On the proof of a variant of Lindel\"of's hypothesis

TL;DR

This paper proves an analogue of Lindel"of's hypothesis for a variant of the transcendental sum related to the Riemann zeta function's asymptotics, using a novel Riemann-Hilbert problem approach and integral equation analysis.

Contribution

It introduces a new method involving Riemann-Hilbert problems to analyze the asymptotics of sums related to the zeta function, proving an analogue of Lindel"of's hypothesis for a modified sum.

Findings

Proved an analogue of Lindel"of's hypothesis for a variant sum.

Developed a Riemann-Hilbert problem approach for asymptotic analysis.

Analyzed integral equations to obtain rigorous estimates.

Abstract

The leading asymptotic behaviour as $t\to \infty$ of the celebrated Riemann zeta function $\zeta(s), \ s = \sigma + it, \quad 0<\sigma<1, \quad t>0 , \ t\to\infty,$ can be expressed in terms of a transcendental sum. The sharp estimation of this sum remains one of the most important open problems in mathematics with a long and illustrious history. Lindel\"of's hypothesis states that for $\sigma=1/2$, this sum is of order $O\left(t^\varepsilon\right)$ for every $\varepsilon>0$. We have recently introduced a novel approach for estimating such transcendental sums: we have first embedded the Riemann zeta function in a certain Riemann-Hilbert problem and we have began the analysis of the large $t$-asymptotics of the associated integral equation. The asymptotic analysis of the resulting integral equation requires the further splitting of the relevant interval of integration into four subintervals which are defined in terms of the small positive numbers $\{\delta_j\}_1^4$. The rigorous asymptotic analysis of the first two relevant integrals, $I_1$ and $I_2$, was performed in [F]. Here, the rigorous analysis is performed of the last two integrals, $I_3$ and $I_4$. The combination of the above results yields a proof for the analogue of Lindel{\"o}f's hypothesis for a slight variant of the transcendental sum characterising the large $t$-asymptotics of $|\zeta(s)|^2$, namely for a sum which differs from the latter sum only in the occurrence of a logarithmic term which is larger than $\frac{1}{2}\ln t$ and smaller than $t^{\varepsilon}$. Interestingly, the parameter $\varepsilon$ in Lindel\"of's hypothesis is explicitly defined in terms of $\delta_3$.