Dedekind eta function identity involving two complex variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:03:28Z http://mathoverflow.net/feeds/question/106925 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106925/dedekind-eta-function-identity-involving-two-complex-variables Dedekind eta function identity involving two complex variables Tito Piezas III 2012-09-11T15:49:06Z 2012-09-11T15:49:06Z <p>Given the <em>Dedekind eta function</em> $\eta(\tau)$ and complex numbers <em>a,b</em> with imaginary part > 0, anybody knows how to prove the proposed identity,</p> <p>$$\sum_{k=0}^{p-1} e^{2\pi i k/4}\eta^3\big(\tfrac{a+k}{p}\big)\eta^3\big(\tfrac{b+k}{p}\big) = p^3\eta^3(p a)\eta^3(p b)$$</p> <p>where <em>p</em> is ANY prime of form $p = 4n-1$. </p> <p>(This was inspired by Berndt and Hart's paper "<em><a href="http://www.math.uiuc.edu/~berndt/articles/345.pdf" rel="nofollow">An Identity for the Dedekind eta function involving two independent complex variables</a></em>" (2007) wherein they discussed the case p = 3 but not p = 7, 11, etc.)</p>