[Paper Review] Super-positivity of a family of L-functions

Dorian Goldfeld, Bingrong Huang|arXiv (Cornell University)|Dec 30, 2016

Analytic Number Theory Research11 references3 citations

TL;DR

This paper proves that there are infinitely many modular forms for $SL(2,\mathbb{Z})$ whose associated $L$-functions exhibit super-positivity—meaning all derivatives of the completed $L$-function at $s=1/2$ are non-negative. The proof relies on mollification techniques and moment estimates to show that a positive proportion of $L$-functions in the family satisfy the super-positivity condition, with explicit bounds on the density of such forms.

ABSTRACT

Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value $s = 1/2$ should be non-negative. They proved that the Riemann hypothesis implies super-positivity for self dual cuspidal automorphic L-functions on $GL(n)$. Super-positivity of the Riemann zeta function was established by Pólya in 1927 and since then many other cases have been found by numerical computation. In this paper we prove, for the first time, that there are infinitely many L-functions associated to modular forms for $SL(2, \mathbb{Z})$ each of which has the super-positivity property. Our proof also establishes that all derivatives of the completed L-function at any real point $σ> 1/2$ must be positive.

Motivation & Objective

• To establish the existence of infinitely many self-dual cuspidal automorphic $L$-functions on $GL(2,\mathbb{A}_\mathbb{Q})$ with the super-positivity property.
• To prove that all derivatives of the completed $L$-function at $s=1/2$ are non-negative for these $L$-functions.
• To extend super-positivity to all real $\sigma > 1/2$, showing all derivatives are strictly positive there.
• To provide a constructive method to verify super-positivity for individual $L$-functions using zero-free regions.

Proposed method

• Mollification of $L$-functions using a family of smooth cutoff functions to control the size of the $L$-function and its derivatives.
• Use of the approximate functional equation and integral transforms to relate $L$-function values to averages over spectral families.
• Application of moment estimates for $|L(\sigma + it, f)|^2$ to control the size of the mollified $L$-function and its derivatives.
• Estimation of the density of $L$-functions with no zeros in the critical strip near $\sigma > 1/2$ via zero-density bounds and integral majorization.
• Use of the functional equation and symmetry to relate values at $s=1/2$ to values in the right half-plane, enabling derivative sign analysis.
• Numerical evaluation of integrals involving hyperbolic and trigonometric functions to derive explicit upper bounds on the proportion of $L$-functions violating super-positivity.

Experimental results

Research questions

• RQ1 Are there infinitely many modular $L$-functions for $SL(2,\mathbb{Z})$ such that all derivatives of the completed $L$-function at $s=1/2$ are non-negative?
• RQ2 Can super-positivity be established for $L$-functions at all real $\sigma > 1/2$, not just at $s=1/2$?
• RQ3 What proportion of $L$-functions in the family of holomorphic cusp forms of level 1 and even weight satisfy the super-positivity condition?
• RQ4 Can the super-positivity property be verified for individual $L$-functions using zero-free region estimates and derivative sign analysis?

Key findings

• There are infinitely many holomorphic cusp forms of level 1 and even weight for which all derivatives of the completed $L$-function at $s=1/2$ are non-negative.
• For these $L$-functions, all derivatives of the completed $L$-function at any real $\sigma > 1/2$ are strictly positive.
• The proportion of $L$-functions in the family with no zeros in the region $\sigma \in (1/2,1)$, $|t| \leq \sigma - 1/2$ is at least $27\%$, implying a positive density of super-positive $L$-functions.
• Explicit bounds were computed: $\mathcal{N}_1(K,\Phi)/\mathcal{A}(K,\Phi) \leq 0.19441$, $\mathcal{N}_2(K,\Phi)/\mathcal{A}(K,\Phi) \leq 0.03891$, and $\mathcal{N}_3(K,\Phi)/\mathcal{A}(K,\Phi) \leq 0.00989$, with the tail sum $\sum_{j=14}^{J-1} \mathcal{N}_j(K,\Phi)/\mathcal{A}(K,\Phi) \leq 0.01212$.
• The method confirms that the set of $L$-functions violating super-positivity has density at most $63\%$, so at least $27\%$ are super-positive.