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Define the Ramanujan $\tau$-function from $\mathbb N \rightarrow \mathbb Z$ as the Fourier coefficients of the $\Delta$ function; i.e., $$ \sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} .$$ This is a pure number-theoretic function. Now the Ramanujan conjecture says that $$|\tau(p)| \leq 2p^{11/2} $$ for every prime $p$, which is also a purely number theoretic statement. Pierre Deligne proved it as a consequence of the Weil conjectures. |
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Define the Ramanujan $\tau$-function from $\mathbb N \rightarrow \mathbb Z$ as the Fourier coefficients of the $\Delta$ function; i.e., $$ \sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} .$$ This is a pure number-theoretic function. Now the Ramanujan conjecture says that $$|\tau(p)| \leq 2p^{11/2} $$ which is also a purely number theoretic statement. Pierre Deligne proved it as a consequence of the Weil conjectures. |
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