Mycielski, Jan Primes and power-primes. (English) Zbl 0693.10034 Colloq. Math. 58, No. 1, 145-150 (1989). Define \[ R(x)=\sum^{\infty}_{k=1}\frac{\mu (k)}{k}\text{ li}(x^{1/k}). \] Since \(\pi (x)=R(x)+\Omega (x^{-\delta})\) for every \(\delta >0\), the author asks whether \(R(x)\) has any more accurate arithmetic significance. To answer this he uses “power-primes”, namely those numbers \(P\) not of the form \(a^b\) with \(b>1\). He then shows that \[ \sum_{P\leq x}\frac{1}{\log P}=R(x)+O(\log \log x). \] Reviewer: D. R. Heath-Brown (Oxford) MSC: 11N05 Distribution of primes Keywords:Möbius function; logarithmic integral; power-primes PDFBibTeX XMLCite \textit{J. Mycielski}, Colloq. Math. 58, No. 1, 145--150 (1989; Zbl 0693.10034) Full Text: DOI