Odd moments and adding fractions

Technical Summary of "Odd Moments and Adding Fractions" by Thomas F. Bloom and Vivian Kuperberg

Problem Statement
This paper addresses two primary problems in analytic number theory concerning "odd moments," which have historically received less attention than their even counterparts. The first problem concerns the distribution of coprime residues in short intervals. Specifically, the authors investigate the $k$-th central moment $M_k(q, h)$ of the proportion of integers in an interval of length $h$ that are coprime to a modulus $q$. While Montgomery and Vaughan established sharp bounds for even $k$, their bounds for odd $k$ were not optimal. The authors aim to confirm a conjecture by Montgomery and Vaughan that the bound for odd $k$ should be significantly stronger, specifically reducing the exponent of the secondary term from $\frac{k}{2} - \frac{1}{7k}$ to $\frac{k-1}{2}$.

The second problem involves the distribution of prime $k$-tuples. The authors study the average of the "refined singular series" $S_0(d_1, \dots, d_k)$ over distinct $k$-tuples in a range $[1, h]$, denoted as $R_k(h)$. For even $k$, Montgomery and Soundararajan provided an asymptotic formula. For odd $k$, previous bounds were weaker than the conjectured growth rate of $h^{(k-1)/2} (\log h)^{(k+1)/2}$. The paper seeks to prove near-optimal upper bounds for $R_k(h)$ when $k$ is odd.

Methodology and Key Ingredient
The central innovation of the paper is a new, near-optimal upper bound for the number of solutions to a rational linear equation with an odd number of summands. This result is of independent interest and serves as the primary engine for the main theorems.

The authors prove a bound for the number of solutions to:
$$\frac{a_1}{q_1} + \dots + \frac{a_k}{q_k} \in \mathbb{Z}$$
where $k$ is an odd integer, $\gcd(a_i, q_i) = 1$, $|a_i| \le n$, and $q_i \le Q$. The bound obtained is $\ll (\log Q)^{O(1)} n^{(k+1)/2} Q^{(k-1)/2}$. This represents a saving of a factor of $(n/Q)^{1/2}$ compared to the "trivial" bound derived from Cauchy–Schwarz and even-$k$ results.

To achieve this, the authors employ the technique of relative greatest common divisors. This method parameterizes a tuple of integers $(q_1, \dots, q_k)$ using a set of $2^k$ integers $(g_I)_{I \subseteq \{1, \dots, k\}}$ such that $q_i = \prod_{I \ni i} g_I$ and $\gcd(g_I, g_J) = 1$ unless $I \subseteq J$ or $J \subseteq I$. This parameterization allows for a refined counting of solutions by decomposing the denominators and analyzing the constraints on the numerators modulo specific divisors. The proof involves a careful partitioning of the indices into sets $X$ and $Y$ to minimize the product of the parameterized components, followed by an inductive argument to fix variables step-by-step while controlling the divisor function growth.

Main Results

1. Distribution of Reduced Residues (Theorem 1):
   The authors prove that for any odd $k \ge 3$, $q \ge 1$, and $h \ge 2$:
   $$M_k(q, h) \ll (\log h)^{O(1)} \left( \frac{\phi(q)}{q}h + \left(\frac{\phi(q)}{q}h\right)^{\frac{k-1}{2}} \right) q$$
   This confirms the conjecture of Montgomery and Vaughan up to a power of $\log h$. The authors note that while they cannot prove an asymptotic formula (due to the lack of a dominant "diagonal term" as seen in the even case), they believe the main term is of the order indicated, making the bound essentially best possible.

2. Average of the Refined Singular Series (Theorem 2):
   For any odd integer $k \ge 3$ and $h \ge 2$, the average of the refined singular series satisfies:
   $$R_k(h) \ll (\log h)^{O(1)} h^{\frac{k-1}{2}}$$
   This matches the conjectured growth rate up to logarithmic factors. The proof relies on expressing $R_k(h)$ as a linear combination of moments $V_j(q, h)$ (related to $M_j$) using an inclusion-exclusion argument over partitions of the $k$-tuple, and then applying Theorem 1.

3. Adding Fractions (Theorem 3 and 4):
   The paper establishes the near-optimal bound for the number of solutions to the rational linear equation mentioned in the methodology. This result generalizes previous work by Blomer, Brüdern, Heath-Brown, and others, which were restricted to specific cases (e.g., $n=Q$, $k=3$, or even $k$).

Significance and Claims
The paper claims that the bound for adding fractions (Theorem 3) is the first result to address the case of general $n$ and general odd $k$, a generality crucial for the applications to moments. The authors state that this bound is the "driving force" behind the proofs of Theorems 1 and 2.

Regarding the significance of the results on moments:

• The authors confirm the conjectured shape of the bounds for odd moments, which had previously been an open problem with weaker known bounds.
• They highlight that the three problems of estimating odd moments for primes in short intervals, $M_k(q, h)$ for odd $k$, and $R_k(h)$ for odd $k$ are "strongly linked."
• The authors are modest about the limitations: they do not provide asymptotic formulas for $M_k(q, h)$ or $R_k(h)$ when $k$ is odd, noting that the leading constants are not yet clear and that the problem of determining the exact main term remains open. They suggest that an asymptotic understanding of $R_k(h)$ is essentially equivalent to an asymptotic understanding of $M_k(q, h)$ and the moments of primes in short intervals.

The paper concludes by noting that while their bounds are near-optimal (losing only logarithmic factors), improving the error terms for even moments or obtaining precise asymptotics for odd moments remains a challenge, potentially requiring a deeper understanding of non-degenerate solutions to the rational linear equations discussed.