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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Analytic continuation of Eisenstein series

Author: Joseph Lewittes
Journal: Trans. Amer. Math. Soc. 171 (1972), 469-490
MSC: Primary 10K20
MathSciNet review: 0306148
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Abstract: The classical Eisenstein series are essentially of the form $ {\Sigma '_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{m} ,n}}{((m + {r_1})z + n + {r_2})^{ - s}},m,n$ ranging over integer values, $ \operatorname{Im} z > 0,{r_1},{r_2}$ rational and $ s$ an integer $ > 2$. In this paper we show that if $ s$ is taken to be complex the series, with $ {r_1},{r_2}$ any real numbers, defines an analytic function of $ (z,s)$ for $ \operatorname{Im} z > 0,\operatorname{Re} s > 2$. Furthermore this function has an analytic continuation over the entire $ s$ plane, exhibted explicitly by a convergent Fourier expansion. A formula for the transformation of the function when $ z$ is subjected to a modular transformation is obtained and the special case of $ s$ an integer is studied in detail.

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Keywords: Analytic continuation, Eisenstein series, modular group, zeta function
Article copyright: © Copyright 1972 American Mathematical Society

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