Improvement of an estimate of H. Müller involving the order of 2(mod u) II

Abstract

Let m 1 be an arbitrary fixed integer and let N m (x) count the number of odd integers u x such that the order of 2 modulo u is not divisible by m. In case m is prime, estimates for N m (x) were given by Müller that were subsequently sharpened into an asymptotic estimate by the present author. Müller on his turn extended the author's result to the case where m is a prime power and gave bounds in the case m is not a prime power. Here an asymptotic for N m (x) is derived that is valid for all integers m. We also generalize to other base numbers than 2. A further analysis of Müller's method leads us to study and solve a certain Diophantine equation.

On the distribution of the orders of 2(mod u) for odd u

“…q n l(u 2 ), that is u 1 , u 2 ∈ N q n . R e m a r k. The case n = 1 was already considered in [4]. Lemma 1 is no longer true if q has at least two different prime divisors.…”

“…However, P. Moree (see [4]) pointed out that there are already sharper and also more general results in this direction in the literature. So R. W. K. Odoni (see Theorem 2 in [6]) showed…”

“…As in [4] we will use a result of P. Moree (see Theorem 4 in [3]) which quantifies asymptotic results studied first by E. Wirsing (see [9]): …”

“…First we consider the set of odd natural numbers u satisfying q n l(u) with a fixed prime number power q n ∈ N. Unfortunately this set is not completely multipicative, as one sees in the case n = 1 by the example x = q (not x = q a as mentioned in [4]) and y = q a , where the exponent a is determined by the congruences 2 q−1 ≡ 1(mod q a ) and 2 q−1 ≡ 1(mod q a+1 ).…”

Artin's Primitive Root Conjecture – A Survey

“…In case m = q is an odd prime H. Müller [364] proved that x log 1/(q−1) x ≪ N q (x) ≪ x log 1/q x . This was improved in Moree [336] to…”