Utilizing Smoothing Techniques to Bound $|ζ(1+it)|$

Abstract

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