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The prototype of L-functions is the Riemann zeta-function defined by \[ \zeta(s)=\sum_{n=1}^\infty \frac {1}{n^s} \] for $\Re s>1$. Euler was interested in this function and discovered the beautiful fact that $\zeta(2)=\frac{\pi^2}{6}.$ He also found the fundamental identity \[ \zeta(s)=\prod_p \left(1-\frac{1}{p^s}\right)^{-1} \] and used it to prove that the series $\sum_p \frac{1}{p}$ diverges. Euler played with other divergent series and noticed a connection between values of $\zeta$ at $s$ and $1-s$, linked by powers of $\pi$ and Bernoulli numbers, such as the interpretations $$1+2+3+\dots =-\frac{1}{12}$$ and $$1+4+9+16+\dots =0.$$ Euler found hints of many of the remarkable features of L-functions that make them such worthy objects of study: the Euler product, special values, and the functional equation. It was left to Riemann to discover perhaps the most remarkable property of all, one that hasn't been proven yet: the Riemann Hypothesis that each non-real zero of $\zeta(s)$ has real part equal to $\frac 12 $.

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  • Last edited by Andrew Sutherland on 2018-12-13 05:42:27
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