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In 1939 R. A. Rankin [MR:0000411] studied Ramanujan's conjecture about the size of $\tau(n)$ and was led to consider the analytic properties of $g(s)=\sum_{n=1}^\infty \tau(n)^2 n^{-s}$. He proved that $\zeta(2s-22) g(s)$ is analytic everywhere except for a simple pole at $s=12$ and satisfies a functional equation. This was the beginning of the Rankin - Selberg convolution which we now realize was a hugely important event in the theory of L-functions. Selberg [MR:0002626] did the same calculation around the same time.

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• Last edited by Brian Conrey on 2016-05-10 02:35:26
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