The Nicolas and Robin inequalities with sums of two squares

Abstract

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ (n) < e γ n log log n holds for every integer n > 5040, where σ (n) is the sum of divisors function, and γ is the Euler-Mascheroni constant. We exhibit a broad class of subsets S of the natural numbers such that the Robin inequality holds for all but finitely many n ∈ S. As a special case, we determine the finitely many numbers of the form n = a 2 +b 2 that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/ϕ(n) < e γ log log n; since σ (n)/n < n/ϕ(n) for n > 1 our results for the Robin inequality follow at once.

A Totient Function Inequality

“…Let n k = 2 · 3 · · · p k be the product of the first k ≥ 1 primes. Then n k ϕ(n k ) > e γ log log n k (2) for all large k ≥ 1.…”

“…Some related and earlier works on this topic include the works of Ramanujan, Erdos, and other on abundant numbers, see [25], [1], and recent related works appeared in [29], [2], [32], [35], and [36]. The first few sections cover some background materials focusing on some finite sums over the prime numbers and some associated and products.…”

On Robin's inequality

“…This criterion on the Riemann hypothesis is called Robin's criterion and the inequality (1.3) is called Robin's inequality. Robin's inequality is proved to hold in many cases (see, for instance, Banks et al [5], Briggs [6], Grytczuk [11], and Grytczuk [12]), but remains open in general. In particular, Robin's inequality has been proven for several the t-free integers.…”

Analogues of the Robin-Lagarias Criteria for the Riemann Hypothesis

“…In particular, Robin's inequality has been studied from various points of view and for certain families of integers. See, for example, [2], [5], [12], [18].…”