The least primary factor of the multiplicative group

2024/03/04 by Greg S. Martin, Martin, Greg, Chàu Nguyen +1 · 1 citation
Mathematics · #Analytic Number Theory Research #Finite Group Theory Research #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2403.02548

Abstract

Let S(n) denote the least primary factor in the primary decomposition of the multiplicative group M_n = (\Bbb Z/n\Bbb Z)^×. We give an asymptotic formula, with order of magnitude x/(log x)^1/2, for the counting function of those integers n for which S(n) ≠ 2. We also give an asymptotic formula, for any prime power q, for the counting function of those integers n for which S(n) = q. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory.

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