The simplest Eisenstein series
نویسنده
چکیده
We explain some essential aspects of the simplest Eisenstein series for SL2(Z) on the upper half-plane H. There are many different proofs of meromorphic continuation and functional equation of the simplest Eisenstein series for Γ = SL2(Z). We will follow [Godement 1966a] rewriting of a Poisson summation argument that appeared in [Rankin 1939], if not earlier. This argument is the most elementary and least messy of all the meromorphic continuation proofs I know, but is less informative than arguments that engage more seriously with the spectral theory itself. Nevertheless, it is best to obtain decisive information in this simple case. Arguments based on Fourier expansions do unnecessary work, and risk confusion over peripheral details.
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