Über die Primteiler-Anzahl \(\omega\) (n). (On the prime divisor function \(\omega\) (n)) (Q2639897)

It is shown by analytical means that, if one assumes the Riemann hypothesis, the asymptotic formula \[ \sum_{n\leq x}\omega (n)=x \ln \ln x+B-x\int^{x^{1/2}}_{1}\frac{\{t\}}{t^ 2(\ln x-\ln t)}dt+O(x^{1/2+\epsilon}) \] holds. This improves a result of \textit{B. Saffari} [Enseign. Math., II. Sér. 14, 205-224 (1970; Zbl 0211.380)], who got a weaker error term by using the Dirichlet ``hyperbola method''. The above formula, in turn, implies the Riemann hypothesis.