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

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

让我们表示由d（Ñ）的约数的数量Ñ，通过\（{{\，\ mathrm {利} \，}}（T）\）的对数积分吨，由\（\测试_2 \）数\（\ frac {\ log 3/2} {\ log 2} = 0.584 \ ldots \）并通过R（t）函数\（t \ mapsto \ frac {2 \ sqrt {t} + \ sum _ \ rho t ^ \ rho / \ rho ^ 2} {\ log ^ 2 t} \），其中\（\ rho \）在Riemann \（\ zeta \）函数的非平凡零上运行。在Rmanmann假设下，Ramanujan在有关高度合成数的PHD论文中证明了：

当n趋于无穷大时成立。本文的目的是为Ramanujan的上述渐近结果提供一种有效的形式。