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

### MSC:

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

Zbl 0237.10009
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### References:

 [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
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