Analogues of the Robin–Lagarias criteria for the Riemann hypothesis

Abstract

Robin’s criterion states that the Riemann hypothesis is equivalent to [Formula: see text] for all integers [Formula: see text], where [Formula: see text] is the sum of divisors of [Formula: see text] and [Formula: see text] is the Euler–Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that [Formula: see text] for all odd numbers [Formula: see text]. Lagarias’s criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to [Formula: see text] for all integers [Formula: see text], where [Formula: see text] is the [Formula: see text]th harmonic number. We establish an analogue to Lagarias’s criterion for the Riemann hypothesis by creating a new harmonic series [Formula: see text] and demonstrating that the Riemann hypothesis is equivalent to [Formula: see text] for all odd [Formula: see text]. 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 [Formula: see text] and its behavior in Robin’s inequality.