## 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

Zbl 0329.10035
Full Text:

### References:

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