Paper detail

Weighted fourth moments of Hecke zeta functions with groessencharacters

We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |ζ(1/2+it,λ^d)|^4| \sum_{0<|μ|^2\leq M} A(μ)λ^d((μ)) |μ|^{-2it}|^2 {\rm d}t$, where $λ^d$ is the groessencharacter satisfying $λ^d((α)) = λ^d(α{\Bbb Z}[i]) = (α/|α|)^{4d}$, for $0\neqα\in{\Bbb Z}[i]$, and $ζ(s,λ^d)$ is the Hecke zeta function that satisfies $ζ(s,λ^d) =(1/4)\sum_{0\neqα\in{\Bbb Z}[i]} λ^d((α)) |α|^{-2s}$ 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.