Explicit bounds of $|\zeta (1+it)|$

Abstract

In this work, we study a bound of the form $|\zeta (1+it)|\le v\log t$ for $t\ge t_0$. We show that the exponential sum method with second-order derivatives can achieve any $v>\frac{1}{2}$ as long as $t_0$ is sufficiently large. Using the Riemann–Siegel formula and numerical computations, we show that when $t\ge e$, $|\zeta (1+it)|\le \frac{1}{2}\log t+0.6633$. This allows us to show that $|\zeta (1+it)|\le 0.6443\log t$ when $t\ge e$. This is the best possible result of the form $|\zeta (1+it)|\le v\log t$ that holds for all $t\ge e$, as the equality is achieved when $t=17.7477$.