@article{22191,
  abstract     = {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.},
  author       = {Kuperberg, Vivian Zieve},
  issn         = {1944-7833},
  journal      = {Algebra & Number Theory},
  number       = {4},
  pages        = {617--666},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Odd moments in the distribution of primes}},
  doi          = {10.2140/ant.2025.19.617},
  volume       = {19},
  year         = {2025},
}