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It is well known that the first Kronecker limit theorem gives the Laurent expansion of the Eisenstein series $E(z,s)$ over $SL(2,Z)$ at $s=1$; see, for example, Serge Lang's book Elliptic Curves, Section 20.4.

My question is, is there an analogous formula for the Eisenstein series over congruence subgroups? This seems a natural question and I believe that it must be hidden somewhere in the literature, but I cannot find a reference.

Any help will be appreciated. Thanks!

Remark: Thanks to Anweshi's help, we can find the following papers.

To be more precise, the above mentioned papers are

  1. MR0318065 (47 #6614) Goldstein, Larry Joel, Dedekind sums for a Fuchsian group. I. Nagoya Math. J. 50 (1973), 21--47. 10D10 (10G05). [This paper gives the generalization of Kronecker's first limit formula to Eisenstein series over a Fuchsian group of the first kind at any cusp.]

MR0347739 (50 #241) Goldstein, Larry Joel, Errata for ``Dedekind sums for a Fuchsian group. I'' (Nagoya Math. J. 50 (1973), 21--47). Nagoya Math. J. 53 (1974), 235--237. 10D15 (10G05)

  1. MR0347740 (50 #242) Goldstein, Larry Joel, Dedekind sums for a Fuchsian group. II. Nagoya Math. J. 53 (1974), 171--187. 10D15 (10G05). [This paper gives the generalization of Kronecker's second limit formula for "generalized Eisenstein series".]
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I think you're looking for the theory of automorphic forms. – Steve Huntsman Jan 20 '10 at 20:20
Yes, but in a very classical language. – qiaozi Jan 21 '10 at 15:46
The book of Rademacher on analytic number theory may help with background on Dedekind sums. – Anweshi Jan 21 '10 at 17:38
up vote 2 down vote accepted

Yes, there is a generalization. It was done by Larry Joel Goldstein, in the paper "Dedekind sums for a Fuchsian group". The paper has two parts and was published in the Nagoya Journal in around 1973-74.

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Thanks, Anweshi! – qiaozi Jan 21 '10 at 15:41

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