A New Upper Bound for the Sum of Divisors Function

Abstract

Robin’s criterion states that the Riemann hypothesis is true if and only if $\unicode[STIX]{x1D70E}(n)<e^{\unicode[STIX]{x1D6FE}}n\log \log n$ for every positive integer $n\geq 5041$. In this paper we establish a new unconditional upper bound for the sum of divisors function, which improves the current best unconditional estimate given by Robin. For this purpose, we use a precise approximation for Chebyshev’s $\unicode[STIX]{x1D717}$-function.

Large values of $n/\varphi (n)$ and $\sigma (n)/n$

“…< e γ log log n + α 0 (log log n) 2 for n ≥ M (0) , where (1.19) α 0 = 0.0094243 × e γ = 0.0167853 . .…”

“…If we combine (1.17) with (1.18), we can see that (1.22) σ(n) n < e γ log log n + α 0 (log log n) 2 for n > 5040.…”

“…However, the ratios σ(n)/n and n/φ(n) cannot be too large. Rosser and Schoenfeld [30,Theorem 15] showed that 2 for n ≥ 10 10 10 . Robin [28] proved that if Robin's inequality (1.8) is satisfied for two consecutive colossally abundant numbers N and N ′ (see Sect.…”