Computational estimation of the constant $\beta (1)$ characterizing the order of $\zeta (1+it)$

Abstract:

The paper describes a computational estimation of the constant $\beta (1)$ characterizing the bounds of $\left \vert \zeta (1+it)\right \vert$. It is known that as $t\rightarrow \infty$ \begin{equation*} \frac {\zeta (2)}{2\beta (1)e^{\gamma }\left [ 1+o(1)\right ] \log \log t}\leq \left \vert \zeta (1+it)\right \vert \leq 2\beta (1)e^{\gamma }\left [ 1+o(1) \right ] \log \log t \end{equation*} with $\beta (1)\geq \frac {1}{2}$, while the truth of the Riemann hypothesis would also imply that $\beta (1)\leq 1$. In the range $1<t\leq 10^{16}$, two sets of estimates of $\beta (1)$ are computed, one for increasingly small minima and another for increasingly large maxima of $\left \vert \zeta (1+it)\right \vert$. As $t$ increases, the estimates in the first set rapidly fall below $1$ and gradually reach values slightly below $0.70$, while the estimates in the second set rapidly exceed $\frac {1}{2}$ and gradually reach values slightly above $0.64$. The obtained numerical results are discussed and compared to the implications of recent theoretical work of Granville and Soundararajan.