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Thursday, September 29, 2016

**Abstract:** Let $\zeta(s)$ denote the Riemann zeta function. If $n$ is a positive integer, a famous formula of Euler provides an elegant evaluation of $\zeta(2n)$. However, little is known about $\zeta(2n+1)$. In Ramanujan's earlier notebooks, we find a formula for $\zeta(2n+1)$ which is a natural analogue of Euler's formula. We provide its history, indicate why it is "interesting," and show its connections with other mathematical objects such as the Dedekind eta function, Eisenstein series, and period polynomials.