×

Distribution of values of Euler’s function. (RĂ©partition des valeurs de la fonction d’Euler.) (French) Zbl 0684.10002

Denote by \(\varphi\) Euler’s function and let \(F(x)\) be the number of values of the \(\varphi\)-function less than or equal to \(x\). The best known result for the error term \(R(x)\) defined by \[ F(x) = \zeta (2)\zeta (3)/\zeta (6)\cdot x + R(x) \] is due to P. T. Bateman [Acta Arith. 21, 329–345 (1972; Zbl 0217.31902)]. Utilizing methods of complex analysis he showed \[ R(x) \ll x\cdot \exp (-(1-\varepsilon)( \log x \log \log x)^{1/2}) \] for any \(\varepsilon >0\). Using only elementary means J. L. Nicolas [Enseign. Math., II. Ser. 30, 331–338 (1984; Zbl 0553.10036)] proved \(R(x) \ll x/\log x\). Generalizing this method the author improves the result to \(R(x) \ll x/\log^2 x\).
Reviewer’s remark. The author conjectured that elementary methods lead to an estimation \(R(x) \ll x/\log^k x\) for any \(k>0\). In the meantime M. Balazard and the author have even proved \[ R(x) \ll x\cdot \exp (-c\sqrt{\log x}) \] which is very close to the above mentioned result of P. T. Bateman.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11N37 Asymptotic results on arithmetic functions
PDFBibTeX XMLCite