Highly composite numbers and the Riemann hypothesis
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{itemize} Although the line of reasoning follows that used by Ramanujan, hard-working and new approximation are still required in translating from an implicit to explicit estimate like in this work. The author shows a very careful calculation leading to an interesting and beautiful theorem. He not only obtains the main results as stated above but he also made some nice observations on highly composite numbers, superior highly composite numbers, and other functions related to primes. The ingredients of the proof of the main results consist of Ramanujan's original argument, the author's nice observations, the author's results in the past to the present, and some other formulas and inequalities obtained by various mathematicians. The reviewer believes that this article will be of high interest to many number theorists who are working on explicit results in number theory.
scientific article; zbMATH DE number 7472661
scientific article