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

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 $$|S(t)|\le \left(\frac{1}{4}+o(1)\right)\frac{\log t}{\log \log t}$$ 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 $f(x)=\arctan \left(\frac{1}{x}\right)-\frac{x}{1+x^2}$. This draws upon recent work by Carneiro and Littmann.

References in corpus (1)

• Bounding $\zeta (s)$ in the critical strip

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