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Mathematics of Computation

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Computations of Eisenstein series on Fuchsian groups


Author: Helen Avelin
Journal: Math. Comp. 77 (2008), 1779-1800
MSC (2000): Primary 11F72; Secondary 11F03, 11F06, 11Y35.
DOI: https://doi.org/10.1090/S0025-5718-08-02092-9
Published electronically: January 31, 2008
MathSciNet review: 2398794
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Abstract: We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series $ E(z;s)$ on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of $ E(z;s)$ as $ \operatorname{Re} s=1/2$, $ \operatorname{Im} s\to \infty$ and also, on non-arithmetic groups, a complex Gaussian limit distribution for $ E(z;s)$ when $ \operatorname{Re} s>1/2$ near $ 1/2$ and $ \operatorname{Im} s\to \infty$, at least if we allow $ \operatorname{Re} s\to 1/2$ at some rate. Furthermore, on non-arithmetic groups and for fixed $ s$ with $ \operatorname{Re} s \ge 1/2$ near $ 1/2$, our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.


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Additional Information

Helen Avelin
Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
Email: helen.avelin@math.uu.se

DOI: https://doi.org/10.1090/S0025-5718-08-02092-9
Keywords: Automorphic forms, spectral theory, computational number theory, Fourier coefficients, explicit machine computations, Phillips-Sarnak conjecture, $K$-Bessel function, Teichm\"{u}ller space.
Received by editor(s): September 21, 2006
Received by editor(s) in revised form: May 16, 2007
Published electronically: January 31, 2008
Article copyright: © Copyright 2008 American Mathematical Society