Explicit Bound of \pmb|ζ(1+it)|

2024/12/01 by Eunice Hoo Qingyi, Lee-Peng Teo, Qingyi, Eunice Hoo +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2412.00766

openalex publication_date 2024/12/01 · openalex created_date 2024/12/05 · openalex updated_date 2026/07/28

Abstract

In this work, we show that for all t≥ e, |ζ(1+it)|≤ 0.6443 log t. The equality is achieved when t=17.7477. We also use the Riemann-Siegel formula and numerical computations to show that |ζ(1+it)|≤(1)/(2)log t+0.6633\hspace1cmwhen t≥ e.When t≥ 100, the bound (1)/(2)log t+0.6633 is better than the bound 0.6443log t.

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