Utilizing Smoothing Techniques to Bound $\mathrm{\mid }\zeta (1+it)\mathrm{\mid }$

We demonstrate an improved explicit upper bound of $\mathrm{\mid }\zeta (1+it)\mathrm{\mid }$ for $3\le t\le {10}^9$ using smoothing techniques. Our method sharpens previous bounds relying on the Riemann–Siegel formula and the triangle inequality. In particular, we prove that for $t\ge 3$, \begin{align*} |ζ(1+it)| \leq \frac{1}{2}\log t + 1.57 \end{align*} and for $t\ge {10}^8$, $$\mathrm{\mid }\zeta (1+it)\mathrm{\mid }\le \frac{1}{3}\log t+2\log \log t-\mathrm{1.16.}$$