On Dirichlet L-functions with periodic coefficients and Eisenstein series

We prove a Lipschitz type summation formula with periodic coefficients. Using this formula, representations of the values at positive integers of Dirichlet L-functions with periodic coefficients are obtained in terms of Bernoulli numbers and certain sums involving essentially the discrete Fourier transform of the periodic function forming the coefficients. The non-vanishing of these L-functions at s = 1 are then investigated. There are additional applications to the Fourier expansions of Eisenstein series over congruence subgroups of ({SL_2(mathbb{Z})}) and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given.