Weighted fourth moments of Hecke zeta functions with groessencharacters

Abstract

We use recently obtained bounds for sums of Kloosterman sums to bound the sum ${\sum }_{-D\le d\le D}{\int }_{-D}^D|\zeta (1/2+it,{\lambda }^d){|}^4|{\sum }_{0<|\mu {|}^2\le M}A(\mu ){\lambda }^d((\mu ))|\mu {|}^{-2it}{|}^2\mathrm{d}t$, where ${\lambda }^d$ is the groessencharacter satisfying ${\lambda }^d((\alpha ))={\lambda }^d(\alpha \mathbb{Z}[i])=(\alpha /|\alpha |{)}^{4d}$, for $0\ne \alpha \in \mathbb{Z}[i]$, and $\zeta (s,{\lambda }^d)$ is the Hecke zeta function that satisfies $\zeta (s,{\lambda }^d)=(1/4){\sum }_{0\ne \alpha \in \mathbb{Z}[i]}{\lambda }^d((\alpha ))|\alpha {|}^{-2s}$ for $\Re (s)>1$, while the numbers $D,M\in (0,\infty )$ and function $A:\mathbb{Z}[i]-\{0\}\to \mathbb{C}$ 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.

Changes since version 1: the use of different fonts and numerous minor corrections and improvements of the presentation of subject matter. In particular, the statement (in (1.6)) of the bound of Deshouillers and Iwaniec for S(T,N) has now been corrected

References in corpus (1)

• Spectral large sieve inequalities for Hecke congruence subgroups of SL(2,Z[i])

Cited by in corpus (1)

• Spectral large sieve inequalities for Hecke congruence subgroups of SL(2,Z[i])