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 N δ ≤ H ≤ N 1−δ . Their work depends on showing that sums Rk (h) of k-term singular series are µk (−h log h + Ah) k/2 + Ok (h k/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.