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The Distribution of Totatives

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http://dx.doi.org/10.4153/CMB-2005-019-9
Canad. Math. Bull. 48(2005), 211-220
Published:2005-06-01
Printed: Jun 2005
• Jam Germain
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Abstract

The integers coprime to $n$ are called the {\it totatives} \rm of $n$. D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th 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 $\chi \pmod \kappa$ (where $\kappa:=k/\gcd(k,n/\prod_{p|n} p)$) there exists a prime $p=p_\chi$ dividing $n$ for which $\chi(p)=1$.

MSC Classifications:  11A05 - Multiplicative structure; Euclidean algorithm; greatest common divisors 11A07 - Congruences; primitive roots; residue systems 11A25 - Arithmetic functions; related numbers; inversion formulas 20C99 - None of the above, but in this section
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