Highly composite numbers and the Riemann hypothesis

Abstract

Let us denote by d(n) the number of divisors of n, by ${{\,\mathrm{li}\,}}(t)$ the logarithmic integral of t, by $\beta _2$ the number $\frac{\log 3/2}{\log 2}=0.584\ldots$ and by R(t) the function $t\mapsto \frac{2\sqrt{t} +\sum _\rho t^\rho /\rho ^2}{\log ^2 t}$, where $\rho$ runs over the non-trivial zeros of the Riemann $\zeta$ function. In his PHD thesis about highly composite numbers, Ramanujan proved, under the Riemann hypothesis, that

$$\begin{aligned}\frac{\log d(n)}{\log 2} \leqslant {{\,\mathrm{li}\,}}(\log n)+\beta _2{{\,\mathrm{li}\,}}(\log ^{\beta _2} n)- \frac{\log ^{\beta _2} n}{\log \log n} -R(\log n)+ {\mathcal {O}}\left( \frac{\sqrt{\log n}}{(\log \log n)^3}\right) \end{aligned}$$

holds when n tends to infinity. The aim of this paper is to give an effective form to the above asymptotic result of Ramanujan.