## On the exponential divisor function.(English)Zbl 0898.11034

The authors employ various elementary and analytic methods to obtain several interesting results concerning the exponential divisor function $$\tau^{(e)}(n)$$. This function, introduced by M. Subbarao in 1972, is a multiplicative function of $$n$$ defined by the relation ($$p^\nu\| n$$ means that $$p^\nu$$ exactly divides $$n$$ and $$p$$ denotes primes) $\tau^{(e)}(n) = \prod_{p^\nu\| n}\tau(\nu),$ where $$\tau(\nu)$$ is the ordinary number of divisors function. In analogy with the classical Titchmarsh divisor problem they prove $\sum_{p\leq x}\tau^{(e)}(p - 1) = C\text{ li } x + O_A\left({x\over\log^Ax}\right)$ for any fixed constant $$A > 0$$, where $$C > 0$$ is an explicit constant and as usual $$\text{li }x = \int_2^x{dt\over\log t}$$. The proof depends on the Siegel-Walfisz form of the prime number theorem for arithmetic progressions.
Next they prove a result concerning the distribution of $$\tau^{(e)}(n)$$ by using results and methods connected with the function $$\Psi(x,y)$$, the number of integers $$n \leq x$$, all of whose prime factors are $$\leq y$$. An interesting problem in the theory of arithmetic functions is to determine, if it exists, the minimal and maximal order of a given arithmetic function. The authors prove ($$\omega(n)$$ and $$\Omega(n)$$ denote as usual the number of distinct and the total number of prime factors of $$n$$, respectively, and $$\log_kn = \log(\log_{k-1}n$$))
(i) A maximal order for $$\Omega(\tau^{(e)}(n))$$ is $$\log n/(2\log_2n)$$.
(ii) A maximal order for $$\omega(\tau^{(e)}(n))$$ is $$\log_2n/((\log 2)\log_3n)$$.
(iii) As $$n \to \infty$$ we have $$\log\tau^{(e)}(\tau^{(e)}(n)) \leq (1 + o(1)) (\frac{\log_2n}{\log_3n})^2$$, while for infinitely many $$n$$ one has $\log\tau^{(e)}(\tau^{(e)}(n)) \geq (\log 2 + o(1)){\log_2n\over\log_3n}.$ The rest of the paper is devoted to the comparison of the distribution of values of $$\tau^{(e)}(n)$$ and $$\omega(n)$$, which is of interest since $$\omega(n)$$ is well-known to possess normal order $$\log_2n$$. The results of the paper render an accurate picture of the distribution of values of $$\tau^{(e)}(n)$$, and as stated in the introduction, the results can be generalized to a certain class of prime-independent, multiplicative functions.

### MSC:

 11N37 Asymptotic results on arithmetic functions 11N64 Other results on the distribution of values or the characterization of arithmetic functions
Full Text: