Mean values of the logarithmic derivative of the Riemann zeta-function near the critical line

TL;DR

Under the assumption of the Riemann Hypothesis and small zero gaps, the paper proves a conjecture about the asymptotic mean values of the logarithmic derivative of the Riemann zeta-function near the critical line, confirming a specific formula for all positive integers K.

Contribution

The paper proves a conjecture relating to the mean values of the logarithmic derivative of the zeta-function near the critical line under certain hypotheses, extending previous results.

Findings

Confirmed the conjecture for all positive integers K.

Derived explicit asymptotic formulas for the mean values.

Connected the results to earlier work by Goldston, Gonek, and Montgomery.

Abstract

Assume the Riemann Hypothesis and a hypothesis on small gaps between zeta zeros, we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, which states that for any positive integer $K$ and real number $a>0$, \begin{align*} \lim_{a \to 0^+}\lim_{T \to \infty} \frac{(2a)^{2K-1}}{T (\log T)^{2K}} \int_{T}^{2T} \left|\frac{\zeta'}{\zeta}\left(\frac{1}{2}+\frac{a}{\log T}+it\right)\right|^{2K} dt = \binom{2K-2}{K-1}. \end{align*} When $K=1$, this was essentially a result of Goldston, Gonek and Montgomery.