A conjecture for the regularized fourth moment of Eisenstein series

TL;DR

This paper formulates a new conjecture for the regularized fourth moment of Eisenstein series using Zagier's inner product and proves an asymptotic formula linking it to L-functions, advancing understanding of Eisenstein series moments.

Contribution

It introduces a novel conjecture based on Zagier's regularized inner product and derives an asymptotic formula connecting the fourth moment to L-functions.

Findings

Established a new conjecture for the regularized fourth moment.

Derived an asymptotic formula expressing the moment as a mean value of L-functions.

Improved upon previous approaches by avoiding truncated Eisenstein series.

Abstract

We formulate a version of the Random Wave Conjecture for the fourth moment of Eisenstein series which is based on Zagier's regularized inner product. We prove an asymptotic formula expressing the regularized fourth moment as a mean value of L-functions. This is an advantage over previous work in the literature, which has approached the fourth moment problem through truncated Eisenstein series and not yielded a suitable expression in terms of L-functions.