Analogues of the Robin-Lagarias Criteria for the Riemann Hypothesis

Abstract

Robin's criterion states that the Riemann hypothesis is equivalent to $\sigma (n)<e^{\gamma }n\log \log n$ for all integers $n\ge 5041$, where $\sigma (n)$ is the sum of divisors of $n$ and $\gamma$ is the Euler-Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that $\sigma (n)<\frac{e^{\gamma }}{2}n\log \log n$ for all odd numbers $n\ge 3^4\cdot 5^3\cdot 7^2\cdot 11\cdots 67$. Lagarias's criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to $\sigma (n)<H_n+\exp H_n\log H_n$ for all integers $n\ge 1$, where $H_n$ is the $n$th harmonic number. We establish an analogue to Lagarias's criterion for the Riemann hypothesis by creating a new harmonic series $H_n^{\prime }=2H_n-H_{2n}$ and demonstrating that the Riemann hypothesis is equivalent to $\sigma (n)\le \frac{3n}{\log n}+\exp H_n^{\prime }\log H_n^{\prime }$ for all odd $n\ge 3$. We prove stronger analogues to Robin's inequality for odd squarefree numbers. Furthermore, we find a general formula that studies the effect of the prime factorization of $n$ and its behavior in Robin's inequality.

24 pages, 1 figure; To appear in IJNT