Mean values of L-functions and symmetry

TL;DR

This paper explores how the symmetry group associated with families of L-functions influences their mean values and mollified mean-squares, extending the symmetry paradigm to these statistical properties.

Contribution

It provides a comprehensive description of mean values of L-functions using recent work, linking them to symmetry groups and special functions like Barnes-Vignéras Gamma_2.

Findings

Mean values are governed by the symmetry group.

Connection established with Barnes-Vignéras Gamma_2-function.

Evidence supports symmetry group's influence on L-function statistics.

Abstract

Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L-functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L-functions. We consider the mean-values of the L-functions and the mollified mean-square of the L-functions and find evidence that these are also governed by the symmetry group. We use recent work of Keating and Snaith to give a complete description of these mean values. We find a connection to the Barnes-Vign\'eras $\Gamma_2$-function and to a family of self-similar functions.