Weighted fourth moments of Hecke zeta functions with groessencharacters

TL;DR

This paper derives bounds for weighted fourth moments of Hecke zeta functions with Groessencharacters, utilizing recent bounds on Kloosterman sums, with potential applications to Gaussian prime distribution.

Contribution

It introduces new bounds for sums involving Hecke zeta functions and Groessencharacters, especially when the parameter M is small relative to D, advancing understanding of their moments.

Findings

Established bounds for sums of Hecke zeta functions with Groessencharacters.

Connected bounds to potential improvements in Gaussian prime distribution results.

Utilized recent Kloosterman sum bounds to achieve these results.

Abstract

We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |\zeta(1/2+it,\lambda^d)|^4| \sum_{0<|\mu|^2\leq M} A(\mu)\lambda^d((\mu)) |\mu|^{-2it}|^2 {\rm d}t$, where $\lambda^d$ is the groessencharacter satisfying $\lambda^d((\alpha)) = \lambda^d(\alpha{\Bbb Z}[i]) = (\alpha /|\alpha|)^{4d}$, for $0\neq\alpha\in{\Bbb Z}[i]$, and $\zeta(s,\lambda^d)$ is the Hecke zeta function that satisfies $\zeta(s,\lambda^d) =(1/4)\sum_{0\neq\alpha\in{\Bbb Z}[i]} \lambda^d((\alpha)) |\alpha|^{-2s}$ for $\Re(s)>1$, while the numbers $D,M\in(0,\infty)$ and function $A:{\Bbb Z}[i]-\{0\}\rightarrow{\Bbb C}$ are arbitrary (though it is only in respect of cases in which $M$ is relatively small, compared to $D$, that our results are new and interesting). One of our new bounds may have an application in enabling a certain improvement of a result of P.A. Lewis on the distribution of Gaussian primes.