Oscillations d'un terme d'erreur lié à la fonction totient de Jordan
Journal de Théorie des Nombres de Bordeaux, Volume 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.

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     author = {Y.-F. S. P\'etermann},
     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},
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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, Volume 3 (1991) no. 2, pp. 311-335. doi : 10.5802/jtnb.53. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.53/

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