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Y.-F. S. Pétermann
Oscillations d'un terme d'erreur lié à la fonction totient de Jordan
Journal de théorie des nombres de Bordeaux, 3 no. 2 (1991), p. 311-335, doi: 10.5802/jtnb.53
Article PDF | Analyses MR 1149800 | Zbl 0749.11041

Résumé - Abstract

Let $J_k(n) := n^k \prod _{p \mid n}(1 - p^{-k})$ (the $k$-th Jordan totient function, and for $k = 1$ the Euler phi function), and consider the associated error term

$$ E_k (x) := \sum _{n \le x}\ J_k (n) - \frac{x^{k+1}}{(k + 1)\zeta (k + 1)}. $$

When $k \ge 2$, both $i_k := E_k(x)x^{- k }$ and $s_k := \limsup E_k(x)x^{-k}$ are finite, and we are interested in estimating these quantities. We may consider instead

$$ I_k := \liminf _{n \in \mathbb{N}, n \rightarrow \infty } \sum _{d \ge 1} \frac{\mu (d)}{d^k} {\left( \frac{1}{2} - \left\rbrace \frac{n}{d}\right\lbrace \right)}, $$

since from [AS] $i_k = I_k - ( \zeta (k + 1))^-1$ and from the present paper $s_k = - i_k$. We show that $I_k$ belongs to an interval of the form

$$ \left(\frac{1}{2 \zeta (k)} - \frac{1}{(k-1)N^{k-1}}, \frac{1}{2 \zeta (k)}\right),$$

where $N = N(k) \rightarrow \infty $ as $k\rightarrow \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 $.46196896 < I_4 < .46196916$.


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