Fourier coefficients of cusp forms

Abstract

Let ψ₁,ψ₂,ψ₃,... be an orthonormal basis of the space of cusp forms of weight zero for the full modular group. Let

be the Fourier series expansion. The following theorem is proved: Let σ∈(1/4, 1/2); letf be a holomorphic function on the strip |Res|≦σ, satisfyingf(−s)=f(s) and

$$f(s) = \mathcal{O}(|\tfrac{1}{4} - s^2 |^{ - 2} |cos \pi s|^{ - 1} )$$

on this strip; letm andn be non-zero integers, then

$$\sum\limits_{j = 1}^\infty {f(s_j )\bar \gamma _{jm} \gamma _{jn} } $$

converges and is equal to

$$\begin{gathered} - (2\pi i)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)c_{00} ( - s)c_{0|m|} (s)c_{0|n|} (s)ds} \hfill \\ + (2\pi i)^{ - 1} (4\pi |m|)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)c_{mn} (s)2sds} \hfill \\ - \delta _{mn} (2\pi i)^{ - 1} (4\pi |m|)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)\sin \pi s2sds.} \hfill \\ \end{gathered} $$

The functionsc ₀₀(s) andc _0|m|(s) are coefficients occurring in the Fourier series expansion of the Eisenstein series; the functionc _mn(s) is a coefficient in the Fourier series expansion of a Poincaré series.

The theorem is applied to obtain some asymptotic results concerning the Fourier coefficients γ_jn. Under additional conditions on the functionf the formula in the theorem is modified in such a way that the Fourier coefficients of holomorphic cusp forms appear.