On the integral of the fourth Jacobi theta function

TL;DR

This paper generalizes a classical formula to the q-loggamma function and reveals a connection between the integral of the fourth Jacobi theta function's logarithm, the partition function, and the Riemann zeta function.

Contribution

It introduces a generalization of the Raabe-formula to the q-loggamma function and links the integral of the theta function to fundamental mathematical functions.

Findings

Integral of the log of the fourth Jacobi theta function relates to the partition function.

Connection established between the integral and the Riemann zeta function.

Generalization of the Raabe-formula to the q-loggamma function.

Abstract

We generalize the Raabe-formula to the $q$-loggamma function. As a consequence, we get that the integral of the logarithm of the fourth Jacobi theta function between its least imaginary zeros is connected to the partition function and the Riemann zeta function.