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Primes and power-primes. (English) Zbl 0693.10034

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
\[ \sum_{P\leq x}\frac{1}{\log P}=R(x)+O(\log \log x). \]

MSC:

11N05 Distribution of primes
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