Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1972-0306148-1
MathSciNet review: 0306148
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] E. Hecke, Theorie der Eisensteinschen Reihe hoherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Sem. Univ. Hamburg 5 (1927), 199-224.
  • [2] J. Lewittes, Analytic continuation of the series $ \Sigma {(m + nz)^{ - s}}$, Trans. Amer. Math. Soc. 159 (1971), 594-598. MR 0279286 (43:5009)
  • [3] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press, New York, 1969 (reprint). MR 1424469 (97k:01072)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10K20

Retrieve articles in all journals with MSC: 10K20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0306148-1
Keywords: Analytic continuation, Eisenstein series, modular group, zeta function
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society