The smallest invariant factor of the multiplicative group

Abstract

Let λ 1 (n) denote the least invariant factor in the invariant factor decomposition of the multiplicative group M n = (Z/nZ) × . We give an asymptotic formula, with order of magnitude x/ √ log x, for the counting function of those integers n for which λ 1 (n) = 2. We also give an asymptotic formula, for any even q ≥ 4, for the counting function of those integers n for which λ 1 (n) = q. These results require a version of the Selberg-Delange method whose dependence on certain parameters is made explicit, which we provide in an appendix. As an application, we give an asymptotic formula for the counting function of those integers n all of whose prime factors lie in an arbitrary fixed set of reduced residue classes, with implicit constants uniform over all moduli and sets of residue classes.

On the Parity of the Order of Appearance in the Fibonacci Sequence

“…Our last auxiliary lemma is a particular case of a theorem due to Chang and Martin [15]. We state this more precisely in the next lemma.…”

“…Lemma 5 (Theorem 3.4 of [15]). For any integer q ≥ 3, there exists a positive absolute constant C such that uniformly for q ≤ (log x) 1/3 , we have…”

Distribution mod $p$ of Euler's totient and the sum of proper divisors

“…In this section, we supply the promised proofs of Lemmas 3.2 and 5.2, by the method of Landau-Selberg-Delange. We use a recent formulation of that method due to Chang and Martin [CM20], which is based on Tenenbaum's treatment in [Ten15, Chapter II.5] but (crucially for us) more explicit about the dependence on certain parameters.…”

“…Setup. We follow [CM20] in setting log + y = max{0, log y}, with the convention that log + 0 = 0. We write complex numbers s as s = σ + iτ .…”

“…While similar analytic methods have been applied in such problems before (such as in the work of Narkiewicz mentioned above), the modulus was always fixed. To allow p to grow with x, we apply a version of the Landau-Selberg-Delange method enunciated recently by Chang and Martin [CM20]. Interestingly, this part of the argument uses crucially that nontrivial Jacobi sums over F p are bounded by √ p in absolute value; the trivial bound of p would only allow the method to work for p up to about log log x, just shy of what is required for overlap with our second range.…”