preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202409.1972.v2 (registering DOI)

Preprint Article Version 2 This version is not peer-reviewed

Version 1 : Received: 24 September 2024 / Approved: 25 September 2024 / Online: 25 September 2024 (12:09:06 CEST)
Version 2 : Received: 16 October 2024 / Approved: 21 October 2024 / Online: 21 October 2024 (13:03:10 CEST)

Robin's criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \cdot n \cdot \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The possible smallest counterexample $n > 5040$ of the Robin inequality implies that $(N_{m})^{Y_{m}} > n$, $(\log q_{m}) \cdot \left(1 + \frac{1}{\log^{2}(q_{m})}\right) > \log \log n$ and $\left(1 + \frac{0.2}{\log^{3}(q_{m})}\right) > \frac{\log \log N_{m}}{\log q_{m}}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$, $q_{m}$ is the largest prime divisor of $n$ and $Y_{m} = \frac{e^{\frac{0.2}{\log^{2}(q_{m})}}}{\left(1 - \frac{1}{\log^{3}(q_{m})}\right)}$. By combining these results, we present a proof of the Riemann hypothesis. This work is an expansion and refinement of the article "Robin's criterion on divisibility", published in The Ramanujan Journal.

Riemann hypothesis; Robin inequality; Sum-of-divisors function; Prime numbers; Riemann zeta function

Computer Science and Mathematics, Algebra and Number Theory

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