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Grandes déviations pour certaines fonctions arithmétiques. (Large deviations for certain arithmetic functions). (French) Zbl 0745.11041

It is well known that the mean value of the divisor function \(\tau(n)\) for \(n\leq x\) is \(\log x\); Steinig proposed the natural problem of estimating \(S(x)\), the number of positive integers \(n\leq x\) for which \(\tau(n)\geq\log x\), and in Ill. J. Math. 20, 681-705 (1976; Zbl 0329.10035) K. K. Norton showed that \[ R(x)=x^{-1}S(x)(\log x)^ \delta(\log \log x)^{1\over2}, \] where \(\delta=1-(1+\log \log 2)/\log 2\), is bounded above and below by positive constants. As a corollary to their main results, the present authors obtain an asymptotic formula for \(R(x)\) involving a function of the fractional part \(\{{{\log \log x}\over{\log 2}}\}\) that has countably many jump discontinuities.
More generally, they establish an asymptotic formula for \(N(x,\lambda;f)\), the number of positive integers \(n\leq x\) for which \(f(x)\geq\lambda \log \log x\) if \(\lambda>1\) or \(f(n)<\lambda \log \log x\) if \(\lambda<1\), where \(f(x)\) is a real-valued additive function of the form \(f(n)=\omega(n)+\rho(n)\) with \(\rho(p)=0\) for all primes, \(p\), \(\omega(n)\) denotes as usual the number of distinct prime factors of \(n\), and \(\lambda\) is restricted to an interval depending on the function \(\rho\). They also obtain an asymptotic estimate for the number of positive integers \(n\leq x\) for which \(\omega(n)=k\) and \(\rho(n)\geq y\) that is uniform for \(1\leq k\ll\log \log x\) and \(y\in\mathbb{R}\), with an analogous result when \(\rho(n)<y\) holds instead.

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11K65 Arithmetic functions in probabilistic number theory
11N37 Asymptotic results on arithmetic functions

Citations:

Zbl 0329.10035
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References:

[1] Elliott, P. D.T. A., (Probabilistic Number Theory (1979), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · Zbl 0431.10029
[2] Erdős, P.; Nicolas, J.-L, Sur la fonction: Nombre de facteurs premiers de \(n\), L’Ens. Math., 27, 3-27 (1981) · Zbl 0466.10037
[3] Hardy, G. H.; Wright, E. M., (An Introduction to the Theory of Numbers (1979), Oxford Univ. Press) · Zbl 0423.10001
[4] Kubilius, J., Probabilistic methods in the theory of numbers, (American Mathematical Society Translations, Vol. 11 (1964)), Providence, RI · Zbl 0133.30203
[5] Norton, K. K., On the number of restricted prime factors of an integer, I, Illinois J. Math., 20, 681-705 (1976) · Zbl 0329.10035
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