Improvement of an estimate of H. Müller involving the order of \(2 \pmod u\). II (Q2501158)

Let \(m\geq1\) be a fixed integer and let \(N_m(x)\) count the number of odd integers \(u\leq x\) such that the order of 2 mod u is not divisible by \(m\). In case \(m\) is a prime or a power of a prime the reviewer has proven some asymptotic estimates of \(N_m(x)\), which are sharpened and extended in the note under review. The author shows an asymptotic formula for \(N_m(x)\) which is valid for all integers \(m\). It turns out that these formulas depend essentially on \(N_{p^e}(x)\) where the prime powers \(p^e\) divide \(m\). Furthermore, he generalizes to other base numbers than \(2\). Deep results of K. Wiertelak on the distribution of certain primes are used. A further analysis leads to study and solve a Diophantine equation. For Part I, see Arch. Math. 71, No. 3, 197--200 (1998; Zbl 1006.11056).