Character analogue of the Boole summation formula with applications

Authors: MÜMÜN CAN, MUHAMMET CİHAT DAĞLI

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

Keywords: Boole summation formula, Dirichlet $L$-function, Hardy--Berndt sum, Bernoulli and Euler polynomials,

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