Two-variable zeta and L-functions

We define several generalizations of zeta and L-functions. Firstly we define two variable Hurwitz zeta function and Dirichlet L-function. Then we define a two-variable Dirichlet series associated with two aritmetic functions. These functions and series are related to the Riemann zeta function, Dirichlet L-function, Dirichlet series associated to the harmonic numbers, and truncated multiple zeta functions. Using Euler-Maclaurin and periodic Euler-Maclaurin summation formulas we obtain representations in terms of known and ordinary Dirichlet series, which lead to explicit evaluations of their values at nonpositive integers. We also find corresponding reciprocity formulas, which provide some symmetric formulas involving Bernoulli polynomials, and Bernoulli numbers and associated numbers.