Initial blocks and multiplicity of representations of even integers as sums of two Gaussian primes

We study representations of an even integer N as a sum of two Gaussian primes with positive real parts and opposite imaginary parts, N = (a + bi) + (N − a − bi). We prove two results. First, adapting Rankin's covering construction, we show that there are infinitely many N = 2n for which n² + b² is composite for every 1 ≤ b ≤ c·log N·log₂N·log₄N/(log₃N)²; thus the symmetric representations are blocked on an initial interval of heights at Rankin's scale. Second, combining a combinatorial lower bound for the total number of representations, a uniform upper-bound sieve and the moments of the singular series, we show that a positive proportion of even N have at least ≫ N²/(log N)² representations. We also record that the existence of a representation, even a symmetric one, for almost all even N follows from a theorem of Merikoski on Gaussian primes in sparse sets.