Eisenstein series and the Kronecker limit theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T05:28:46Z http://mathoverflow.net/feeds/question/12443 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12443/eisenstein-series-and-the-kronecker-limit-theorem Eisenstein series and the Kronecker limit theorem qiaozi 2010-01-20T20:07:44Z 2010-01-21T15:46:26Z <p>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. </p> <p>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. </p> <p>Any help will be appreciated. Thanks!</p> <p>Remark: Thanks to Anweshi's help, we can find the following papers.</p> <p>To be more precise, the above mentioned papers are</p> <ol> <li>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.]</li> </ol> <p>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)</p> <ol> <li>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".]</li> </ol> http://mathoverflow.net/questions/12443/eisenstein-series-and-the-kronecker-limit-theorem/12458#12458 Answer by Anweshi for Eisenstein series and the Kronecker limit theorem Anweshi 2010-01-20T22:10:03Z 2010-01-20T22:18:40Z <p>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.</p>