Surpassing the Ratios Conjecture in the 1-level density of Dirichlet -functions
We study the -level density of low-lying zeros of Dirichlet -functions in the family of all characters modulo , with . For test functions whose Fourier transform is supported in , we calculate this quantity beyond the square-root cancellation expansion arising from the -function Ratios Conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a new lower-order term which is not predicted by this powerful conjecture. This is the first family where the 1-level density is determined well enough to see a term which is not predicted by the Ratios Conjecture, and proves that the exponent of the error term in the Ratios Conjecture is best possible. We also give more precise results when the support of the Fourier Transform of the test function is restricted to the interval . Finally we show how natural conjectures on the distribution of primes in arithmetic progressions allow one to extend the support. The most powerful conjecture is Montgomery’s, which implies that the Ratios Conjecture’s prediction holds for any finite support up to an error .
Key words and phrases:Dirichlet -functions, low-lying zeros, primes in progressions, random matrix theory, ratios conjecture
2010 Mathematics Subject Classification:11M26, 11M50, 11N13 (primary), 11N56, 15B52 (secondary)
- 1 Introduction
- 2 Background and New Results
- 3 The Explicit Formula and Needed Sums
- 4 Unconditional Results (Theorems 2.1 and 2.3)
- 5 Results Under GRH (Theorems 1.2 and 2.6)
- 6 Results under De-averaging Hypothesis (Theorem 2.8)
- 7 Results under Montgomery’s Hypothesis (Theorem 2.13)
In this paper we study the 1-level density of Dirichlet -functions with modulus . The goal is to compute this statistic for large support and small error terms, providing a test of the predictions of the lower order and error terms in the -function Ratios Conjecture. In this introduction we assume the reader is familiar with low-lying zeros of families of -functions and the Ratios Conjecture, and briefly describe our results. For completeness we provide a brief review of the subject in §2.1, and state our results in full in §2.2 to §2.4.
We let be an even real function such that is and has compact support. Denoting by the non-trivial zeros of (i.e., ) and choosing a scaling parameter close to , the 1-level density is111Since is , we have that for large , hence the sum over the zeros is absolutely convergent. While most of the literature uses as the test function, since we will use Euler’s totient function extensively we use .
throughout this paper a sum over always means a sum over all characters, including the principal character. If we assume GRH then the are real. As is defined for complex values of , it makes sense to consider (1.1) for complex , in case GRH is false (in other words, GRH is only needed to interpret the 1-level density as a spacing statistic arising from an ordered sequence of real numbers, allowing for a spectral interpretation). We also study the average of (1.1) over the moduli , which is easier to understand in general:
The powerful Ratios Conjecture of Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] yields a formula for which is believed to hold up to an error . While there have been several papers [CS1, CS2, DHP, GJMMNPP, HMM, Mil3, Mil4, MilMo] showing agreement between various statistics involving -functions and the Ratios Conjecture’s predictions, evidence for this precise exponent in the error term is limited; the reason this exponent was chosen is the “philosophy of square-root cancelation”. While some of the families studied have 1-level densities that agree beyond square-root cancelation, it is always for small support (). Further, in no family studied were non-zero lower order terms beyond square-root cancelation isolated in the -level density.
The motivation of this paper was to resolve these issues. As the ratios conjecture is used in a variety of problems, it is important to test its predictions in the greatest possible window. Our key findings are the following.
We uncover new, non-zero lower-order terms in the -level density for our families of Dirichlet characters. These terms are beyond what the Ratios Conjecture can predict, and suggest the possibility that a refinement may be possible and needed.
We show (unconditionally) that the natural limit of accuracy of the -function Ratios Conjecture is . Thus the error term cannot be improved for a general family of -functions, though of course its veracity for all families is still open.
The existence of lower-order terms beyond the Ratios Conjecture’s prediction in statistics of -functions is not without precedent. Indeed such terms have been isolated in the second moment of by Heath-Brown [HB], and for a more general shifted sum by Conrey [C].
Before stating our main result, we give the Ratios Conjecture’s prediction. This prediction is done for a slightly different family in [GJMMNPP], but it is trivial to convert from their formulation to the one below (we discuss the conversion in §2.2).
