Abstract

Let p be an odd prime. An integer z is a quadratic residue (respectively, non-residue) of p if there is (respectively, is not) an integer whose square is congruent to z modulo p. It is a theorem going all the way back to Euler that precisely half of the integers in the set {1, 2,..., p-1} are quadratic residues of p, and the study of the various ways that these residues are distributed in this set has fascinated practitioners of number theory ever since. In this talk I will discuss some aspects of the distribution of quadratic residues and and non-residues among the arithmetic progressions in {1, 2,..., p-1}. A typical question among the ones that I will emphasize asks for the sharp asymptotic behavior as p tends to infinity of the number of arithmetic progressions with a fixed constant term, difference, and length which are sets of quadratic residues of p inside {1, 2,..., p-1}. This continues work begun by Harold Davenport in the 1930's.