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On a sum involving the Mangoldt function. (English) Zbl 1488.11151

Summary: Let \(\Lambda(n)\) be the von Mangoldt function, and \([t]\) be the integral part of real number \(t\). In this note we prove that the asymptotic formula \[\sum_{n\le x}\Lambda\left(\left[\frac{x}{n}\right]\right)=x\sum_{d\ge 1}\frac{\Lambda(d)}{d(d+1)}+O_{\varepsilon}\left(x^{35/71+\varepsilon}\right)\] holds as \(x\to\infty\) for any \(\varepsilon>0\).

MSC:

11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas
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References:

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