Applications of the L-functions ratios conjectures

TL;DR

This paper explores the applications of the ratios conjectures for L-functions, providing new insights into zero statistics, mollified moments, and averages over zeros, simplifying complex number theory problems.

Contribution

It demonstrates how ratios conjectures can be applied to derive results in zero statistics, mollified moments, and averages over zeros of L-functions, advancing theoretical understanding.

Findings

Derived lower order terms in zero statistics of L-functions

Computed mollified moments using ratios conjectures

Analyzed discrete averages over zeros of the Riemann zeta function

Abstract

In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted L-functions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of L-functions, mollified moments of L-functions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult computations.