On Robin’s inequality

Abstract

AbstractLet $$\sigma (n)$$ σ ( n ) denote the sum of divisors function of a positive integer n. Robin proved that the Riemann hypothesis is true if and only if the inequality $$\sigma (n) < \textrm{e}^{\gamma }n \log \log n$$ σ ( n ) < e γ n log log n holds for every integer $$n > 5040$$ n > 5040 , where $$\gamma $$ γ is the Euler–Mascheroni constant. In this paper we establish a new family of integers for which Robin’s inequality $$\sigma (n) < \textrm{e}^{\gamma }n \log \log n$$ σ ( n ) < e γ 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 $$\vartheta $$ ϑ -function and for some product defined over prime numbers.