A variant of the prime number theorem

2021/05/23 by Liu, Kui, Wu, Jie, Yang, Zhishan
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2105.10844

Abstract

Let Λ(n) be the von Mangoldt function, and let [t] be the integral part of real number t. In this note, we prove that for any ε>0 the asymptotic formula ∑_{n≤ x} Λ([(x)/(n)]) = x∑_{d≥ 1} (Λ(d))/(d(d+1)) + O_ε(x^9/19+ε) (x→∞) holds. This improves a recent result of Bordellès, which requires (97)/(203) in place of (9)/(19).

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