The Herglotz–Zagier function, double zeta functions, and values of L-series
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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 α : ℤ→ℂ 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(α, s) = ∑ n=1∞ α(n)h(n n-s, Re(s)>1, and the Dirichlet L-function. Second, we prove that H(α,s) has a meromorphic continuation to ℂ and evaluate H(α,s) at s = -2l for each integer l≥0 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(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. © 2004 Elsevier Inc. All rights reserved.
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