On Robin’s inequality

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