Ratios conjecture for quadratic Hecke $L$-functions in the Gaussian field

TL;DR

This paper extends the ratios conjecture to quadratic Hecke L-functions in the Gaussian field and derives an asymptotic formula for their central value moments under GRH.

Contribution

It develops the ratios conjecture with one shift for quadratic Hecke L-functions and provides an asymptotic formula for their first moment.

Findings

Ratios conjecture adapted for quadratic Hecke L-functions in Gaussian field.

Asymptotic formula for the first moment with error term O(X^{1/2+ε}).

Results obtained under the generalized Riemann hypothesis.

Abstract

We develope the $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic Hecke $L$-functions in the Gaussian field using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain an asymptotical formula for the first moment of central values of the same family of $L$-functions, obtaining an error term of size $O(X^{1/2+\varepsilon})$.