ArticleOpen Access http://dx.doi.org/10.26855/jamc.2025.03.003
A Short Proof of Littlewood’s Theorem on Riemann Zeta Function
Binjie Chang
Department of Mathematics and Three Gorges Mathematics Research Center, China Three Gorges University, Yichang 443002, Hubei, China.
*Corresponding author:Binjie Chang
Published: April 11,2025
Abstract
This note primarily focuses on investigating the upper bound of the Riemann zeta function on the 1-line. Firstly, integrating Winston Heap's lemma from σ=1 to 9/8, we can get the logarithmic formula for the Riemann zeta function. Then in order to further simplify the above integral, we need to divide the sum over nontrivial zeros into three intervals to sum and combine a short calculation with N(t) to reduce three intervals, further analyzing the sum of three intervals that it can be divided into more summation terms, meanwhile the number of terms in the inner sum can also be reduced. Finally, choosing X to satisfy the error terms of the formula are all o(1), we can get the Euler product form of the Riemann zeta function on the 1-line by the definition of von Mangoldt's function, revealing the relationship between the Euler product formula and the Riemann zeta function. Moreover, we need to derive an estimate of the maximum of the Riemann zeta function on the 1-line, as an application, thus, a simple proof of Littlewood's theorem was given by Mertens' theorem.
Keywords
Riemann zeta function; Euler product; Mertens theorem
References
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How to cite this paper
A Short Proof of Littlewood’s Theorem on Riemann Zeta Function
How to cite this paper: Binjie Chang. (2025) A Short Proof of Littlewood’s Theorem on Riemann Zeta Function. Journal of Applied Mathematics and Computation, 9(1), 21-25.
DOI: http://dx.doi.org/10.26855/jamc.2025.03.003