Multiple Hurwitz zeta functions

Abstract

Abstract. After giving a brief overview of the theory of multiple zeta functions, we derive the analytic continuation of the multiple Hurwitz zeta functionusing the binomial theorem and Hartogs' theorem. We also consider the cognate multiple L-functions,where χ 1 , ..., χ r are Dirichlet characters of the same modulus.

Remarks on double zeta values of level 2

“…These real values can be regarded as a kind of Euler sums (see Section 2) or multiple Hurwitz series (see [8]), which are well studied, but apparently it is believed that the form (1) matches the theory of modular forms. The relationship between double zeta values and modular forms was originally studied in [5].…”

Functional equations for double series of Euler type with coefficients

“…In particular, applying Theorem 5.1 to the special case a(n) = 1 for only one fixed n, and a(n) = 0 for all other n, we obtain a functional equation for the single series [20] and [17]. Also Ram Murty and Sinha [25] studied analytic properties of this type of function and its generalizations. Now we consider the case when a(n)'s are Fourier coefficients of modular forms.…”

On universality for linear combinations of L-functions

“…In this section, we consider the universality of Euler-Zagier double L-functions and Euler-Zagier-Hurwitz type of double zeta functions defined by the absolute convergent series Noteworthly is the fact that the meromorphic continuation of these functions is a simple consequence of the meromorphic continuation of the multiple zeta function defined in the 2-dimensional case as ζ(s 1 , s 2 ; 1, 1) (see [14]). …”