The Distribution of Totatives

• Autores: Jam Germain
• Localización: Canadian mathematical bulletin, ISSN 0008-4395, Vol. 48, Nº 2, 2005, págs. 211-220
• Idioma: inglés
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• Resumen

  • The integers coprime to n are called the totatives of n. D. H. Lehmer and Paul Erdös were interested in understanding when the number of totatives between in/k and (i+1)n/k are 1/kth of the total number of totatives up to n. They provided criteria in various cases. Here we give an "if and only if" criterion which allows us to recover most of the previous results in this literature and to go beyond, as well to reformulate the problem in terms of combinatorial group theory. Our criterion is that the above holds if and only if for every odd character chi mod kappa (where kappa := k/gcd(k,n/prodp|n p)) there exists a prime p = pchi dividing n for which chi(p) = 1.