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Square-full divisors of square-full integers. (English) Zbl 1209.11087

A positive integer \(n\) is square-full if \(p^2\mid n\) for every prime \(p\mid n\). Let \[ \sigma_{\text{square-full},\alpha,\beta}(n)= \sum_{\substack{ d_1d_2= n,\\ d_1,d_2\text{\,square-full}}} d^\alpha_1 d^\beta_2,\quad \sigma_{\text{square},\alpha,\beta}(n)= \sum_{\substack{ d_1d_2= n,\\ d_1\text{\,square},d_2\text{\,square-full}}} d^\alpha_1 d^\beta_2.\tag{\(*\)} \] Both sums equal zero unless \(n\) is square-full. For square-full \(n\) and \(\alpha= 0=\beta\), these sums count the number of square-full divisors of \(n\) and the number of square divisors of \(n\), respectively. In Theorem 1 the authors establish asymptotic formulae for the weighted moments \[ M_{\text{square-full},\alpha,\beta}(x)= \sum_{\substack{ n\leq x,\\ n\text{\,square-full}}} \Biggl(1-{n\over x}\Biggr) \sigma_{\text{square-full},\alpha,\beta}(n), \]
\[ M_{\text{square}, \alpha,\beta}(x)= \sum_{\substack{ n\leq x,\\ n\text{\,square-full}}} \Biggl(1-{n\over x}\Biggr) \sigma_{\text{square}, \alpha, \beta}(n) \] for any \(\alpha\geq 0\), \(\beta\geq 0\). They deduce that \[ \lim_{x\to \infty}\,{M_{\text{square},\alpha,\beta}(x)\over M_{\text{square-full},\alpha,\beta}(x)}\geq {\zeta(3)\over\zeta(3/2)} \] with equality if and only if \(\alpha= \beta\). The two Dirichlet series with coefficients the quantities on the left of the equations in \((*)\) can be expressed in terms of several Riemann zeta-functions with different variables involving \(\alpha\) or \(\beta\), and then the results are proved using standard analytic methods and properties of \(\zeta(s)\).

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11N37 Asymptotic results on arithmetic functions
11M99 Zeta and \(L\)-functions: analytic theory
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