The Herglotz–Zagier function, double zeta functions, and values of L-series

In this paper, we define L-series generalizing the Herglotz–Zagier function (Ber. Verhandl. Sächsischen Akad. Wiss. Leipzig 75 (3–14) (1923) 31, Math. Ann. 213 (1975) 153), and the double zeta function (First European Congress of Mathematics, Vol. II, Paris, 1992, pp. 497–512; Progress in Mathematics, Vol. 120, Birkhäuser, Basel, 1994) and evaluate them after meromorphic continuation at integer points in their extended domains. This is accomplished in three steps. First, when is a periodic function and h(n)=∑j=1n j−1 are the harmonic numbers, we establish identities relating these series to the L-series and the Dirichlet L-function. Second, we prove that H(α,s) has a meromorphic continuation to and evaluate H(α,s) at s=−2l for each integer l 0 in terms of Hurwitz zeta functions, generalized Eulerʹs constants, a finite sum, and zeta functions closely resembling the Hurwitz zeta function. Third, we combine steps one and two with well-known facts concerning the Dirichlet L-function to obtain the desired evaluations. Our results for H(α,s) generalize previous work of Apostol and Vu (J. Number Theory 19 (1) (1984) 85) in the case α is identically one with period one.