article published yes Vivian Zieve Kuperberg author c3bac823-112d-11f0-a3f5-c264f852e697 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. Mathematical Sciences Publishers2025 eng 1937-0652 1944-783310.2140/ant.2025.19.617 194617-666 yes V.Z. Kuperberg, Algebra &#38; Number Theory 19 (2025) 617–666. Kuperberg VZ. 2025. Odd moments in the distribution of primes. Algebra &#38; Number Theory. 19(4), 617–666. V. Z. Kuperberg, “Odd moments in the distribution of primes,” <i>Algebra &#38; Number Theory</i>, vol. 19, no. 4. Mathematical Sciences Publishers, pp. 617–666, 2025. Kuperberg VZ. Odd moments in the distribution of primes. <i>Algebra &#38; Number Theory</i>. 2025;19(4):617-666. doi:<a href="https://doi.org/10.2140/ant.2025.19.617">10.2140/ant.2025.19.617</a> Kuperberg, Vivian Zieve. “Odd Moments in the Distribution of Primes.” <i>Algebra &#38; Number Theory</i>, vol. 19, no. 4, Mathematical Sciences Publishers, 2025, pp. 617–66, doi:<a href="https://doi.org/10.2140/ant.2025.19.617">10.2140/ant.2025.19.617</a>. Kuperberg, V. Z. (2025). Odd moments in the distribution of primes. <i>Algebra &#38; Number Theory</i>. Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/ant.2025.19.617">https://doi.org/10.2140/ant.2025.19.617</a> Kuperberg, Vivian Zieve. “Odd Moments in the Distribution of Primes.” <i>Algebra &#38; Number Theory</i>. Mathematical Sciences Publishers, 2025. <a href="https://doi.org/10.2140/ant.2025.19.617">https://doi.org/10.2140/ant.2025.19.617</a>. 221912026-06-29T12:56:09Z2026-07-14T10:52:02Z