On Ramanujan's formula for $\zeta(1/2)$ and $\zeta(2m+1)$

Abstract: Euler's remarkable formula for $\zeta(2m)$ immediately tells us that even zeta values are transcendental. However, the algebraic nature of odd zeta values is yet to be determined. Page 320 and 332 of Ramanujan's Lost Notebook contains an intriguing identity for $\zeta(2m+1)$ and $\zeta(1/2)$, respectively. Many mathematicians have studied these identities over the years.

In this talk, we shall discuss transformation formulas for a certain infinite series, which will enable us to derive Ramanujan's formula for $\zeta(1/2),$ Wigert's formula for $\zeta(1/k)$, as well as Ramanujan's formula for $\zeta(2m+1)$. We also obtain a new identity for $\zeta(-1/2)$ in the spirit of Ramanujan.

This is joint work with Anushree Gupta.

classical analysis and ODEscombinatoricsnumber theory

Audience: researchers in the topic

Special Functions and Number Theory seminar

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