On Robin's inequality
Abstract
Let $\sigma(n)$ denotes the sum of divisors function of a positive integer $n$. Robin proved that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma}n \log \log n$ holds for every positive integer $n \geq 5041$, where $\gamma$ is the Euler-Mascheroni constant. In this paper we establish a new family of integers for which Robin's inequality $\sigma(n) < e^{\gamma}n \log \log n$ hold. Further, we establish a new unconditional upper bound for the sum of divisors function. For this purpose, we use an approximation for Chebyshev's $\vartheta$-function and for some product defined over prime numbers.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2021
- DOI:
- 10.48550/arXiv.2110.13478
- arXiv:
- arXiv:2110.13478
- Bibcode:
- 2021arXiv211013478A
- Keywords:
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- Mathematics - Number Theory;
- 11A25 (Primary);
- 11N56 (Secondary)
- E-Print:
- v3: A typo in Theorem 1.4 is fixed