On the fractional parts of $x/n$ and related sequences. II

Bahman Saffari; R. C. Vaughan

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The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H (x) ≪ {(log x)}^{2 / 3} {(log log x)}^{4 / 3}$ for the error term $H (x) = \sum_{n \leq x} \frac{φ (n)}{n} - \frac{6}{π^{2}} x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H (x) ≪ {(log x)}^{2 / 3} {(log log x)}^{1 + ϵ}$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical -...

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Displaying similar documents to “On the fractional parts of x / n and related sequences. II”

Displaying similar documents to “On the fractional parts of $x / n$ and related sequences. II”