Real moments of the restrictive factor
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.
scientific article; zbMATH DE number 5644695