Character analogue of the Boole summation formula with applications

TL;DR

This paper develops a character analogue of the Boole summation formula, deriving integral representations for the alternating Dirichlet L-function, evaluating its derivative at zero, and establishing new reciprocity formulas for related arithmetic sums.

Contribution

Introduces a novel character analogue of the Boole summation formula, enabling new integral representations and reciprocity formulas for Dirichlet L-functions and related sums.

Findings

Derived integral representation for the alternating Dirichlet L-function.

Evaluated the derivative of the L-function at s=0.

Proved reciprocity formulas for new arithmetic sums and Hardy--Berndt sums.

Abstract

In this paper, we present the character analogue of the Boole summation formula. Using this formula, an integral representation is derived for the alternating Dirichlet $L-$function and its derivative is evaluated at $s=0$. Some applications of the character analogue of the Boole summation formula and the integral representation are given about the alternating Dirichlet $L-$function. Moreover, the reciprocity formulas for two new arithmetic sums, arising from utilizing the summation formulas, and for Hardy--Berndt sum $S_{p}\left( b,c:\chi\right) $ are proved.