Odd moments in the distribution of primes
Year: 2025
Type: article
Abstract: Montgomery and Soundararajan showed that the distribution of ψ(x+H) - ψ(x), for 0 $\leq$ x $\leq$ N, is approximately normal with mean ~ H and variance ~ H log(N/H), when N$^δ$ $\leq$ H $\leq$ 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$^{k/2-1/(7k)+ε}$), 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.
Source: Algebra & Number Theory
Author: Vivian Kuperberg
Institution: ETH Zurich
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funder: National Science Foundation
OpenAlex ID: W3196737800