In analytic number theory

Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form

Modular form

In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...

. It is a specialization of the more general Kuznetsov trace formula.

In its simplest form the Petersson trace formula is as follows. Let be an orthonormal basis of , the space of cusp forms of weight on . Then for any positive integers we have

where is the Kronecker delta function, is the Kloosterman sum and is the Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

of the first kind.