In analytic number theory, the '''Petersson trace formula''' is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trace formula.
In its simplest form the Petersson trace formula is as follows. Let <math>\mathcal{F}</math> be an orthonormal basis of <math>S_k(\Gamma(1))</math>, the space of cusp forms of weight <math>k>2</math> on <math>SL_2(\mathbb{Z})</math>. Then for any positive integers <math>m,n</math> we have
:<math> \frac{\Gamma(k-1)}{(4\pi \sqrt{mn})^{k-1}} \sum_{f \in \mathcal{F}} \bar{\hat{f}}(m) \hat{f}(n) = \delta_{mn} + 2\pi i^{-k} \sum_{c > 0}\frac{S(m,n;c)}{c} J_{k-1}\left(\frac{4\pi \sqrt{mn}}{c}\right), </math> where <math>\delta</math> is the Kronecker delta function, <math>S</math> is the Kloosterman sum and <math>J</math> is the Bessel function of the first kind.
== References ==
<references/> * Henryk Iwaniec: ''Topics in Classical Automorphic Forms''. Graduate Studies in Mathematics 17, American Mathematics Society, Providence, RI, 1991. *{{cite book |doi=10.1090/amsip/037/04|chapter=Petersson and Kuznetsov trace formulas |title=Lie Groups and Automorphic Forms |series=AMS/IP Studies in Advanced Mathematics |year=2006 |volume=37 |pages=147–168 |isbn=9780821841983 }} Category:Theorems in analytic number theory
{{Numtheory-stub}} Category:Automorphic forms