Ratios conjecture for primitive quadratic Hecke $L$-functions

TL;DR

This paper extends the ratios conjecture to primitive quadratic Hecke L-functions over imaginary quadratic fields, providing new asymptotic formulas for their central value moments under GRH.

Contribution

It introduces a ratios conjecture with one shift for these L-functions and derives unconditional asymptotic formulas for their first moments.

Findings

Ratios conjecture adapted for primitive quadratic Hecke L-functions

Unconditional asymptotic formulas for first moments obtained

Error terms are square root of main term size

Abstract

We develop the ratios conjecture with one shift in the numerator and denominator in certain ranges for families of primitive quadratic Hecke $L$-functions of imaginary quadratic number fields with class number one using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain unconditional asymptotic formulas for the first moments of central values of these families of $L$-functions with error terms of size that is the square root of that of the primary main terms.