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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
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