Oscillations d'un terme d'erreur lié à la fonction totient de Jordan

Pétermann, Y.-F. S.

Journal de théorie des nombres de Bordeaux, Série 2, Tome 3 (1991) no. 2, pp. 311-335

Résumé

Let $J_k\left(n\right):=n^k{\prod }_{p\mid n}(1-p^{-k})$ (the $k$-th Jordan totient function, and for $k=1$ the Euler phi function), and consider the associated error term $E_k\left(x\right):={\sum }_{n\le x}{J}_k\left(n\right)-\frac{x^{k+1}}{(k+1)\zeta (k+1)}$. When $k\ge 2$, both $i_k:=E_k\left(x\right)x^{-k}$ and $s_k:=lim\; sup{E}_k\left(x\right)x^{-k}$ are finite, and we are interested in estimating these quantities. We may consider instead $I_k:={lim\; inf}_{n\in \mathbb{N},n\to \infty }{\sum }_{d\ge 1}\frac{\mu \left(d\right)}{d^k}\left(\frac{1}{2}-\left\{\frac{n}{d}\right\}\right),$ since from [AS] $i_k=I_k-{\left(\zeta (k+1)\right)}^{-}1$ and from the present paper $s_k=-i_k$. We show that $I_k$ belongs to an interval of the form $\left(\frac{1}{2\zeta \left(k\right)}-\frac{1}{(k-1)N^{k-1}},\frac{1}{2\zeta \left(k\right)}\right)$, where $N=N\left(k\right)\to \infty$ as $k\to \infty$. From a more practical point of view we describe an algorithm capable of yielding arbitrary good approximations of $I_k$. We apply this algorithm to the small values of $k$ and obtain $.29783<I-2<.29877,.415891<I_3<.415923,$ and $.46196896<I_4<.46196916$.