Odd moments in the distribution of primes

Abstract

Montgomery and Soundararajan showed that the distribution of $ψ(x+H) - ψ(x)$, for $0 \le x \le N$, is approximately normal with mean $ \sim H$ and variance $\sim H \log (N/H)$, when $N^δ \le H \le N^{1-δ}$. Their work depends on showing that sums $R_k(h)$ of $k$-term singular series are $μ_k(-h \log h + Ah)^{k/2} + O_k(h^{k/2-1/(7k) + \varepsilon})$, where $A$ is a constant and $μ_k$ are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when $k$ is odd, $R_k(h) \asymp h^{(k-1)/2}(\log h)^{(k+1)/2}$. 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.