, and we soon see that the total contribution here is .
Now we consider the off-diagonal case; by symmetry we can take . Then the indicator functions restrict to the interval . On the other hand, as a consequence of the Weil conjectures for curves one can show that
for any ; indeed one can use the Chinese remainder theorem and the square-free nature of to reduce to the case when is prime, in which case one can apply (for instance) the original paper of Weil to establish this bound, noting also that and are coprime since is squarefree. Applying the method of completion of sums (or the Parseval formula), this shows that
Summing in (using Lemma 5 from this previous post) we see that the total contribution to the off-diagonal case is
which simplifies to . The claim follows.
A modification of the above argument (using more complicated versions of the Weil conjectures) allows one to replace the summand by more complicated summands such as for some polynomials or rational functions of bounded degree and obeying a suitable non-degeneracy condition (after restricting of course to those for which the arguments are well-defined). We will not detail this here, but instead turn to the question of estimating slightly longer exponential sums, such as
where should be thought of as a little bit larger than .
It will be convenient to introduce a little bit of formal algebraic notation. For the narrow task of establishing a new estimate on an incomplete Kloosterman-type sum of the type used in Zhang’s paper, one does not really need all this formalism, but it may be useful to have this formalism around for future generalisations and strengthenings of this estimate (e.g. when trying to set up a Burgess-style argument).
Define an integral rational function to be a formal rational function in a formal indeterminate where are polynomials with integer coefficients, and is monic; in particular any polynomial can be identified with the integral rational function . For minor technical reasons we do not equate integral rational functions under cancelling, thus for instance we consider to be distinct from ; we need to do this because the domain of definition of these two functions is a little different (the former is not defined when , but the latter can still be defined here). Because we refuse to cancel, we have to be a little careful how we define algebraic operations: specifically, we define
Note that the denominator always remains monic with respect to these operations. This is not quite a ring with a derivation (the subtraction operation does not quite cancel the addition operation due to the inability to cancel) but this will not bother us in practice. (On the other hand, addition and multiplication remain associative, and the latter continues to distribute over the former, and differentiation obeys the usual sum and product rules.)
In applications to bounded gaps between primes, we will deal with integral rational functions of the explicit form
for various integers , but much of the theory below extends without much difficulty to more general integral rational functions, so we shall state it in that level of generality.
Note that if is an integral rational function, we can localise it modulo for any modulus to obtain a rational function that is the ratio of two polynomials in , with the denominator monic and hence non-vanishing. We can define the algebraic operations of addition, subtraction, multiplication, and differentiation on integral rational functions modulo by the same formulae as above, and we observe that these operations are completely compatible with their counterparts over (even without the ability to cancel), thus for instance . We say that is divisible by , and write , if the numerator of has all coefficients divisible by .
If is an integral rational function and , then is well defined as an element of except when is a zero divisor in . We adopt the convention that when is a zero divisor in , thus is really shorthand for ; by abuse of notation we view both as a function on and as a -periodic function on . Thus for instance
Note that if , then for all for which is well defined. We define to be the largest factor of for which ; in particular, if is square-free, we have
We recall the following Chinese remainder theorem:
for all integers , where is the inverse of in and similarly for .
When there is no chance of confusion we will write for (though note that does not qualify as an integral rational function since the constant is not monic).
Proof: See Lemma 7 of this previous post.
We need the following algebraic version of the fundamental theorem of calculus:
Lemma 3 (Fundamental theorem of calculus) Let be a rational function, and let be a positive integer. If and all prime factors of are sufficiently large depending on the degrees of , then there exists such that .
Proof: By definition, the constraint may be rewritten as
By abuse of notation, we view as polynomials over (i.e. all occurrences of in what follows should actually be if we did not abuse notation). If has degree larger than , then by inspecting the coefficient of this equation we obtain a contradiction if all prime factors of are large enough. Thus we may assume that ; since is monic, this implies that for some with . We can then rewrite the previous constraint as
If is non-zero, then inspecting the coefficient of this equation leads to a contradiction if all prime factors of are large enough. Thus is zero, and so as required.
Now we give an estimate for complete exponential sums, which combines both Ramanujan sum bounds with Weil conjecture bounds.
More generally, if we set for some integers , we obtain Weil’s Kloosterman sum bound
and so forth.
The bound in Proposition 4 is quite sharp, with one exception of relevance to the polymath8 project, namely the estimation of the degenerate Kloosterman-type sum
with integers. Proposition 4 gives a bound of in this case, but one can do better by performing the change of variables that transforms this sum into a Ramanujan sum, yielding the superior bound
One could try to modify the argument below to incorporate this improvement, but it leads to a significantly more complicated right-hand side for the estimate, so we will not do so here.
Proof: By the Chinese remainder theorem and the divisor bound, it suffices to verify this when is a single prime and with replaced by , thus we wish to show that
otherwise. We may assume that is sufficiently large depending on the degrees of as the claim is trivial otherwise.
