Moments and Non-vanishing of central values of Quadratic Hecke $L$-functions in the Gaussian Field

Abstract

We evaluate the first three moments of central values of a family of qudratic Hecke L-functions in the Gaussian field with power saving error terms. In particular, we obtain asymptotic formulas for the first two moments with error terms of size O(X 1/2+ε ). We also study the first and second mollified moments of the same family of L-functions to show that at least 87.5% of the members of this family have non-vanishing central values.

Real zeros of quadratic Hecke $L$-functions

“…Quadratic residue symbol and Gauss sum. It is well-known that K = Q(i) has class number one and that every ideal in O K co-prime to 2 has a unique generator which is ≡ 1 mod (1 + i) 3 . Such a generator is called primary.…”

“…Also, let P (x) be a polynomial satisfying P (0) = P ′ (0) = 0 and P (b) = 1, P ′ (b) = 0. We define a sequence {λ(n)} n∈OK such that when n ≡ 1 mod (1 + i) 3 and N (n) ≤ M , we have…”

“…We now move the line of integration in (5.7) to ℜs = − 1 4 + ǫ to encounter simple poles at s = ±τ , ±δ in the process. We note the following convexity bounds for ζ K (s) from [9, Exercise 3, p. 100]: 3 on the new line. Moreover, we have…”

“…r,s≡1 mod (1+i) 3 rs=l λ(r)λ(s) N (r) Using the fact that λ is supported on square-free elements in O K , we now write r = αa, s = αb with α, a, b being primary, square-free and (a, b) = 1. This implies that l = α 2 ab, l 1 = ab, and l 2 = α so that the sum over l in (6.8) can be written as…”

“…For this, we take our parameters exactly as those used by Conrey and Soundararajan in their proof of [2, Theorem 1]. Precisely, we take κ = 10 −10 , M = X 3 . Let σ 0 be given as in Proposition 3.2 and let Φ to be a smooth function supported in (1, 2) satisfying 0 ≤ Φ(t) ≤ 1 for all t, Φ(t) = 1 for t ∈ (1 + ǫ, 2 − ǫ), and |Φ (ν) (t)| ≪ ν,ε 1.…”