Odd moments in the distribution of primes

Abstract

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