Residue pattern for infinitely many primes

Let S be a nonempty finite subset of N and theta : S ->{-1, 1} be an arbitrary map (choice of signs for S). We will say that S has residue pattern theta modulo p if (s/p) - theta(s) for all s is an element of S, where (./p) is the Legendre symbol mod p. For a given nonempty finite subset S of N with a choice of signs theta and a real number 1 <= c < 12/11, we obtain an asymptotic formula for the number of primes p of the form p = left perpendicularn(c)right perpendicularsuch that S has residue pattern theta modulo p which also satisfies p equivalent to f ( Mod d) where f,d are integers with 1 <= f <= d and (f,d) = 1. For an irrational alpha > 1 and a real beta >= 0, we also obtain an asymptotic formula for the same but primes p of the form p = left perpendicular alpha n + beta right perpendicular under certain assumption on alpha.

Automated summary

The paper discusses the asymptotic formula for the number of prime numbers p that have a residue pattern with a given finite set S. It also provides a similar formula for primes p that satisfy certain conditions involving an irrational number alpha and a real number beta. The author introduces a mapping function theta and auxiliary conditions to derive these formulas and extends the results under certain assumptions.