Character analogue of the Boole summation formula with applications

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.