Real moments of the restrictive factor
type: Item
instance of: scholarly article
title: Real moments of the restrictive factor
zbMATH Open document ID: 1239.11104
author: Alexandru Zaharescu
author: Andrew Ledoan
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2009-12-09T00:00:00Z
review text: In the reviewer's papers [Irrational factor: definition, properties and problems, Notes Number Theory Discrete Math. 2, No. 3, 42--44 (1996)] and [Restrictive factor: definition, properties and problems, Notes Number Theory Discrete Math. 8, No. 4, 117--118 (2002)], two arithmetic functions were introduced \[ I(n) = \prod_{\nu =1}^k \root{\alpha_{\nu}}\of {p_{\nu}} \] and \[ R(n) = \prod_{\nu =1}^k p_{\nu}^{\alpha_{\nu}-1}, \] where \(n = \prod_{\nu =1}^k p_{\nu}^{\alpha_{\nu}}\) is the prime factorization of \(n\). They were called, respectively, irrational factor function and restrictive factor function. In the paper the following is proved: \textbf{Theorem} Let \(\lambda\) be a real number such that \(0 < \lambda < 1\) and let \(\varepsilon > 0\). Also, define \[ R^{\lambda}(n) = \prod_{\nu =1}^k p_{\nu}^{\lambda(\alpha_{\nu}-1)}. \] Then, there exists a positive absolute constant \(c_2\) such that for all large \(x\) \[ \begin{multlined}\sum_{n \leq x} R^{\lambda}(n)\left(1-\frac{n}{x} \right)= \frac{\zeta(2 - \lambda)}{2\zeta(2)} \prod_{p \text{ prime}} \frac{\left ( 1 + \frac{1}{p - p^{\lambda}} \right ) \left ( 1 - \frac{1}{p^{2-\lambda}} \right )}{1 + \frac{1}{p}} x \\ + \frac{2\zeta\left (\frac{1 + \lambda}{2} \right )}{(\lambda^2 + 4 \lambda + 3)\zeta(1 + \lambda)} \prod_{p \text{ prime}} \frac{\left ( 1 + \frac{1}{p^{\frac{1+\lambda}{2}} - p^{\lambda}} \right ) \left ( 1 - \frac{1}{p} \right )}{1 + \frac{1}{p^{\frac{1+\lambda}{2}}}} x^{\frac{1+\lambda}{2}}\\ + \begin{cases} O_{\lambda}(x^{\frac{1}{2}}\exp(-c_2(\log x)^{\frac{3}{5}}(\log \log x)^{-\frac{1}{5}})), & \text{ if } 0 < \lambda < \frac{1}{4} \\ O_{\lambda,\epsilon}(x^{\frac{1+2\lambda}{3}+\epsilon}), & \text{ if } \frac{1}{4} \leq \lambda < 1 ,\end{cases}\end{multlined} \] where \(\zeta\) denotes the Riemann zeta-function.
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 5644695
zbMATH Keywords: strong restrictive factor
zbMATH Keywords: arithmetic functions
zbMATH Keywords: Dirichlet series
zbMATH Keywords: Riemann zeta-function
reviewed by: Krassimir Atanassov
MaRDI profile type: Publication
full work available at URL: s12044-009-0039-7
OpenAlex ID: W2055575840
cites work: Asymptotic behavior of the irrational factor
cites work: Q4819631
cites work: Q4198832
cites work: Q3861492
cites work: Some remarks on the mean value of the Riemann zetafunction and other Dirichlet series. III
cites work: AN EXAMPLE IN THE THEORY OF THE SPECTRUM OF A FUNCTION
cites work: Q3735790
DOI: 10.1007/S12044-009-0039-7
description: scientific article; zbMATH DE number 5644695