Highly composite numbers and the Riemann hypothesis
type: Item
instance of: scholarly article
title: Highly composite numbers and the Riemann hypothesis
zbMATH Open document ID: 1500.11072
published in: The Ramanujan Journal
publication date: 2022-02-11T00:00:00Z
review text: Let $d(n)$ be the number of positive divisors of $n$. Let $\mathrm{li}(x)$ be the logarithmic integral of $x$ defined by \[ \mathrm{li}(x) = \lim_{\varepsilon\to 0^+}\left(\int_0^{1-\varepsilon}\frac{1}{\log t}dt+\int_{1+\varepsilon}^x\frac{1}{\log t}dt\right)\text{ for }x>1 \] and \[ \mathrm{li}(x) = \gamma_0+\log(-\log x)+\sum_{k=1}^\infty \frac{(\log x)^k}{k\cdot k!}\text{ for }x\in (0,1), \] where $\gamma_0$ is the Euler constant. In addition, let \[ \beta_2 = \frac{\log(3/2)}{\log 2}\text{ and }R(t) = \frac{2\sqrt t+\sum_{\rho}\frac{t^\rho}{\rho^2}}{(\log t)^2}, \] where $\rho$ runs over the nontrivial zeros of the Riemann zeta function. In his PhD Thesis about highly composite numbers, Ramanujan proved under the Riemann hypothesis that \begin{align*} \frac{\log d(n)}{\log 2}&\leq\mathrm{li}(\log n)+\beta_2\mathrm{li}\left((\log n)^{\beta_2}\right)-\frac{\left(\log n\right)^{\beta_2}}{\log\log n}-R(\log n)\\ &+O\left(\frac{\sqrt{\log n}}{(\log\log n)^3}\right)\;\text{ as }n\to\infty. \tag{1} \end{align*} The main purpose of this paper is to give an effective form to the above asymptotic result of Ramanujan. In particular, the author proves that Theorem. For each $t>1$, let $F(t) =\mathrm{li}(t)+\beta_2\mathrm{li}(t^{\beta_2})-\frac{t^{\beta_2}}{\log t}$. Then the following statements hold. \begin{itemize} \item[(i)] Under the Riemann hypothesis, for $n>183783600$, \[ \frac{\log d(n)}{\log 2}\leq F(\log n)-R(\log n)-5.12\frac{\sqrt{\log n}}{(\log\log n)^3}+\frac{1.52(\log n)^{\beta_3}}{\log\log n}\tag{2} \] with $\beta_3=(\log(4/3))/\log 2 = 0.415\ldots$. \item[(ii)] If the Riemann hypothesis is not true, there exists infinitely many $n$ for which (1) does not hold. In other words, (i) is equivalent to the Riemann hypothesis. \item[(iii)] Independently of the Riemann hypothesis, there exists infinitely many $n$ such that \[ \frac{\log d(n)}{\log 2}\geq F(\log n)-R(\log n)-25.3\frac{\sqrt{\log n}}{(\log\log n)^3}-\frac{1.45(\log n)^{\beta_3}}{\log\log n}. \] \end…
reviewed by: Prapanpong Pongsriiam
Mathematics Subject Classification ID: 11N56
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 7472661
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: divisor function
zbMATH Keywords: highly composite numbers
Wikidata QID: Q114223576
author: Jean-Louis Nicolas
MaRDI profile type: Publication
full work available at URL: s11139-021-00392-0
OpenAlex ID: W3154490010
cites work: Q5558293
cites work: On Highly Composite and Similar Numbers
cites work: Nombres 2-hautement composes
cites work: An analytic method for bounding 𝜓(𝑥)
cites work: Number Theory
cites work: The largest prime factor of Landau's function
cites work: Explicit estimates of some functions over primes
cites work: Q4074972
cites work: Q3762388
cites work: On Highly Composite Numbers
cites work: On large values of the divisor function
cites work: Q5520004
cites work: Ordre maximal d'un élément du groupe $S_n$ des permutations et «highly composite numbers»
cites work: Répartition Des Nombres Hautement Composés de Ramanujan
cites work: Répartition des nombres largement composés
cites work: Q3795740
cites work: Nombres hautement composés
cites work: Estimates of \(\mathrm{li}(\theta (x))-\pi (x)\) and the Riemann hypothesis
cites work: Majorations Explicites Pour le Nombre de Diviseurs de N
cites work: On the first sign change of \(\theta(x) -x\)
cites work: Highly composite numbers. Annotated by Jean-Louis Nicolas and Guy Robin
cites work: Q3312306
cites work: Sur la différence $Li\left( \theta \left(x \right)\right) - \pi \left( x \right)$
cites work: Approximate formulas for some functions of prime numbers
cites work: Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x). II
DOI: 10.1007/S11139-021-00392-0
description: scientific article; zbMATH DE number 7472661