Character analogues of the poisson and Euler-MacLaurin summation formulas with applications

Character versions of the Poisson and Euler-Maclaurin summation formulas are derived. Instead of Bernoulli numbers and polynomials which appear in the classical Euler-Maclaurin formula, generalized Bernoulli numbers and polynomials now appear. Many applications of the results are given. In particular, applications are made to the evaluation of L-functions, the derivation of a character analogue of the Lipschitz summation formula, and the examination of a new character version of the classical gamma function.