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.     

The eighth moment of Dirichlet L-functions

We prove an asymptotic for the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q  , over all moduli q?Q$q?Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2)$\mathit{GL}(2)$L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24 024 appears as a factor in the leading order term of the eighth moment.