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arXiv · 2607.01424

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

Andrew Christensen·Kyle Pratt
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Abstract

We demonstrate an improved explicit upper bound of $|\zeta(1+it)|$ 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*} |\zeta(1+it)| \leq \frac{1}{2}\log t + 1.57 \end{align*} and for $t \geq 10^8$, \[ |\zeta(1+it)|\leq \frac{1}{3}\log t + 2\log \log t -1.16 . \]

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Andrew Christensen, Kyle Pratt. 2026-07-01. Utilizing Smoothing Techniques to Bound $|\zeta(1+it)|$. https://arxiv.org/abs/2607.01424

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