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Odd moments in the distribution of primes

Vivian Kuperberg

TL;DR

This work investigates odd moments of the prime-distribution in short intervals via refined control of sums of singular series. Building on Montgomery–Soundararajan’s framework, it analyzes $R_k(h)$, the sum of $k$-term singular series, and conjectures that odd $k$ yield $R_k(h) \asymp h^{(k-1)/2}(\log h)^{(k+1)/2}$, supported by numerical evidence. The paper proves a sharp upper bound for $k=3$ in the integer setting, and establishes analogous results for $k=3,5$ in the function-field setting, including a fifth-moment bound and corollaries for sums of singular series. It also provides numerical data that align with the conjectured growth rates and discusses open problems and toy models to guide future research on the delicate balance of diagonal and off-diagonal contributions in singular-series sums.

Abstract

Montgomery and Soundararajan showed that the distribution of $ψ(x+H) - ψ(x)$, for $0 \le x \le N$, is approximately normal with mean $ \sim H$ and variance $\sim H \log (N/H)$, when $N^δ \le H \le N^{1-δ}$. Their work depends on showing that sums $R_k(h)$ of $k$-term singular series are $μ_k(-h \log h + Ah)^{k/2} + O_k(h^{k/2-1/(7k) + \varepsilon})$, 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.

Odd moments in the distribution of primes

TL;DR

This work investigates odd moments of the prime-distribution in short intervals via refined control of sums of singular series. Building on Montgomery–Soundararajan’s framework, it analyzes , the sum of -term singular series, and conjectures that odd yield , supported by numerical evidence. The paper proves a sharp upper bound for in the integer setting, and establishes analogous results for in the function-field setting, including a fifth-moment bound and corollaries for sums of singular series. It also provides numerical data that align with the conjectured growth rates and discusses open problems and toy models to guide future research on the delicate balance of diagonal and off-diagonal contributions in singular-series sums.

Abstract

Montgomery and Soundararajan showed that the distribution of , for , is approximately normal with mean and variance , when . Their work depends on showing that sums of -term singular series are , where is a constant and are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when is odd, . We prove an upper bound with the correct power of when , and prove analogous upper bounds in the function field setting when and . We provide further evidence for this conjecture in the form of numerical computations.

Paper Structure

This paper contains 13 sections, 35 theorems, 220 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Three-term integer sums: Proof of Theorem \ref{['thm:threetermintegersums']}
  3. Preparing for the proof of Theorem \ref{['threetermintegerreducedresidues']}
  4. Bounding $T_1$: terms with $gx \ge h$
  5. Bounding $T_2$: terms with $gx,gy, gz$ small and $a_2,a_3$ large
  6. Bounding $T_3$: terms with $gx,gy, gz$ small and each $a_i$ small
  7. Function Field Analogues: Proof of Theorem \ref{['thm:mvfuncfield']}
  8. The analog of MontgomeryVaughanReducedResidues in the function field setting
  9. The fifth moment of reduced residues in the function field setting
  10. Proof of Theorem \ref{['thm:funcfieldfifth']}
  11. Proof of Corollary \ref{['cor:singseriesff']}: Bounds on $R_k(q^h)$
  12. Numerical Evidence for Odd Moments
  13. Toy Models and Open Problems

Key Result

Theorem 1.2

For $h \ge 4$ and $R_3$ defined in eq:rkh,

Figures (14)

  • Figure 1: $\frac{1}{6} R_3(h)$ for $3 \le h \le 20000$
  • Figure 2: $\frac{1}{6} R_3(h) - Ah(\log h)^2$ for $3 \le h \le 20000$
  • Figure : $\delta = 0.25$
  • Figure : $\delta = 0.25$
  • Figure : $\delta = 0.25$
  • ...and 9 more figures

Theorems & Definitions (67)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 57 more