×

On a conjecture about sums of multiplicative functions. (Russian) Zbl 0719.11060

Let f(n),g(n) be real-valued multiplicative functions and f satisfy conditions \(\sum_{p\leq x}| f(p)| \leq C_ 1 x/\log x,\quad | f(p^ r)| \leq C_ 2p^{rC_ 3},\quad r\geq 2,\quad 0\leq c_ 3\leq 1/2,\quad \sum_{p\leq x}f(p)(\log p)/p\sim \tau_ f \log x.\)
The author proves the following result: Let E be a set of prime numbers, f(p)\(\leq g(p)\) for \(p\in E\), \[ \sum_{p\leq x,p\in E}g(p)\log p\sim \tau '_ g(E)x,\quad \sum_{p\leq x,p\in E}f(p)(\log p)/p\sim \tau_ f(E) \log x \] and \(\tau_ f(E)\tau '_ g(E)>0\). Then as \(x\to \infty\) \[ \sum_{n\leq x}f(n)=\frac{x}{\log x}\frac{e^{-\gamma \tau_ f}}{\Gamma (\tau_ f)}\Pi_{f}(x)+o(\frac{x}{\log x}\Pi_{| f|}(x)) \] where \(\Pi_ f(x)=\prod_{p\leq x}(1+\sum^{\infty}_{r=1}f(p^ r)/p^ r).\)
This includes a conjecture of B. V. Levin and A. S. Fajnlejb.

MSC:

11N37 Asymptotic results on arithmetic functions
PDFBibTeX XMLCite
Full Text: EuDML