Prove or Disprove the Gaussian Prime Count Estimate (and Determine Error Term)

Prove or disprove the asymptotic estimate T_G(r) ~ (1/2) r^2 / ln r for T_G(r), the number of Gaussian primes a+bi in Z[i] with a, b ≥ 0 and sqrt(a^2 + b^2) ≤ r; if the estimate holds, determine an accompanying error term.

Background

In Section 4, the authors paper primes in the Gaussian integers Z[i] and define T_G(r) as the count of Gaussian primes in the first quadrant (a, b ≥ 0) within the Euclidean circle of radius r. Based on empirical data, they suggest the estimate T_G(r) ~ (1/2) r^2 / ln r, analogous to the classical Prime Number Theorem in Z.

Open Problem 4.4 requests a rigorous resolution of this estimate, including an error term if the estimate is true.