[Paper Review] One level density of low-lying zeros of quadratic and quartic Hecke $L$-functions

Peng Gao, Liangyi Zhao|arXiv (Cornell University)|Aug 5, 2017

Analytic Number Theory Research34 references21 citations

TL;DR

This paper establishes one-level density results for low-lying zeros of quadratic and quartic Hecke $L$-functions over the Gaussian field $ vert\mathbb{Q}(i)$. Using two-dimensional Poisson summation and character sum estimates, it proves that at least 94.27% of quadratic family members and 5% of quartic family members do not vanish at the central point, supporting the Katz-Sarnak density conjecture for these families in the symplectic symmetry regime.

ABSTRACT

In this paper, we prove some one level density results for the low-lying zeros of famliies of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that, respectively, at least $94.27 \%$ and $5\%$ of the members of the quadratic family and the quartic family do not vanish at the central point.

Motivation & Objective

• To extend the Katz-Sarnak density conjecture to families of quadratic and quartic Hecke $L$-functions over the Gaussian field.
• To determine the distribution of low-lying zeros in these families using one-level density analysis.
• To establish non-vanishing results at the central point for a positive proportion of $L$-functions in each family.
• To apply higher-dimensional Poisson summation techniques to improve character sum estimates and extend the support of test functions in the density formula.
• To confirm that the families exhibit symplectic symmetry, consistent with random matrix theory predictions.

Proposed method

• Utilizes the explicit formula to convert sums over zeros into sums over prime ideals in the ring of Gaussian integers $\mathbb{Z}[i]$.
• Applies two-dimensional Poisson summation over $\mathbb{Z}[i]$ to transform and bound character sums involving quartic and quadratic residue symbols.
• Employs weighted sums with smooth cutoff functions and the Mellin transform to handle oscillatory terms in the density trace formula.
• Estimates character sums $S_M(X,Y;\hat{\phi},\Phi)$ via partial summation and bounds on exponential sums over ideals in $\mathbb{Z}[i]$.
• Uses the dual lattice structure from Poisson summation to shorten the length of character sums, enabling better error control.
• Applies the $\tilde{W}$-function and smooth weight functions to localize the sums and control the error terms uniformly in the parameter $X$.

Experimental results

Research questions

• RQ1 What is the one-level density of low-lying zeros for families of quadratic and quartic Hecke $L$-functions over $\mathbb{Q}(i)$?
• RQ2 To what extent do the zero distributions of these $L$-functions align with the predictions of random matrix theory for the symplectic group $\mathrm{USp}$?
• RQ3 What proportion of $L$-functions in these families do not vanish at the central critical point?
• RQ4 How can higher-dimensional Poisson summation improve the support of the Fourier transform of test functions in one-level density results?
• RQ5 Can unconditional results be obtained for the one-level density in these families without assuming the generalized Riemann hypothesis?

Key findings

• The one-level density of low-lying zeros for the quadratic Hecke $L$-function family over $\mathbb{Q}(i)$ converges to the symplectic density kernel $W_{\mathrm{USp}}(x) = 1 - \frac{\sin(2\pi x)}{2\pi x}$ as $X \to \infty$, confirming symplectic symmetry.
• For the quartic Hecke $L$-function family, the one-level density similarly converges to the same symplectic kernel, indicating the same symmetry type.
• At least 94.27% of the members of the quadratic family do not vanish at the central point, derived from the one-level density result.
• At least 5% of the members of the quartic family do not vanish at the central point, as a corollary of the one-level density analysis.
• The support of the Fourier transform of the test function in the one-level density formula is extended to $(-2,2)$ for the quadratic family and to a comparable range for the quartic family, via improved character sum estimates.
• The error terms in the density trace formula are shown to be $o(X \log X)$ when $U = \log \log X$ and $Z = \log^5 X$, ensuring convergence to the expected density.