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Determining optimal test functions for 2-level densities

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Abstract

Katz and Sarnak conjectured a correspondence between the n-level density statistics of zeros from families of L-functions with eigenvalues from random matrix ensembles. In many cases the sums of smooth test functions, whose Fourier transforms are finitely supported, over scaled zeros in a family converge to an integral of the test function against a density \(W_{n, G}\) depending on the symmetry G of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of L-functions. We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger n and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when \(n=2\), minimizing

$$\begin{aligned} \frac{1}{\Phi (0, 0)} \int _{{{\mathbb {R}}}^2} W_{2,G} (x, y) \Phi (x, y) dx dy \end{aligned}$$

over test functions \(\Phi : {{\mathbb {R}}}^2 \rightarrow [0, \infty )\) with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form \(\phi (x) \psi (y)\) for some fixed admissible \(\psi (y)\) and \({\text {supp}}{{\widehat{\phi }}} \subseteq [-1, 1]\). Extending results from the 1-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal \(\phi \) for appropriately chosen fixed test function \(\psi \). The solution allows us to deduce strong estimates for the proportion of newforms of rank 0 or 2 in the case of \(\mathsf {SO(even)}\), rank 1 or 3 in the case of \(\mathsf {SO(odd)}\), and rank at most 2 for \({\textsf{O}}\), \(\textsf{Sp}\), and \({\textsf{U}}\); our estimates are a significant strengthening of the best known estimates obtained with the 1-level density. As a representative example, the previous best 1-level analysis yields a lower bound of 0.7839 for vanishing to order at most 2 for the \(\mathsf {SO(even)}\) family of cuspidal newforms, and our 2-level work improves this to 0.952694. We conclude by discussing further improvements on estimates by the method of iteration.

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Notes

  1. We only need the test function to be non-negative if we wish to obtain bounds on the order of vanishing. As the zeros are symmetrically distributed on the critical line, there is no loss in taking \(\Phi \) to be even.

  2. stellali110204@gmail.com, Emma Willard School.

  3. Often the analysis leads to bounds on the order of vanishing that exceed 100%, and of course there are easier ways to get such estimates!

  4. We are optimistic that such larger support will be known soon, see [8].

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Authors and Affiliations

  1. Department of Mathematics, University of California Berkeley, Berkeley, CA, 94720, USA

    Elżbieta Bołdyriew, Fangu Chen & Jason Zhao

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Charles Devlin VI

  3. Department of Mathematics and Statistics, Williams College, Williamstown, MA, 01267, USA

    Steven J. Miller

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  1. Elżbieta Bołdyriew
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  2. Fangu Chen
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  4. Steven J. Miller
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This research was supported by NSF Grant DMS1947438 and Williams College. The fourth listed author was supported by NSF Grant DMS1561945. We also thank John Haviland, Fernando Trejos Suárez and Jiahui Yu for their comments on the problem during our many fruitful conversations, as well as the referee for their helpful suggestions. No data sets were used in this analysis; however, some Mathematica code was written to estimate certain integrals, and is available upon request by emailing S. J. Miller

Appendix A: Bounds from the fourth moment

Appendix A: Bounds from the fourth moment

By Jiahui (Stella) LiFootnote 2 and Steven J. Miller

As remarked, there are several avenues for future research to obtain better estimates on the order of vanishing. One of course is to continue to try to improve the test functions, though this paper and earlier work show that simple choices are close to the optimal results. We may also study higher level densities (or equivalently centered moments). This paper is the first along this line, and concentrates on the 2-level density. We can of course continue and consider n-level densities and moments, and in those calculations important ingredients are good choices for fixed test functions, so that once again we can reduce the complexity of the optimization problem. Thus our results here can be fed in to higher n. There is a balancing act here; the larger n give better bounds as the rank grow, but worse results for small ranksFootnote 3. Thus it is important to pursue both avenues, namely optimizing the test function and exploring larger results coming from larger n.

Calculations for bounds from higher level density are computationally difficult as they include multiple n-dimensional determinant integrals. A closely related statistic, the \(n{\textrm{th}}\) centered moment, was used instead by Hughes and Miller [36]. The \(n{\textrm{th}}\) centered moment replaces the \(n \times n\) determinant expansions of the n-level density with a one-dimensional integral through a clever change of variable that reduces a n-dimensional integral involving Bessel functions and n test functions to a related one-dimensional integral with just one Bessel function against a new test function.

Below we report on some work in progress that uses the results of this paper; see [47] for details. As a first step, the results of [36] are generalized to allow the test function to be a product of n distinct even Schwartz test functions, which is essential in order to be able to use the results of this paper as fixed test functions for some of the choices, leaving the other test functions free to be varied. Comparing the results with the bounds obtained using the 1- and 2-level densities, we see significant improvements for bounds on modest rank, but worse results for bounds on small rank. Thus both methods, increasing n and optimizing the test functions, have their utility. Below we report on ranks where the fourth moment bounds are better.

When calculating bounds using the 4th centered moment, if we assume calculations hold for greater support, we find that the combination of a pair of the optimal 2-level test function found in this paper and a pair of the naive test function yields better results than just 4 naive test functions for higher ranks of SO(even) functionsFootnote 4 The calculated bounds are illustrated in the tables below. Table 2 shows an application where the test functions found in this paper yield better results than the naive functions, while Tables 3 and 4 show the significant improvement in upper bounds for modest ranks.

Table 2 Upper bounds for vanishing to order at least r for SO(even) from various approaches
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Table 3 Upper bounds for vanishing to order at least r for SO(even)
Full size table
Table 4 Upper bounds of vanishing to order at least r for SO(odd)
Full size table

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Bołdyriew, E., Chen, F., Devlin, C. et al. Determining optimal test functions for 2-level densities. Res. number theory 9, 32 (2023). https://doi.org/10.1007/s40993-022-00367-0

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