On the reciprocity law for the twisted second moment of Dirichlet $L$-functions

TL;DR

This paper extends the reciprocity law for the twisted second moment of Dirichlet L-functions, providing continuous error terms, exact formulas involving shifted moments, and connections to continued fractions and the Estermann function.

Contribution

It introduces a continuous extension of the reciprocity law's error term, derives exact formulas with shifted moments, and links these to continued fractions and the Estermann function.

Findings

Error term extended to a continuous function of a/q

Derived exact formulas involving shifted moments

Connected shifted moments with the Estermann function

Abstract

We investigate the reciprocity law, studied by Conrey~\cite{Con07} and Young~\cite{You11a}, for the second moment of Dirichlet L-functions twisted by $\chi(a)$ modulo a prime $q$. We show that the error term in this reciprocity law can be extended to a continuous function of $a/q$ with respect to the real topology. Furthermore, we extend this reciprocity result, proving an exact formula involving also shifted moments. We also give an expression for the twisted second moment involving the coefficients of the continued fraction expansion of $a/q$, and, consequently, we improve upon a classical result of Selberg on the second moment of Dirichlet L-functions with two twists. Finally, we obtain a formula connecting the shifted second moment of the Dirichlet $L$-functions with the Estermann function. In particular cases, this result can be used to obtain some simple explicit exact formulae for the moments.