Conjecture 1.1 (Ratios Conjecture).
The 1-level density (from (1.1) with scaling parameter ) equals
The 1-level density (from rescaling222 To rescale we multiply (1.3) by , replace with and average over . The term averages to , explaining the “additional” term . Moreover the average of over this range is easily shown to be . (1.3)) equals
Surprisingly, our techniques are capable of not only verifying this prediction, but we are able to determine the 1-level density beyond what even the Ratios Conjecture predicts. In Theorem 1.2 we obtain a new (arithmetical) term of order , which is not predicted by the Ratios Conjecture.
Assume GRH. If the Fourier Transform of the test function is supported in , then equals
We can give a more precise formula for the term : see Remark 2.5. While Theorem 1.2 is conditional on GRH, in Theorem 2.1 we prove a more precise and unconditional result for test functions whose Fourier transform has support contained in .
The first two terms in (1.5) agree with the Ratios Conjecture’s Prediction. As for the term , its presence confirms that the error term in the Ratios Conjecture is best possible, and suggests more generally that the -level density of a family ought to contain a (possibly oscillating) arithmetical term of order , a statement which should be tested in other families. Interestingly this new term contains the factors and , and is zero when is supported in . In this case we give a more precise estimate for the -level density in Theorem 2.1, in which a lower-order term of order appears, where . This term is a genuine lower-order term, and shows that for such test functions the Ratios Conjecture’s prediction is not best possible. We thus show that a transition happens when is near . Indeed looking at the difference between the -level density and the Ratios Conjecture’s prediction, that is defining
our results imply that333For , this holds for test functions for which either or (see Theorem 1.2); see Theorem 2.1 if . If vanishes in a small interval around , then Theorem 2.6 gives the correct answer. , where
We conjecture that should equal for all , and that our new lower-order term should persist in this extended range.
Theorem 1.2 holds for test functions whose Fourier transform has arbitrarily large finite support .
In Figure 1, the solid curve represents our results (Theorems 1.2 and 2.1), and the dashed line represents Conjecture 1.3; note the resemblance between this graph and the one appearing in Montgomery’s pair correlation conjecture [Mon2]. We prove in Theorem 2.13 that Montgomery’s Conjecture on primes in arithmetic progressions implies that for all .
We believe that this phenomenon is universal and should also happen in different families, in the sense that we believe that the Ratios Conjecture’s prediction should be best possible for , and should not be for . For example, in [Mil4] it is shown that if the Fourier transform of the involved test function is supported in , then the Ratios Conjecture’s prediction is not best possible and one can improve the remainder term; however, in this region of limited support there are no new, non-zero lower order terms unpredicted by the Ratios Conjecture. These results confirm the exceptional nature of the transition point , as is the case in Montgomery’s Pair Correlation Conjecture [Mon2]. Indeed if this last conjecture were known to hold beyond the point , then this would imply the non-existence of Landau-Siegel zeros [CI].
Our plan of attack is to use the explicit formula to turn the -level density into an average of the various terms appearing in this formula. The bulk of the work is devoted to carefully estimating the contribution of the prime sum, which when summing over becomes a sum over primes in the residue class , averaged over . Accordingly, the proof of Theorem 1.2 is based on ideas used in the recent results of the first named author [Fi1], which improve on results of Fouvry [Fo], Bombieri, Friedlander and Iwaniec [BFI], Friedlander and Granville [FG2] and Friedlander, Granville, Hildebrandt and Maier [FGHM]. Theorem 1.1 of [Fi1] cannot be applied directly here, since this estimate is only valid when looking at primes up to modulo with , where is not too close to . Additional estimates are needed, including a careful analysis of the range , which required a combination of divisor switching techniques and precise estimates on the mean value of smoothed sums of the reciprocal of Euler’s totient function. Additionally, in our analysis of the -level density after using the explicit formula and executing the sum over the family we obtain a sum over primes in the arithmetic progressions ; this is one of the cases in which one obtains an asymptotic in Theorem 1.1 of [Fi1], which explains the occurrence of the lower-order term in Theorem 1.2.
