On the Balasubramanian-Ramachandra method close to Re(s)=1

TL;DR

This paper advances the understanding of lower bounds for integrals of the Riemann zeta function near the line Re(s)=1, employing kernel functions and Bessel functions to refine the Balasubramanian-Ramachandra method.

Contribution

It introduces the use of Bessel functions and Paley-Wiener kernel functions to improve lower estimate techniques for zeta integrals close to Re(s)=1.

Findings

Kernel functions simplify the analysis of zeta integrals.

Fourier transform of Ramachandra's kernel is a Bessel function.

Enhanced method provides better lower bounds near Re(s)=1.

Abstract

We study the problem on how to get good lower estimates for the integral $$ \int_T^{T+H} |\zeta(\sigma+it)| dt, $$ when $H \ll 1$ is small and $\sigma$ is close to $1$, as well as related integrals for other Dirichlet series, by using ideas related to the Balasubramanian-Ramachandra method. We use kernel-functions constructed by the Paley-Wiener theorem as well as the kernel function of Ramachandra. We also notice that the Fourier transform of Ramachandra's Kernel-function is in fact a $K$-Bessel function. This simplifies some aspects of Balasubramanian-Ramachandra method since it allows use of the theory of Bessel-functions.