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On integers \(\leq x\) having more than \(\log x\) divisors. (Sur les entiers inférieurs à \(x\) ayant plus de \(\log(x)\) diviseurs.) (French) Zbl 0839.11041

The authors are interested in the asymptotic behavior of the number of integers \(\leq x\) with more than \(\log (x)\) prime factors and, more generally, the behavior of the function \[ S_\lambda (x) = \# \bigl\{ n \leq x : \tau (n) \geq (\log x)^{\lambda \log 2} \bigr\}, \] where \(\tau (n)\) is the divisor function and \(\lambda\) a positive real number. In a recent paper [J. Number Theory 40, 146–164 (1992; Zbl 0745.11041)] M. Balazard, J.-L. Nicolas, C. Pomerance, and G. Tenenbaum showed that for fixed \(\lambda \geq 1\), \(S_\lambda (x)\) has order of magnitude \[ f (\lambda, x) = {x \over (\log x)^{\lambda \log \lambda - \lambda + 1} \sqrt {\log \log x}}; \] more precisely, the behavior of \(S_\lambda (x)\) is given by \[ S_\lambda (x) = c (\lambda) K(\lambda \log \log x) f(\lambda, x) \left( 1 + O \left( {1 \over \log \log x} \right) \right), \tag \(*\) \] where \(c (\lambda)\) is a positive constant and the function \(K (\theta)\) is bounded from above and below, left-continuous, and periodic with period 1.
In the paper under review, the authors investigate the behavior of the function \(K (\theta)\) in \((*)\). Their main result shows that in the case \(1 \leq \lambda \leq 2\) \[ \inf_{\theta} K (\theta) = K(0+), \quad \sup_\theta K (\theta) = K(0). \] Combining this result with numerical computations, they obtain numerical values for the lower and upper limits for the ratio \(S_\lambda (x)/f (\lambda, x) \), as \(x \to \infty\). For example, in the case \(\lambda = 1/ \log 2\) these limits are \(0.93827 \dots\) and \(1.148126 \dots\), respectively.

MSC:

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

Citations:

Zbl 0745.11041
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References:

[1] Balazard, M., Nicolas, J.-L., Pomerance, C., Tenenbaum, G., Grandes déviations pour certaines fonctions arithmétiques, J. Number Theory40 (1992), 146-164. · Zbl 0745.11041
[2] Comtet, L., Analyse combinatoire, Tomes 1 et 2. Presses universitaires de France, 1970. · Zbl 0221.05002
[3] Crapo, H.H., Permanent by Möbius inversion, Journal of Combinatorial Theory4 (1968), 198-200. · Zbl 0162.03201
[4] Davis, H.T., Table of the higher mathematical functions, The principia Press, Bloomington, Indiana, 1935, vol. 2. · Zbl 0013.21603
[5] Deléglise, M., Applications des ordinateurs à la théorie des nombres, Thèse Université de Lyon1, 1991.
[6] Elliot, P.D.T.A., Probabilistic number theory, vol I and II, Grundlehren der Mathematischen Wissenschaften239-240, Springer-Verlag, 1979. · Zbl 0431.10029
[7] Flageolet, P., Vardi, I., Numerical evaluation of Euler products, Prepublication.
[8] Glaisher, W.L., On the sums of the inverse powers of the prime numbers, Quartely Journal of Math25 (1891), 347-362.
[9] Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers, Oxford at the Clarendon Press1962. · Zbl 0086.25803
[10] Norton, K.K., On the number of restricted prime factors of an integer, Illinois J. Math20 (1976), 681-705. · Zbl 0329.10035
[11] Riesel, H., Prime numbers and computer methods for factorization, Birkhaüser, 1985. · Zbl 0582.10001
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