On a certain class of dirichlet series

Abstract

For 0<a<1 and Re(s) > 1, let L(s,a) and L∗(s,a) be the Dirichlet series L(s,a) = ∑ : cos (2πna)n^-s and L∗ [001](s,a) = ∑ sin (2πna)n^-s. We show that L(s,a) and L∗(s,a[001]) have holomorphic extension in the whole complex plane. Values of L(s,a)andL∗(s,a) at the negative integers are given. Moreover values of L∗(s,a) at the intergers 0,2,4,... and values of L^∗(s,a)at the integers 1,3,5,... are obtained. An exponential sums of certain recursion formulas are obtained by means of bernoulli numbers and Bernoulli polynomials

Keywords:

• Dirichlet series
• analytic function
• Hurwitz zeta-function