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

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 $t\rightarrow \infty$

$$\frac{\zeta (2)}{2\beta (1)e^{\gamma }\left[ 1+o(1)\right] \log \... ... (1+it)\right\vert \leq 2\beta (1)e^{\gamma }\left[ 1+o(1) \right] \log \log t$$

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.