Density of consecutive prime quadratic nonresidues among twin primes (assuming the twin prime conjecture)

Determine, assuming the twin prime conjecture, the asymptotic density among twin prime pairs (n, n+2) of those pairs for which both n and n+2 are quadratic nonresidues modulo a large prime p. Specifically, evaluate or estimate the weighted count S_p(x) = (1/4) ∑_{2 ≤ n ≤ x} (1 − (n/p)) (1 − ((n+2)/p)) Λ(n) Λ(n+2), where (·/p) denotes the Legendre symbol and Λ is the von Mangoldt function, to quantify this density relative to the set of twin primes in F_p.

Background

The paper studies small prime quadratic and kth power residues and nonresidues in arithmetic progressions modulo a large prime p, providing new characteristic function representations and asymptotic existence results. In Section 7, the author lists research problems related to finer distributional properties of residues and nonresidues, including patterns across consecutive integers and densities within special prime constellations.

One such problem asks for the density of consecutive prime quadratic nonresidues among twin prime pairs, conditional on the twin prime conjecture. The author indicates a specific weighted sum involving the von Mangoldt function and Legendre symbols that captures the count of twin primes where both components are quadratic nonresidues, and seeks an evaluation or estimate that would yield the desired density.