Residue pattern for infinitely many primes

Abstract

Let [Formula: see text] be a nonempty finite subset of [Formula: see text] and [Formula: see text] be an arbitrary map (choice of signs for [Formula: see text]). We will say that [Formula: see text] has residue pattern [Formula: see text] modulo [Formula: see text] if [Formula: see text], where [Formula: see text] is the Legendre symbol mod [Formula: see text]. For a given nonempty finite subset [Formula: see text] of [Formula: see text] with a choice of signs [Formula: see text] and a real number [Formula: see text], we obtain an asymptotic formula for the number of primes [Formula: see text] of the form [Formula: see text] such that [Formula: see text] has residue pattern [Formula: see text] modulo [Formula: see text] which also satisfies [Formula: see text] where [Formula: see text] are integers with [Formula: see text] and [Formula: see text]. For an irrational [Formula: see text] and a real [Formula: see text], we also obtain an asymptotic formula for the same but primes [Formula: see text] of the form [Formula: see text] under certain assumption on [Formula: see text].