On error term estimates à la Walfisz for mean values of arithmetic functions

2018/11/06 by Suzuki, Yuta
#11L07 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1811.02556

Abstract

Walfisz (1963) proved the asymptotic formula ∑_{n≤ x}φ(n) = (3)/(π²)x²+O(x(log x)^(2)/(3)(loglog x)^(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 ∑_{n≤ x}φ(n) = (3)/(π²)x²+O(x(log x)^(2)/(3)(loglog x)^(1)/(3)). We generalize Liu's result to a certain class of arithmetic functions and improve the result of Balakrishnan and Pétermann (1996). To this end, we provide a refined version of Vinogradov's combinatorial decomposition available for a wider class of multiplicative functions.

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