The first bound trivially follows from the triangle inequality, so we turn to the second bound. Since , we conclude from Lemma 3 that there is such that ; since , must be non-zero. Since , another application of Lemma 3 shows that there is such that . Thus whenever is well-defined. Since the denominator of has only bounded number of zeroes, we conclude that for all but a bounded number of . The claim then follows (7) from the Fourier identity
Now we prove (8). By abuse of notation, we view and as polynomials over the field rather than over the integers. Let be the (monic) greatest common divisor, thus and with coprime and of bounded degree with monic. As in the previous paragraph, we have for all but a bounded number of . So it suffices to show that
We now use the Weil bound for exponentials of rational functions over a field of prime order, as stated for instance in this paper of Perelmuter or this paper of Cochrane and Pinner (in the case when is polynomial, one can also use the original paper of Weil, who also treats the important case of Kloosterman sums when is a fractional linear transformation). We remark that the Cochrane-Pinner paper uses Stepanov’s method and is thus essentially elementary in nature, whereas the other papers use more sophisticated tools from arithmetic geometry (such as -adic cohomology). In any event, the results in these papers give the required bound except when the rational function has the form
in the field of rational functions over for some and . (The presence of the factor arises because of elements of are all roots of .)
Suppose that (9) holds. Writing for some coprime polynomials with monic, we conclude on multiplying everything out that
in the polynomial ring . By unique factorisation, this forces to divide , which for large enough forces to have degree zero, thus as is monic. We thus have
Inspecting the coefficient of this equation, we obtain (for large enough) a contradiction unless has degree zero, thus is constant. In particular and so is a scalar multiple of , hence is a scalar multiple of and so , contradicting the hypothesis. This covers all cases and we are done.
Now we obtain a bound for incomplete exponential sums to smooth moduli. We will focus attention here on the specific sum we need to bound for the Polymath8 project, namely a sum of the form
where are square-free numbers (not necessarily coprime), are integers, , and is a function supported on an interval of length that obeys the smoothness bounds
for any fixed . The arguments below also apply to more general exponential sums as these, but the specific sum benefits from the additional cancellation (6) that is not available for more general sums.
Proposition 5 Let be as above. Write and .
The estimate (ii) is a slightly improved version of Lemma 11 of Zhang’s paper (which is a key estimate to treat the Type I/II sums); one can also write the second term on the right-hand side more suggestively as . But it is the estimate (iii) which is a more substantial improvement, as for the medium-sized values of that come up in Zhang’s argument, the first two terms that appear in (12) are superior to the main term in (11). (Admittedly, the final term in (12) is worse than the final term in (11), but in practice these terms are not dominant.) The improved bound (12) leads to significantly better values of and for the assertion that we need for bounded gaps between primes; this is sketched in this previous comment and will be detailed in a subsequent post.
where is the function
for all (which, by abuse of notation, we identify with ) and any fixed .
Consider first the term in (14). By the Chinese remainder theorem, we may factor as
Applying (6) to the first two factors and the trivial bound of to the second factor, we conclude that
and so the contribution of the term is acceptable. It remains to establish that
Let be fixed. For the bound is trivial from (15) (for large enough), so by overspill it will suffice to show that
Writing , we may bound the previous expression by
vanishes at infinity, we see that implies . Thus
and hence by the divisor bound
and the claim follows.
As before, we can rewrite as (14). Now set
so . Shifting by for and averaging, we may then rewrite (14) as
From Lemma 2, we have
and so we may bound by
By Proposition 4 (and the coprimality of ), we have
Thus we may bound the previous expression by
Next, let be a small fixed quantity. By (15) as before, the contribution of those outside of the interval is negligible, so it will suffice to show that
Bounding , we may bound this sum by
The inner sum is only non-vanishing when , and in that case we have a bound of as in the proof of (11); note that
so we may replace by . By the divisor bound we may thus bound (18) by
By Cauchy-Schwarz, we may thus bound the left-hand side of (17) by
where is the quantity
We may square out as
Now observe from Fourier analysis that the sum
equals when and , and vanishes otherwise. Thus we may simplify the above expression as
Substituting , this becomes
As is supported in an interval of length , we obtain the bound
We first isolate the diagonal terms . In this case the inner sum is a Ramanujan sum, so by (5) the diagonal contribution is bounded by
Summing in using Lemma 5 from this previous post, this is bounded by
Now we handle the non-diagonal terms . By Proposition 4, this contribution to is
we can bound the preceding expression by
The inner sum vanishes unless . In that case, is restricted to a single residue class modulo , and so the inner sum is . From the divisor bound we conclude that
and thus this contribution to is bounded by
From the explicit formula (13) we see that if , then if and only if
For large enough, this only happens if or . We conclude that
and thus this contribution to is
If we estimate
This contribution to is thus