In the article, we prove that the double inequality [Formula: see text] holds for all [Formula: see text], we present the best possible constants λ and μ such that [Formula: see text] for all [Formula: see text], and we find the value of [Formula: see text] in the interval [Formula: see text] such that [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], ...

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality σ(n) := d|n d < e γ n log log n is satisfied for n ≥ 5041, where γ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin's criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, mor...

The bouquet of circles Bn and dipole graph Dn are two important classes graphs in topological theory. For n ≥ 1, we give an explicit formula for the average genus γavg(Bn) Bn. By this expression, one easily sees = (n − ln γ + 1 2) / 2 o(1), where is Euler-Mascheroni constant. Similar results obtained Dn. Our method mainly depends on technique generating series knowledge ordinary differential eq...

Robin's criterion states that the Riemann hypothesis is true if and only inequality \(\sigma(n) &lt; e^{\gamma} \cdot n \log n\) holds for all natural numbers \(n &gt; 5040\), where \(\sigma(n)\) sum-of-divisors function of \(n\) \(\gamma \approx 0.57721\) Euler-Mascheroni constant. We require properties superabundant numbers, to say left right maxima \mapsto \frac{\sigma(n)}{n}\). In this note...

There are several statements equivalent to the famous Riemann hypothesis. In 2011, Solé and Planat stated that hypothesis is true if only inequality \(\zeta(2) \cdot \prod_{q\leq q_{n}} (1+\frac{1}{q}) &gt; e^{\gamma} \log \theta(q_{n})\) holds for all prime numbers \(q_{n}&gt; 3\), where \(\theta(x)\) Chebyshev function, \(\gamma \approx 0.57721\) Euler-Mascheroni constant, \(\zeta(x)\) zeta f...

There are only aleph-zero rational numbers, while there 2 to the power real numbers. Hence probability that a randomly chosen number would be is 0. Yet proving rigorously any specific, natural, constant irrational usually very hard, witness still no proofs of irrationality Euler–Mascheroni constant, Catalan or \(\zeta (5)\). Inspired by Frits Beukers’ elegant rendition Apéry’s seminal (2)\) and...