The paper is organized as follows. In §2.1 we review previous results on low-lying zeros in families of -functions and describe the motivation for the Ratios Conjecture. See for example [GJMMNPP, Mil4] for a detailed description of how to apply the Ratios Conjecture to predict the 1-level density. We describe our unconditional results in §2.2, and then improve our results in §2.3 by assuming GRH. In previous families there often is a natural barrier, and extending the support is related to standard conjectures (for example, in [ILS] the authors show how cancelation in exponential sums involving square-roots of primes leads to larger support for families of cuspidal newforms). A similar phenomenon surfaces here, where in §2.4 we show that increasing the support beyond is related to conjectures on the distribution of primes in residue classes. We analyze the increase in support provided by various conjectures. These range from a conjecture on the variance of primes in the residue classes, which allow us to reach , to Montgomery’s conjecture for a fixed residue, which gives us any finite support. The next sections contain the details of the proof; we state the explicit formula and prove some needed sums in §3, and then prove our theorems in the remaining sections.
2. Background and New Results
2.1. Background and Previous Results
Assuming GRH, the non-trivial zeros of any nice -function lie on the critical line, and therefore it is possible to investigate statistics of its normalized zeros. These zeros are fundamental in many problems, ranging from the distribution of primes in congruence classes to the class number [CI, Go, GZ, RubSa]. Numerical and theoretical evidence [Hej, Mon2, Od1, Od2, RS] support a universality in behavior of zeros of an individual automorphic -function high above the central point, specifically that they are well-modeled by ensembles of random matrices (see [FM, Ha] for histories of the emergence of random matrix theory in number theory). The story is different for the low-lying zeros, the zeros near the central point. A convenient way to study these zeros is via the 1-level density, which we now describe. Let be an even real function whose Fourier transform
is and has compact support. Let be a (finite) family of -functions satisfying GRH.444We often do not need GRH for the analysis, but only to interpret the results. If the GRH is true, the zeros lie on the critical line and can be ordered, which suggests the possibility of a spectral interpretation. The -level density associated to is defined by
where runs through the non-trivial zeros of . Here is the “analytic conductor” of , and gives the natural scale for the low zeros. As decays, only low-lying zeros (i.e., zeros within a distance of the central point ) contribute significantly. Thus the -level density can help identify the symmetry type of the family. To evaluate (2.2), one applies the explicit formula, converting sums over zeros to sums over primes.
Based in part on the function-field analysis where is the monodromy group associated to the family , Katz and Sarnak conjectured that for each reasonable irreducible family of -functions there is an associated symmetry group (one of the following five: unitary , symplectic USp, orthogonal O, SO(even), SO(odd)), and that the distribution of critical zeros near mirrors the distribution of eigenvalues near . The five groups have distinguishable -level densities. To date, for suitably restricted test functions the statistics of zeros of many natural families of -functions have been shown to agree with statistics of eigenvalues of matrices from the classical compact groups, including Dirichlet -functions, elliptic curves, cuspidal newforms, Maass forms, number field -functions, and symmetric powers of automorphic representations [AM, AAILMZ, DM1, FI, Gao, Gü, HM, HR, ILS, KaSa1, KaSa2, Mil1, MilPe, RR, Ro, Rub, ShTe, Ya, Yo2], to name a few, as well as non-simple families formed by Rankin-Selberg convolution [DM2].
In addition to predicting the main term (see for example [Con, KaSa1, KaSa2, KeSn1, KeSn2, KeSn3]), techniques from random matrix theory have led to models that capture the lower order terms in their full arithmetic glory for many families of -functions (see for example the moment conjectures of [CFKRS] or the hybrid model in [GHK]). Since the main terms agree with either unitary, symplectic or orthogonal symmetry, it is only in the lower order terms that we can break this universality and see the arithmetic of the family enter. These are therefore natural and important objects to study, and can be isolated in many families [HKS, Mil2, Yo1]. We thus require a theory that is capable of making detailed predictions. Recently the -function Ratios Conjecture [CFZ1, CFZ2] has had great success in determining lower order terms. Though a proof of the Ratios Conjecture for arbitrary support is well beyond the reach of current methods, it is an indispensable tool in current investigations as it allows us to easily write down the predicted answer to a remarkable level of precision, which we try to prove in as great a generality as possible.
