Remarks on the number of prime divisors of integers (Q2853316)

Let \(\omega(k)= \sum_{p|k} 1\), \(\Omega(k)= \sum_{p^\alpha\parallel k}\alpha\). Formulae for \(S_1(n)= \sum_{k\leq n}\omega(k)\), \(S_2(n)= \sum_{k\leq n}\Omega(k)\) were established in 1917 by \textit{G. H. Hardy} and \textit{S. Ramanujan} [Q. J. Math. Oxf. 48, 76--92 (1917; JFM 46.0262.03)] with an error term \(O({n\over\log n})\). In [Math. Inequal. Appl. 15, No. 2, 403--407 (2012; Zbl 1268.11140)] the present author derived explicit upper and lower bounds in terms of \(n\) for the error term in \(S_2(n)\).NEWLINENEWLINE In this paper the corresponding problems for the error term in \(S_1(n)\), and as a corollary for \(S_2(n)- S_1(n)\), are considered. Also upper and lower bounds are found for the error term in \(S_1(n)\) under the assumption of the Riemann hypothesis.