The exceptional set in the polynomial Goldbach problem

For each natural number N , let R(N) denote the number of representations of N as a sum of two primes. Hardy and Littlewood proposed a plausible asymptotic formula for R(2N) and showed, under the assumption of the Riemann Hypothesis for Dirichlet Lfunctions, that the formula holds “on average” in a certain sense. From this they deduced (under ERH) that all but O (x1/2+ ) of the even natural numbers in [1, x] can be written as a sum of two primes. We generalize their results to the setting of polynomials over a finite field. Owing to Weil’s Riemann Hypothesis, our results are unconditional.