To study the 1-level density, it suffices to obtain good estimates for
(In the current paper, the parameter plays the role of .) Asymptotic formulas for have been conjectured for a variety of families (see [CFZ1, CS1, CS2, GJMMNPP, HMM, Mil3, Mil4, MilMo]) and are believed to hold up to errors of size for any . The evidence for the correctness of this error term is limited to test functions with small support (frequently significantly less than ), though in such regimes many of the above papers verify this prediction. Many of the steps in the Ratios Conjecture’s recipe lead to the addition or omission of terms as large as those being considered, and thus there was uncertainty as to whether or not the resulting predictions should be accurate to square-root cancelation. The results of the current paper can be seen as a confirmation that this is the right error term for the final predicted answer, at least in this family. Further, the novelty in our results resides in the fact that we are able to go beyond square-root cancelation and we find a smaller term which is unpredicted by the Ratios Conjecture (see Theorem 1.2). For a precise explanation on how to derive the Ratios Conjecture’s prediction in our family, we refer the reader to [GJMMNPP], and also recommend [CS1] for an accessible overview of the Ratios Conjecture.
2.2. Unconditional Results
We now describe our unconditional results. We remind the reader that is a real even function such that is and has compact support.
Suppose that the Fourier transform of the test function is supported on the interval , so . There exists an absolute positive constant (coming from the Prime Number Theorem) such that the 1-level density (from (1.1) with scaling parameter ) equals
The average over of the fourth term in (2.1) can be shown to be , and is therefore negligible when considering (see (3.16)). However, the term involving the second integral in (2.1) is of size , and thus constitutes a genuine lower-order term, smaller than the error term in (1.3) predicted using the Ratios Conjecture.
Theorems 1.2 and 2.1 should be compared to the main result of Goes, Jackson, Miller, Montague, Ninsuwan, Peckner and Pham [GJMMNPP], where they show one can extend the support of to and still get the main term, as well as the lower order terms down to a power savings. They only consider prime, and thus the sum over primes dividing below in Theorem 2.3 is absorbed by their error term. We briefly discuss how one can easily extend their results to the case of general . First note that and have the same zeros in the critical strip if is the primitive character of conductor inducing the non-principal character of conductor . We now have , which can be converted to a sum over primes dividing by the same arguments as in the proof of Proposition 3.1. The rest of the expansion follows from expanding the digamma function in the integral in Theorem 1.3 of [GJMMNPP] and then standard algebra (along the lines of the computations in §3). We use Lemma 12.14 of [MonVa2], which in our notation says that for we have
and the identity
with the Euler-Mascheroni constant. We then extend to by rescaling the zeros by and not and summing over the family; recall the technical issues involved in the rescaling are discussed in Footnote 2.
Theorem 2.3 (Goes, Jackson, Miller, Montague, Ninsuwan, Peckner, Pham [Gjmmnpp]).
If , then the 1-level density (from (1.1) with scaling parameter ) equals
and this agrees with the Ratios Conjecture.
2.3. Results under GRH
We first mention a more precise version of Theorem 1.2.
If in addition to the hypotheses of Theorem 1.2 we assume that the Fourier transform of the test function is times continuously differentiable, then we can give a more precise expression for the term appearing in (1.5):
where the are constants depending (linearly) on the Taylor coefficients of at . In fact, is a truncated linear functional, which composed with the Fourier Transform operator is supported on (in the sense of distributions).
Our next result is an extension of Theorem 1.2, in the case where vanishes in a small interval to the right of .
If is supported in for some (if , then we have the full interval ), then we have that equals
Unless and , the third term of (2.10) goes in the error term.
2.4. Results beyond GRH
As the GRH is insufficient to compute the 1-level density for test functions supported beyond , we explore the consequences of other standard conjectures in number theory involving the distribution of primes among residue classes. Before stating these conjectures, we first set the notation. Let
If we assume GRH, we have that
Our first result uses GRH and the following de-averaging hypothesis, which depends on a parameter .
This hypothesis is trivially true for , and while it is unlikely to be true for , it is reasonable to expect it to hold for any . What we need is some control over biases of primes congruent to . For the residue class , is the variance; the above conjecture can be interpreted as bounding in terms of the average variance.555Note that we only need this de-averaging hypothesis for the special residue class .
