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In 1916 Ramanujan [MR:2280861] made the startling observation that the coefficients of the $\Delta$ function are multiplicative! Ramanujan defined $\tau(n)$ by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau(n)q^n$$ and conjectured that $\tau(mn)=\tau(m)\tau(n)$ whenever $m$ and $n$ are relatively prime. Moreover that $$\sum_{r=0}^\infty \tau(p^r)X^r = (1-\tau(p)X +p^{11}X^2)^{-1}$$ and that $|\tau(p)|< 2p^{11/2}$. The first two of these astounding conjectures were verified by Mordell in 1917 (see “On Mr. Ramanujan's Empirical Expansions of Modular Functions.” Proc. Cambridge Phil. Soc.19, (1917)) and the last by Deligne in 1974 in work for which he won a Fields Medal. It was already known that $$\Delta(z)=\sum_{n=1}^\infty \tau(n)e(nz)$$ is a modular form of weight 12. Thus $L(s)=\sum_{n=1}^\infty \tau(n)n^{-s}$ has a functional equation and (by Mordell) an Euler product. Ramanujan's discovery was the ushering in of a new age of arithmetic L-functions.

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  • Review status: reviewed
  • Last edited by Sally Koutsoliotas on 2017-06-12 15:45:52
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