Ratios of Small Integers in Multiplicative Subgroups of Residue Rings

Abstract

We obtain a new estimate, on average, over primes p in a dyadic interval, on the number of integers u, v of absolute value at most h which fall in a given multiplicative subgroup of the residue ring modulo p. Our result is based on an application of a large sieve inequality. We also present some numerical results comparing the new bound with several previously known bounds.

KEYWORDS:

• ratios of small integers
• multiplicative subgroups
• residue classes
• character sums
• large sieve

2000 AMS Subject Classification:

• 11A07
• 11L40
• 11N36

Funding

The author was supported in part by the ARC Grants DP130100237 and DP140100118.