On error term estimates \`a la Walfisz for mean values of arithmetic functions.
On error term estimates \`a la Walfisz for mean values of arithmetic functions.
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在误差项上，像 Walfisz 那样估计算术函数的平均值。
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Yuta Suzuki
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文献类型：
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作者：
Yuta Suzuki
Walfisz (1963) proved the asymptotic formula \[ \sum_{n\le x}\varphi(n) = \frac{3}{\pi^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{4}{3}}), \] which improved the error term estimate of Mertens (1874) and had been the best possible estimate for more than 50 years. Recently, H.-Q. Liu (2016) improved Walfisz's error term estimate to \[ \sum_{n\le x}\varphi(n) = \frac{3}{\pi^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{1}{3}}). \] We generalize Liu's result to a certain class of arithmetic functions and improve the result of Balakrishnan and P\'etermann (1996). To this end, we provide a refined version of Vinogradov's combinatorial decomposition available for a wider class of multiplicative functions.
Walfisz (1963) proved the asymptotic formula \[ \sum_{n\le x}\varphi(n) = \frac{3}{\pi^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{4}{3}}), \] which improved the error term estimate of Mertens (1874) and had been the best possible estimate for more than 50 years. Recently, H.-Q. Liu (2016) improved Walfisz's error term estimate to \[ \sum_{n\le x}\varphi(n) = \frac{3}{\pi^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{1}{3}}). \] We generalize Liu's result to a certain class of arithmetic functions and improve the result of Balakrishnan and P\'etermann (1996). To this end, we provide a refined version of Vinogradov's combinatorial decomposition available for a wider class of multiplicative functions.
影响因子：
0.7
作者：
Moree Pieter;Saad Eddin Sumaia;Sedunova Alisa;Suzuki Yuta
通讯作者：
Suzuki Yuta