Typically, one expects that there are around x ∏ p6∈P, p≤x(1− 1/p) integers up to x, all of whose prime factors come from the set P . Of course for some choices of P one may get rather more integers, and for some choices of P one may get rather less. Hall [4] showed that one never gets more than e + o(1) times the expected amount (where γ is the Euler-Mascheroni constant), which was improved sl...

as N → ∞, where γ is the Euler-Mascheroni constant. Let us introduce a more refined notion of divisibility. A divisor k of n is unitary if k and n/k are coprime, that is, if gcd(k, n/k) = 1. This condition is often written as k||n. The number d∗(n) of unitary divisors of n is 2, where ω(n) is the number of distinct prime factors of n. This fact is easily seen to be true: If p1 1 p a2 2 · · · pr...

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < eγn 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 determin...

The author continues to study series transformations for the Euler-Mascheroni constant γ . Here, we discuss in detail recently published results of A. I. Aptekarev and T. Rivoal who found rational approximations to γ and γ log q q ∈ Q>0 defined by linear recurrence formulae. The main purpose of this paper is to adapt the concept of linear series transformations with integral coefficients such t...

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) &lt; \textrm{e}^{\gamma }n \log n$$ &lt; ...

We build stochastic models for analyzing Jaya and semi-steady-state algorithms. The analysis shows that (a) the maximum expected value of number worst-index updates per generation is a paltry 1.7 regardless population size; (b) size, expectation best-index decreases monotonically with generations; (c) exact upper bounds as well asymptotics best-update counts can be obtained specific distributio...

This sequence has diverse applications in many areas of mathematics, ranging from classical or numerical analysis to number theory, special functions or theory of probability. There is a huge literature about the sequence (γn)n≥1 and the constant γ. Please refer to [3, 4, 5, 6, 7, 8] and all the references therein. The speed of convergence to γ of sequence (γn)n≥1 is very slowly, if we take int...

It is not known if γ is an irrational or a transcendental number. The question of its irrationality has challenged mathematicians since Euler and remains a famous unresolved problem. By computing a large number of digits of γ and using continued fraction expansion, it has been shown that if γ is a rational number p/q then the denominator q must have at least 242080 digits. Even if γ is less fam...