Bounding $S_n(t)$ on the Riemann hypothesis

Abstract

Let $S(t)=\frac{1}{\pi }\arg \zeta (\frac{1}{2}+it)$ be the argument of the Riemann zeta-function at the point $\frac{1}{2}+it$. For $n\ge 1$ and $t>0$ define its iterates \begin{equation*} S_n(t) = \int_0^t S_{n-1}(τ) \,{\rm d}τ\, + δ_n\,, \end{equation*} where ${\delta }_n$ is a specific constant depending on $n$ and $S_0(t):=S(t)$. In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that $S_n(t)=O(\log t/(\log \log t{)}^{n+1})$. The order of magnitude of this estimate was never improved up to this date. The best bounds for $S(t)$ and $S_1(t)$ are currently due to Carneiro, Chandee and Milinovich. In this paper we establish, under RH, an explicit form of this estimate \begin{equation*} -\left( C^-_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}} \ \leq \ S_n(t) \ \leq \ \left( C^+_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}}\,, \end{equation*} for all $n\ge 2$, with the constants $C_n^{\pm }$ decaying exponentially fast as $n\to \infty$. This improves (for all $n\ge 2$) a result of Wakasa, who had previously obtained such bounds with constants tending to a stationary value when $n\to \infty$. Our method uses special extremal functions of exponential type derived from the Gaussian subordination framework of Carneiro, Littmann and Vaaler for the cases when $n$ is odd, and an optimized interpolation argument for the cases when $n$ is even. In the final section we extend these results to a general class of $L$-functions.

23 pages. To appear in Math. Proc. Cambridge Philos. Soc

References in corpus (2)

• Bounding $\zeta (s)$ in the critical strip
• Extremal functions in de Branges and Euclidean spaces

Cited by in corpus (2)

• Fourier optimization and prime gaps
• Analyzing Riemann's hypothesis