Bounding $S(t)$ and $S_1(t)$ on the Riemann hypothesis

TL;DR

Under the Riemann hypothesis, the paper proves a sharper bound on the argument of the zeta function, improving previous results by a factor of two through two novel methods involving extremal functions.

Contribution

The paper introduces two new proofs for bounding |S(t)|, utilizing extremal functions and the Beurling-Selberg problem, advancing understanding of the zeta function's argument.

Findings

Established a bound |S(t)| ≤ (1/4 + o(1)) (log t / log log t) for large t

Improved previous bounds by a factor of 2 under the Riemann hypothesis

Developed two methods: one based on auxiliary functions and another on extremal problems.

Abstract

Let $\pi S(t)$ denote the argument of the Riemann zeta-function, $\zeta(s)$, at the point $s=\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)| \leq \left(\tfrac{1}{4} + o(1) \right)\tfrac{\log t}{\log \log t} $$ for large $t$. This improves a result of Goldston and Gonek by a factor of 2. The first method consists in bounding the auxiliary function $S_1(t) = \int_0^{t} S(u) du$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log\log t$. The alternative approach bounds $S(t)$ directly, relying on the solution of the Beurling-Selberg extremal problem for the odd function $f(x) = \arctan\left(\tfrac{1}{x}\right) - \tfrac{x}{1 + x^2}$. This draws upon recent work by Carneiro and Littmann.