Odd moments in the distribution of primes

Montgomery and Soundararajan showed that the distribution of ψ(x+H) - ψ(x), for 0 ≤ x ≤ N, is approximately normal with mean ~ H and variance ~ H log (N/H), when Nδ ≤ H ≤ N1-δ. Their work depends on showing that sums Rₖ(h) of k-term singular series are μₖ(-h log h + Ah)k/2 + Oₖ(hk/2-1/(7k) + ε), where A is a constant and μₖ 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(log 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.