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