Michel Balazard

Sur la répartition des valeurs de la fonction d'Euler

Let Φ(x) denote the number of those integers n with ϕ(n)≤ x, where ϕ denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that Φ(x)-Ax ≪ R(x), where A=ζ(2)ζ(3)/ζ(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.

THE MEAN SQUARE OF THE LOGARITHM OF THE ZETA-FUNCTION

We investigate the function R(T,σ) , which denotes the error term in the asymptotic formula for \int_0^T|\log\zeta(σ + it)|^2dt . It is shown that R(T,σ) is uniformly bounded for σ \ge 1 and almost periodic in the sense of Bohr for fixed σ \ge 1 ; hence R(T,σ) = Ω(1) when T \to \infty . In case {1 \over 2} is fixed we can obtain the bound R(T,σ) \ll_e T\,^{(9-2σ)/8+e} .

Grandes déviations pour certaines fonctions arithmétiques

Resume Although the mean value of the number-of-divisors function τ(n) for n ≤ x is log x, there are actually only very few n ≤ x with τ(n) ≥ log x. Let S(x) denote the number of n ≤ x with τ(n) ≥ log x. Partially solving a problem of Steinig, Norton has shown that there are positive constants c1, c2 with c1 R(x) = x −1 S(x)( log x) δ ( loglog x) 1 2 and where δ = 1 − (1 + loglog 2) log 2 . In this paper we show that R(x) does not tend to a limit as x → ∞. Instead we have R(x) ∼ CK({ ( loglog x) log 2 }) where { } denotes the fractional part, C is an explicit, computable constant, and K is an explicit, computable function on [0, 1[ which is continuous but for countably many jump discontinuities. This result is obtained as a corollary to a more general theorem which deals with the distribution of integers for which a multiplicative function g(n) “resembling” τ(n) is far from its normal value. The case where g(n) is the number of square free divisors of n was previously handled by Delange.

UNE REMARQUE SUR LA FONCTION D'EULER

RésuméPar une méthode élémentaire, on améliore un résultat de Bateman (1972) concernant la réparaition des valeurs de la fonction d’Euler.