A problem in comparative order theory

Abstract

Write \(\mathrm {ord}_p(\cdot )\) for the multiplicative order in \({\mathbb {F}}_p^{\times }\). Recently, Matthew Just and the second author investigated the problem of classifying pairs \(\alpha , \beta \in {\mathbb {Q}}^{\times }\setminus \{\pm 1\}\) for which \(\mathrm {ord}_p(\alpha ) > \mathrm {ord}_p(\beta )\) holds for infinitely many primes p. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for \(\alpha ,\beta \) to be order-dominant. Via the large sieve, we show that almost all integer pairs \(\alpha ,\beta \) satisfy our condition, with a power savings on the size of the exceptional set.