Exponential divisor problem and exponentially squarefree integers
An exponential divisor of an integer \(n\geq 2\) with prime factorization \(n= p_1^{\mu_1} \dots p_k^{\mu_k}\) is a divisor of the form \(d= p_1^{\nu_1} \dots p_k^{\nu_k}\) where, for each \(i\), \(\nu_i\) is a divisor of \(\mu_i\); \(n\) is called exponentially squarefree if each \(\mu_i\) is squarefree. Let \(\tau^{(e)} (n)\) be the number of exponential divisors of \(n\), and let \(q^{(e)} (n)\) denote the characteristic function of the exponentially squarefree integers, with the convention \(\tau^{(e)} (1)= q^{(e)} (1) =1\). Improving results of \textit{M. V. Subbarao} [in: Theory of arithmetic functions, Proc. Conf. Western Michigan Univ. 1971, Lect. Notes Math. 251, 247-271 (1972; Zbl 0237.10009)], the author establishes the estimates \[ \sum_{n\leq x} \tau^{(e)} (n)= A_1 x+ A_2 x^{1/2}+ O(x^{2/9} \log x) \] and \[ \sum_{n\leq x} q^{(e)} (n)= Bx+ O(x^{1/4} \exp \{-C(\log x)^{3/5} (\log \log x)^{-1/5} \}), \] where \(A_1\), \(A_2\), \(B\), and \(C\) are suitable constants. The method of proof is different from, and simpler than, that of Subbarao. For example, to establish the first estimate, the author represents the function \(\tau^{(e)}\) as a convolution \(\tau_{1,2} *f\), where \(\tau_{1,2}\) is defined by \(\sum_{n\geq 1} \tau_{1,2} (n) n^{-s}= \zeta (s) \zeta (2s)\) and \(f\) is asymptotically small in the sense that the Dirichlet series \(\sum_{n\geq 1} f(n) n^{-s}\) is absolutely convergent in the half-plane \(\sigma> 1/5\). The desired estimate then follows from known estimates for the sums \(\sum_{n\leq x} \tau_{1,2} (n)\).
scientific article; zbMATH DE number 858020