Identities for the Ramanujan zeta function

TL;DR

This paper derives explicit formulas for special values of the Ramanujan tau zeta function, demonstrating their nature as periods and expressing some as integrals of hypergeometric and algebraic functions.

Contribution

It provides new formulas for special values of the Ramanujan tau zeta function and shows they are periods, with explicit integral representations for specific cases.

Findings

Special values are shown to be periods in the sense of Kontsevich and Zagier.

Explicit integral formulas are derived for specific values of k.

The work connects the Ramanujan tau zeta function to hypergeometric and algebraic functions.

Abstract

We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that $L(\Delta, k)$ is a period in the sense of Kontsevich and Zagier when $k\ge12$. As an illustration, we reduce $L(\Delta, k)$ to explicit integrals of hypergeometric and algebraic functions when $k\in\{12,13,14,15\}$.