Hybrid mean value estimates involving a new additive function. (Chinese. English summary) Zbl 1299.11071
Summary: For any positive integer \(n\), define \(H (1)=1\) and \(H (n)=\frac 1{p_1}+\frac 1{p_2}+\cdots +\frac 1{p_k}\) if \(n>1\), and \(n=p^{\alpha_1}_1 p^{\alpha_2}_2\cdots p^{\alpha_k}_k\), which decomposes \(n\) into prime powers, let \(p (n)\) be the smallest prime divisor of \(n\). The main purpose of this paper is using the elementary methods to study the value distribution properties of \(H (P_d (n))\) and \(H (q_d (n))\), and give four sharp asymptotic formulae for it.
MSC:
11N37 | Asymptotic results on arithmetic functions |
11A25 | Arithmetic functions; related numbers; inversion formulas |