Abstract
Let $\varphi$ denote Euler's totient function. A classical result of Schoenberg asserts that $G(t):=\mathrm{dens}\{n \ge 1 : \varphi(n)/n \le t\}$ is well-defined for every $t\in[0,1]$ and recent results of the second author show that the local behaviour of $G$ around any given $t$ may essentially be described in terms of the variations around $t = 1$. We provide, as $\varepsilon\to 0+$, an asymptotic expansion of $G(1-\varepsilon)$ according to negative powers of $\log(1/\varepsilon)$, together with an evaluation of the coefficients and an explicit bound for the remainder.
Citation
Gèrald Tenenbaum. Vincent Toulmonde. "Sur le comportement local de la rèpartitionde l'indicatrice d'Euler." Funct. Approx. Comment. Math. 35 321 - 338, January 2006. https://doi.org/10.7169/facm/1229442631
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