preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202409.1972.v1

Preprint Article Version 1 This version is not peer-reviewed

Version 1 : Received: 24 September 2024 / Approved: 25 September 2024 / Online: 25 September 2024 (12:09:06 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. We show that the Robin inequality is true for all natural numbers $n > 5040$ that are not divisible by some prime between $2$ and $1771559$. We prove that the Robin inequality holds when $\frac{\pi^{2}}{6} \cdot \log\log n' \leq \log\log n$ for some $n>5040$ where $n'$ is the square free kernel of the natural number $n$. The possible smallest counterexample $n > 5040$ of the Robin inequality implies that $q_{m} > e^{31.018189471}$, $1 < \frac{(1 + \frac{1.2762}{\log q_{m}}) \cdot \log(1.006479799241)}{\log \log n}+ \frac{\log N_{m}}{\log n}$, $(\log n)^{\beta_{n}} < 1.000208229291\cdot\log(N_{m})$ and $n < (1.006479799241)^{m} \cdot N_{m}$, 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 $\beta_{n} = \prod_{i = 1}^{m} \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}$ when $n$ is an Hardy-Ramanujan integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$. 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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