Oscillations d'un terme d'erreur lié à la fonction totient de Jordan
Journal de Théorie des Nombres de Bordeaux, Tome 3 (1991) no. 2, pp. 311-335.

Let J k (n):=n k pn (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 ) : = n x J k ( n ) - x k + 1 ( k + 1 ) ζ ( k + 1 ) . When k2, both i k :=E k (x)x -k and s k :=lim supE k (x)x -k are finite, and we are interested in estimating these quantities. We may consider instead I k : = lim inf n , n d 1 μ ( d ) d k 1 2 - n d , since from [AS] i k =I k -(ζ(k+1)) - 1 and from the present paper s k =-i k . We show that I k belongs to an interval of the form 1 2 ζ ( k ) - 1 ( k - 1 ) N k - 1 , 1 2 ζ ( k ) , where N=N(k) as k. 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.

@article{JTNB_1991__3_2_311_0,
     author = {P\'etermann, Y.-F. S.},
     title = {Oscillations d'un terme d'erreur li\'e \`a la fonction totient de Jordan},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {311--335},
     publisher = {Universit\'e Bordeaux I},
     volume = {3},
     number = {2},
     year = {1991},
     doi = {10.5802/jtnb.53},
     zbl = {0749.11041},
     mrnumber = {1149800},
     language = {fr},
     url = {jtnb.centre-mersenne.org/item/JTNB_1991__3_2_311_0/}
}
Y.-F. S. Pétermann. Oscillations d'un terme d'erreur lié à la fonction totient de Jordan. Journal de Théorie des Nombres de Bordeaux, Tome 3 (1991) no. 2, pp. 311-335. doi : 10.5802/jtnb.53. https://jtnb.centre-mersenne.org/item/JTNB_1991__3_2_311_0/

[AS] S.D. Adhikari and A. Sankaranarayanan, On an error term related to the Jordan totient function Jk(n), J. Number Theory 34 (1990), 178-188. | MR 1042491 | Zbl 0694.10041

[ES] P. Erdös and H.N. Shapiro, The existence of a distribution function for an error term related to the Euler function, Canad. J. Math. 7 (1955), 63-75. | MR 65580 | Zbl 0067.27601

[M] H.L. Montgomery, Fluctuations in the mean of Euler's phi function, Proc. Indian Acad. Sci. (Math.Sci.) 97 (1987), 239-245. | MR 983617 | Zbl 0656.10042

[P1] 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

[P2] Y.-F.S. Pétermann, 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). | MR 1024603 | Zbl 0685.10030

[P3] Y.-F.S. Pétermann, On the average behaviour of the largest divisor of n which is prime to a fixed integer k, prépublication.

[W] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin (1963). | MR 220685 | Zbl 0146.06003