A Short Proof of Littlewood’s Theorem on Riemann Zeta Function

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.