Factorization of factorials and a result of Hardy and Ramanujan (Q2885359)

Let \(\Omega(k)\) denote the total number of prime factors of \(k\). In [Q. J. Math. 48, 76--92 (1917; JFM 46.0262.03)], \textit{G. H. Hardy} and \textit{S. Ramanujan} asserted that NEWLINE\[NEWLINE\Omega(n!)= \sum_{k\leq n} \Omega(k)= n\log\log n+ M'n+ O\Biggl({n\over\log n}\Biggr),\tag{\(*\)}NEWLINE\]NEWLINE where the constant \(M'\) has an explicit representation.NEWLINENEWLINE The aim of this paper is to show that the error term in \((*)\) lies between \(-81492{n\over\log n}\) and \({n\over\log^2n}\). Write \(v_p(k)\) for the power of the prime \(p\) in the factorization of \(k\). Noting that NEWLINE\[NEWLINE\Omega(n!)= \sum_{p\leq n} v_p(n!),NEWLINE\]NEWLINE the starting point of the proof is the author's result in [JIPAM, J. Inequal. Pure Appl. Math. 6, No. 2, Paper No. 29, 7 p. (2005; Zbl 1114.11078)] that for \(p\leq n\), NEWLINE\[NEWLINE{n-p\over p-1}- {\log n\over\log p}< v_p(n!)\leq {n-1\over p-1}.NEWLINE\]