Exponential divisor problem and exponentially squarefree integers. (Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré.) (French) Zbl 0855.11047

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 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 \(\sum_{n\leq x} \tau_{1,2} (n)\).


11N25 Distribution of integers with specified multiplicative constraints
11N37 Asymptotic results on arithmetic functions


Zbl 0237.10009
Full Text: DOI Numdam EuDML EMIS


[1] Delange, H., Sur certaines fonctions additives à valeur entiers, Acta Arith., 16, (1969/70), 195-206. · Zbl 0192.39104
[2] Graham, S.W. and Kolesnik, G.A., On the difference between consecutive squarefree numbers, Acta Arith49 (1988), 435-447. · Zbl 0656.10041
[3] Ivić, A., The Riemann zeta-function, John Wiley &New York, 1985. · Zbl 0556.10026
[4] Richert, H.E., On the difference between consecutive squarefree numbers, J. London Math. Soc. (2) 26 (1951), 16-20. · Zbl 0055.04001
[5] Subbarao, M.N., On some arithmetic convolutions, in The Theory of Arithmetic Functions, No.251, 247 -271, Springer-Verlag, Berlin-Heidelberg- New York, 1972. · Zbl 0237.10009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.