Quadratic Residues and Non-residues in Arithmetic Progression
Abstract
Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set T ∈ S : T ⊂eq 1,...,p-1 and T is a set of quadratic residues (respectively, non-residues) of p. When S is constructed in various ways from the set of all arithmetic progressions of nonnegative integers, we determine the sharp asymptotic behavior of c(p) as p +∞. Generalizations and variations of this are also established, and some problems connected with these results that are worthy of further study are discussed.
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