The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series

TL;DR

This paper proves the ratios conjecture for real Dirichlet characters using multiple Dirichlet series, enabling asymptotic calculations of sums involving shifted logarithmic derivatives under GRH.

Contribution

It extends the ratios conjecture to real Dirichlet characters with one shift, using multiple Dirichlet series, and improves the range by including non-primitive characters.

Findings

Proved ratios conjecture for real Dirichlet characters under GRH.

Extended the range of shifts for which the conjecture holds.

Derived asymptotic formulas for sums of shifted logarithmic derivatives.

Abstract

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions, including moments or statistics about the distribution of zeros. We consider the family of real Dirichlet characters, and use multiple Dirichlet series to prove the ratios conjectures with one shift in the numerator and denominator in some range of the shifts. This range can be improved by extending the family to include non-primitive characters. All of the results are conditional under the Generalized Riemann hypothesis. This extended range is good enough to enable us to compute an asymptotic formula for the sum of shifted logarithmic derivatives for some range of the shift.