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Oscillations d’un terme d’erreur lié à la fonction totient de Jordan. (Oscillations of the error term associated with the Jordan totient function). (French) Zbl 0749.11041

S. D. Adhikari and A. Sankaranarayanan [J. Number Theory 34, 178-188 (1990; Zbl 0694.10041)] studied the error term \[ E_ k(x)=\sum_{n\leq k}J_ k(n)-{x^{k+1}\over (k+1)\zeta(k+1)}\text{ for } k\geq 2, \] where \(J_ k(n)=n^ k\prod_{p\mid n}(1-p^{-k})\) denotes the \(k\)-th Jordan totient function. The present author continues this investigation. Define \(i_ k\), \(s_ k\) to be the lim inf, lim sup, respectively, of \(E_ k(x)x^{-k}\) as \(x\to \infty\) and \[ I_ k=\liminf_{n\in\mathbb{N}\;n\to\infty}\sum^ \infty_{d=1}{\mu(d)\over d^ k}\left({1\over 2}-\left\{{n\over d}\right\}\right). \] In loc. cit. and the present paper, respectively, it is shown that \(i_ k=I_ k- (\zeta(k+1))^{-1}\), \(i_ k=-s_ k\), and so it suffices to estimate \(I_ k\). The author establishes inequalities for \(I_ k\) that imply that \[ {1\over 2\zeta(k)}-{1\over (k-1)N^{k-1}} < I_ k < {1\over 2\zeta(k)} \] where \(N=N(k)\to \infty\) as \(k\to \infty\), and also applies an algorithm, described in the paper, to obtain close numerical upper and lower bounds for \(I_ k\) when \(k=2,3,4\).

MSC:

11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas
11N64 Other results on the distribution of values or the characterization of arithmetic functions

Citations:

Zbl 0694.10041
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References:

[1] Adhikari, S.D. and Sankaranarayanan, A., On an error term related to the Jordan totient function Jk(n), J. Number Theory34 (1990), 178-188. · Zbl 0694.10041
[2] Erdös, P. and Shapiro, H.N., The existence of a distribution function for an error term related to the Euler function, Canad. J. Math.7 (1955), 63-75. · Zbl 0067.27601
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[4] Y.-F.S., Pétermann, Existence of all the asymptotic λ-th means for certain arithmetical convolutions, Tsukuba J. Math.12 (1988), 241-248. · Zbl 0661.10056
[5] Pétermann, Y.-F.S., On the distribution of values of an error term related to the Euler function, Proc. Conf. Théorie des nombres Univ. Laval juillet 1987, 785-797, Walter de Gruyter, Berlin (1989). · Zbl 0685.10030
[6] Pétermann, Y.-F.S., On the average behaviour of the largest divisor of n which is prime to a fixed integer k, prépublication. · Zbl 0758.11039
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