Twisted first moment of central values of primitive quadratic Dirichlet $L$-functions

TL;DR

This paper derives an asymptotic formula for the twisted first moment of central values of primitive quadratic Dirichlet L-functions, using double Dirichlet series and recursive methods, with a notably small error term.

Contribution

It introduces a novel approach combining double Dirichlet series and recursion to evaluate the twisted first moment with improved error bounds.

Findings

Established an asymptotic formula for the twisted first moment.

Achieved an error term with size as the square root of the main term.

Demonstrated the effectiveness of double Dirichlet series in this context.

Abstract

We evaluate the twisted first moment of central values of the family of primitive quadratic Dirichlet $L$-functions using the method of double Dirichlet series together with a recursive argument. Our main result is an asymptotic formula with an error term of size that is the square root of that of the main term.