Odd moments in the distribution of primes

Montgomery and Soundararajan showed that the distribution of (x+H) - (x), for 0 x N, is approximately normal with mean H and variance H (N/H), when N^ H N^1-. Their work depends on showing that sums Rₖ (h) of k-term singular series are ₖ (-h h + Ah) ^k/2 + Oₖ (h^k/2-1/ (7k) +), where A is a constant and ₖ are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when k is odd, Rₖ (h) h^ (k-1) /2 (h) ^ (k+1) /2. We prove an upper bound with the correct power of h when k = 3, and prove analogous upper bounds in the function field setting when k =3 and k = 5. We provide further evidence for this conjecture in the form of numerical computations.