Applying the Resonance Method to $\textrm{Re}\left(e^{-i\theta}\log\zeta(\sigma+it)\right)$

TL;DR

This paper employs the resonance method to analyze the maximal behavior of the real part of a rotated logarithm of the Riemann zeta function within a critical strip, providing new insights into its extremal values.

Contribution

It introduces the application of the resonance method to Montgomery's convolution formula for the zeta function's logarithm in the critical strip, offering novel bounds on its maximum values.

Findings

Provides bounds on maximal values of $ extrm{Re}(e^{-i heta}\log\zeta(\sigma+it))$

Analyzes behavior for $t$ in $[T^{eta}, T]$ for all $eta ext{ in } (0,1)$

Extends understanding of zeta function's extremal behavior in the critical strip

Abstract

We apply the resonance method to Montgomery's convolution formula for $\textrm{Re}\left(e^{-i\theta}\log\zeta(\sigma+it)\right)$ in the strip $1/2 < \sigma < 1$. This gives new insight into maximal values of $\textrm{Re}\left(e^{-i\theta}\log\zeta(\sigma+it)\right)$ for $t \in [T^{\beta},T]$ for all $\beta \in (0,1)$ and real $\theta$.