Under these hypotheses, we show how to extend the support to a wider but still limited range.
Assume GRH and Hypothesis 2.7 for some . The average 1-level density equals
which is asymptotic to provided the support of is contained in .
In Theorem 2.8 we study the weighted 1-level density
which is technically easier to study than the unweighted version
This is similar to many other families of -functions, such as cuspidal newforms [ILS, MilMo] and Maass forms [AAILMZ, AM], where the introduction of weights (arising from the Petersson and Kuznetsov trace formulas) facilitates evaluating the arithmetical terms.
Finally, we show how we can determine the 1-level density for arbitrary finite support, under a hypothesis of Montgomery [Mon1].
Hypothesis 2.10 (Montgomery).
For any such that and , we have
It is by gaining some savings in in the error that we can increase the support for families of Dirichlet -functions. The following weaker version of Montgomery’s Conjecture, which depends on a parameter , also suffices to increase the support beyond .
For any such that and , we have
Fix . We have for that
Note that this is a weighted version of ; that is, we added the weight . The reason for this is that it makes the count smoother, and this makes it easier to analyze in general since the Mellin transform of in the interval is decaying faster in vertical strips than that of .
Amongst the last three hypotheses, Hypothesis 2.12 is the weakest, but it is still sufficient to derive the asymptotic in the -level density for test functions with arbitrary large support.
For whose Fourier Transform has arbitrarily large (but finite) support, we have the following:
Under GRH, the left hand side of (2.20) is . Therefore, if we win by more than a logarithm over GRH, then we have the expected asymptotic for the 1-level density for of arbitrarily large finite support.
We derive the explicit formula for the families of Dirichlet characters in §3, as well as some useful estimates for some of the resulting sums. We give the unconditional results in §4, Theorems 2.1 and 2.3. The proofs of Theorems 1.2 and 2.6 are conditional on GRH, and use results of [FG2] and [Fi1]; we give them in §5. We conclude with an analysis of the consequences of the hypotheses on the distribution of primes in residue classes, using the de-averaging hypothesis to prove Theorem 2.8 in §6 and Montgomery’s hypothesis to prove Theorem 2.13 in §7.
3. The Explicit Formula and Needed Sums
Our starting point for investigating the behavior of low-lying zeros is the explicit formula, which relates sums over zeros to sums over primes. We follow the derivation in [MonVa2] (see also [ILS, RS], and [Da, IK] for all needed results about Dirichlet -functions). We first derive the expansion for Dirichlet characters with fixed conductor , and then extend to . We conclude with some technical estimates that will be of use in proving Theorem 1.2. Here and throughout, we will set . Note that is real and even, and thus so is the case for , and moreover we have .
3.1. The Explicit Formula for fixed
Proposition 3.1 (Explicit Formula for the Family of Dirichlet Characters Modulo ).
Let be an even, twice differentiable test function with compact support. Denote the non-trivial zeros of by . Then the 1-level density equals
We start with Weil’s explicit formula for , with a non-principal character (we add the contribution from the principal character later). We can replace by (where is the primitive character of conductor inducing ), since these have the same non-trivial zeros. Taking in Theorem 12.13 of [MonVa2] (whose conditions are satisfied by our restrictions on ), we find , and
where for the half of the characters with and for the half with . Making the substitution in the integral and summing over , we find
To get (3.1) from (3.1) we added zero by writing as . Summing over all gives if and otherwise; as our sum omits the principal character, the sum of over the non-principal characters yields the sum on the third line above. We also replaced by in the first term, hence the .
We use Proposition 3.3 of [FiMa] for the first term (which involves the sum over the conductor of the inducing character). We then use the duplication formula of the digamma function to simplify the next two terms, namely . As (equation 6.3.3 of [AS]) and (equation 6.3.8 of [AS]), setting yields . We keep the next two terms as they are, and then apply Proposition 3.4 of [FiMa] (with ) for the last term, obtaining that it equals
Writing , this term is zero unless . If , then it is zero unless , where is the largest such that . Therefore this term equals