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 F be an orthonormal basis of S k ( Γ ( 1 ) ) , the space of cusp forms of weight k > 2 on S L 2 ( Z ) . Then for any positive integers m , n we have
Γ ( k − 1 ) ( 4 π m n ) k − 1 ∑ f ∈ F f ¯ ( m ) f ( n ) = δ m n + 2 π i − k ∑ c > 0 S ( m , n ; c ) c J k − 1 ( 4 π m n c ) , where δ is the Kronecker delta function, S is the Kloosterman sum and J is the Bessel function of the first kind.
