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Transactions of the American Mathematical Society

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



Analytic continuation of the series $ \sum \,(m+nz)\sp{-s}$

Author: Joseph Lewittes
Journal: Trans. Amer. Math. Soc. 159 (1971), 505-509
MSC: Primary 30.28
MathSciNet review: 0279286
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Abstract: The series $ \Sigma {(m + nz)^{ - s}},m,n$ ranging over all integers except both zero, for $ s$ an integer greater than two is well known from the theory of elliptic functions and modular forms. In this paper, we show that this series defines an analytic function $ (z,s)$ for $ \operatorname{Im} z > 0$ and $ \operatorname{Re} s > 2$ which has an analytic continuation to all values of $ s$. It is then shown that $ G$ satisfies a functional equation under the transformation $ z \to - 1/z$, and finally as an application some numerical results are obtained.

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Keywords: Analytic continuation, modular form, functional equation
Article copyright: © Copyright 1971 American Mathematical Society

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