arXiv subjects
Publications and source records attributed to Dhir Patel.
In this article we prove an explicit sub-Weyl bound for the Riemann zeta function $ζ(s)$ on the critical line $s = 1/2 + it$. In particular, we show that $|ζ(1/2 + it)| \le 66.7\, t^{27/164}$ for $t \ge 3$. Combined, our results form the sharpest known bounds on $ζ(1/2 + it)$ for $t \ge \exp(61)$.
An explicit subconvex bound for the Riemann zeta function $ζ(s)$ on the critical line $s=1/2+it$ is proved. Previous subconvex bounds relied on an incorrect version of the Kusmin-Landau lemma. After accounting for the needed correction in that lemma, we recover and improve the record explicit bound for $|ζ(1/2 + it)|$.
In this paper we provide an explicit bound for $|ζ(1+it)|$ in the form of $|ζ(1+it)|\leq \min\left(\log t, \frac{1}{2}\log t+1.93, \frac{1}{5}\log t+44.02 \right)$. This improves on the current best-known explicit bound of $|ζ(1+it)|\leq 62.6(\log t)^{2/3}$ up until $t$ of the magnitude $10^{10^7}$.