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Power series and asymptotic series associated with the Lerch zeta-function. (English) Zbl 0937.11035
Let \(s\) be a complex variable, \(\alpha\) and \(\lambda\) real parameters with \(\alpha>0\). We define the Lerch zeta-function \[ \varphi(\lambda, \alpha,s)= \sum_{n=0}^\infty e^{2\pi in\lambda} (n+\alpha)^{-s} \quad (\operatorname {Re}s>1). \] For \(\lambda\in \mathbb{R}\setminus \mathbb{Z}\), this function is continued to an entire function over the \(s\)-plane. If \(\lambda\in \mathbb{Z}\), then this function reduces to the Hurwitz zeta-function \(\zeta(s,\alpha)\). In this paper, based on the Mellin-Barnes type of integral formulae the author studies the power series and asymptotic series for the Lerch zeta-function \(\varphi(\lambda, \alpha,s)\) in the second parameter, and obtains several asymptotic formulae.

MSC:
11M35 Hurwitz and Lerch zeta functions
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[1] T. M. Apostol : On the Lerch zeta function. Pacific J. Math., 1, 161-167 (1951). · Zbl 0043.07103
[2] E. W. Barnes : The theory of the gamma function. Messenger Math., 29, 64-128 (1899). · JFM 30.0389.01
[3] E. W. Barnes : The theory of the double gamma function. Philos. Trans. Roy. Soc. (A), 196, 265-387 (1901). · JFM 32.0442.02
[4] B. C. Berndt: Ramanujan’s Notebooks. Part I, Springer-Verlag, New York (1985). · Zbl 0555.10001
[5] Chr. Deninger : On the analogue of the formula of Chowla and Selberg for real quadratic fields. J. Reine Angew. Math., 351, 172-191 (1984). · Zbl 0527.12009
[6] A. Erdelyi (ed.), W. Magnus, F. Oberhettinger, and F. G. Tricomi: Higher transcendental functions. vol. I, McGraw-Hill, New York (1953). · Zbl 0051.30303
[7] M. Katsurada: Power series with the Riemann zeta-function in the coefficients. Proc. Japan Acad., 72A, 61-63 (1996). · Zbl 0860.11050
[8] M. Katsurada: On Mellin-Barnes type of integrals and sums associated with the Riemann zeta-function. Publ. Inst. Math. (N. S.), Beograd, 62(76), 13-25 (1997). · Zbl 0885.11052
[9] D. Klusch : On the Taylor expansion of the Lerch zeta-function. J. Math. Anal. Appl., 170, 513-523 (1992). · Zbl 0763.11036
[10] M. Lerch: Note sur la fonction K(w,x,s) = 2M;> 0exp27tinx (n + w) s. Acta Math., 11, 19-24 (1887). · JFM 19.0438.01
[11] R. Lipschitz: Untersuchung einer aus vier Elementen gebildeten Reihe. J. Reine Angew. Math., 105, 127-156 (1889). · ERAM 054.1441cj
[12] K. Matsumoto: Asymptotic series for double zeta, double gamma, and Hecke L-functions. Math. Proc. Camb. Phil. Soc, 123, 385-405 (1998). · Zbl 0903.11021
[13] K. Ueno and M. Nishizawa: Quantum groups and zeta-functions ; in ”Quantum Groups : Formalism and Applications.” Proceedings of the XXX-th Karpacz Winter School (eds. J. Lukierski, Z. Popowicz, and J. Sobczyk). Polish Scientific Publishers PWN, pp. 115-126 (1994). · Zbl 0874.17006
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