Odd moments in the distribution of primes

TL;DR

This paper investigates the distribution of prime-related functions, providing new bounds and conjectures for odd moments, supported by theoretical proofs and numerical evidence, extending understanding of prime distribution fluctuations.

Contribution

It introduces a conjecture on the behavior of odd moments of prime distributions and proves upper bounds for specific cases, including in the function field setting.

Findings

Upper bounds for third moments in number fields

Confirmation of conjectured behavior through numerical data

Extension of bounds to function fields for k=3 and 5

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 $\sim H \log (N/H)$, 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 $\mu_k(-h \log h + Ah)^{k/2} + O_k(h^{k/2-1/(7k) + \varepsilon})$, 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, $R_k(h) \asymp h^{(k-1)/2}(\log h)^{(k+1)/2}$. 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.