Character analogue of the Boole summation formula with applications

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.

References in corpus (2)

• On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials
• Character analogues of certain Hardy-Berndt sums

Cited by in corpus (4)

• Reciprocity formulas for Hall-Wilson-Zagier type Hardy-Berndt sums
• Transformation formulas of a character analogue of $\log {\theta }_2(z)$
• Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives
• Asymptotics of generalized partial theta functions with a Dirichlet character