First Moment of Quadratic Hecke $L$-Functions with Lower Order Term

TL;DR

This paper computes the average value of quadratic Hecke L-functions over Gaussian integers, revealing detailed asymptotic behavior including secondary terms under key hypotheses.

Contribution

It introduces a novel application of double Dirichlet series to evaluate the first moment with secondary main terms under RH and Lindel"of hypotheses.

Findings

Derived asymptotic formulas with secondary main terms.

Achieved error terms significantly smaller than the main term.

Validated the method under RH and Lindel"of hypotheses.

Abstract

We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindel\"of hypothesis. We obtain asymptotic formulas with secondary main terms and error terms of size that is one quarter of that of the main term.