ISSN:
1432-1297
Keywords:
Cusp forms
;
Fourier coefficients
;
Modular group
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 γjn. 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01406220
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