A remark about the orders of \(2\pmod u\) where \(u\) is an odd number
Let \(u\) be an odd number, and let \(\ell=\ell(u)\) denote the smallest positive integer with the property \(2^{\ell}\equiv 1\bmod u\). Given a prime \(q\geqslant 3\), the author proves the following estimates concerning the asymptotic behavior of the number \(N_q(x)\) of odd, positive integers \(u\leqslant x\) with \(q\nmid \ell(u)\): \[ {x\over\log^{1/(q-1)}x}\ll N_q(x)\ll {x\over\log^{1/q}x} \qquad\text{for }x\to\infty. \] This is an essential sharpening of theorem 5 in \textit{Z. Franco} and \textit{C. Pomerance} [Math. Comput. 64, 1333-1336 (1995; Zbl 0833.11003)].
prime number theorem for arithmetic progressions
asymptotic results on arithmetic functions
scientific article; zbMATH DE number 1145745