The HerglotzZagier function, double zeta functions, and values of L-series
Academic Article
Overview
Identity
Additional Document Info
Other
View All
Overview
abstract
In this paper, we define L-series generalizing the Herglotz-Zagier function (Ber. Verhandl. Schsischen 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, Birkhuser, 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 H(, s) = n=1 (n)h(n n-s, Re(s)>1, and the Dirichlet L-function. Second, we prove that H(,s) has a meromorphic continuation to and evaluate H(,s) at s = -2l for each integer l0 in terms of Hurwitz zeta functions, generalized Euler's 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(,s) generalize previous work of Apostol and Vu (J. Number Theory 19 (1) (1984) 85) in the case is identically one with period one. 2004 Elsevier Inc. All rights reserved.
