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

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 $$\begin{aligned} |S(t)| \le \left(\frac{1}{4} + o(1) \right)\frac{\log t}{\log \log t} \end{aligned}$$ for large $$t$$ . This improves a result of Goldston and Gonek by a factor of 2. The first method consists of bounding the auxiliary function $$S_1(t) = \int _0^{t} S(u) \> \text{ d}u$$ 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.