Exponential sums over Möbius convolutions with applications to partitions

This paper investigates partitions defined by Möbius convolution weights by establishing upper bounds via exponential sums, deriving asymptotic formulas for odd and even partitions, and relating major arc contributions to the zeros of the Riemann zeta-function.

The Gist

Imagine you have a giant pile of Lego bricks. Your job is to build towers using these bricks, but there's a catch: you can only use specific types of bricks, and some of them have a "special rule" attached to them.

This paper is about a very specific, mathematical game of Lego. The authors are trying to figure out how many different towers you can build with a certain number of bricks, but with a twist involving prime numbers and a mathematical concept called the Möbius function.

Here is the story of their discovery, broken down into simple parts:

1. The Game: Blue vs. Red Bricks

In the world of numbers, every number can be broken down into prime factors (like 12 = 2 × 2 × 3).

• Blue Bricks (Set B): These are numbers made of an even number of prime factors (like 6 = 2×3, or 10 = 2×5).
• Red Bricks (Set R): These are numbers made of an odd number of prime factors (like 2, 3, 5, 7, or 30 = 2×3×5).

The authors created a rule for building towers (partitions):

1. You can use as many Blue bricks as you want.
2. You can use Red bricks, but only once each. You can't use the Red "3" twice in the same tower.

They then asked a big question: As the number of bricks gets huge, do we end up with roughly the same number of "Even" towers (towers with an even number of Red bricks) as "Odd" towers (towers with an odd number of Red bricks)?

2. The Mystery: The Tug-of-War

In math, this is like a tug-of-war.

• If you have 10 bricks, there are 8 "Even" towers and 7 "Odd" towers.
• The difference is small, but it exists.
• The authors wanted to know: As the pile of bricks grows to infinity, does this difference disappear? Do the two sides become perfectly balanced?

The Answer: Yes! The paper proves that as the numbers get huge, the number of Even towers and Odd towers becomes essentially equal. The difference between them is so tiny compared to the total number of towers that it's like a single grain of sand in a desert.

3. The Tool: The Circle Method (The Magic Lens)

To solve this, the authors used a famous mathematical tool called the Hardy-Littlewood Circle Method.

Imagine you are trying to count all the possible towers, but the number is so big you can't count them one by one. Instead, you use a "Magic Lens" (a complex mathematical formula) that turns the problem into a journey around a circle.

• The Major Arcs: These are the "easy" parts of the circle where the math behaves nicely. This is where the main answer lives.
• The Minor Arcs: These are the "chaotic" parts of the circle where the numbers act weirdly. To prove the answer, you have to show that the chaos here is so small it doesn't matter.

4. The Challenge: The "Twisted" Sum

The real difficulty was in the Minor Arcs (the chaotic part). The authors had to prove that a specific type of math sum (an "exponential sum") involving their special "Möbius" rules didn't explode out of control.

Think of it like trying to predict the weather. The "Major Arcs" are the predictable sunny days. The "Minor Arcs" are the stormy, unpredictable days. The authors had to prove that even during the worst storms, the wind wouldn't blow their whole theory away. They developed new, sharper tools (bounds) to measure these storms and showed they were weak enough to ignore.

5. The Big Reveal: The Ghosts of Zeros

One of the most beautiful parts of the paper is what happens on the "Major Arcs" (the sunny days).

When they calculated the main part of the answer, they found a surprising connection to the Riemann Zeta Function. This is a famous mathematical object that holds the secrets of prime numbers. Its "non-trivial zeros" are like hidden ghosts that haunt the number line.

The authors found an explicit formula (a precise recipe) that says:

"The number of these special towers is determined by a sum involving these 'ghost' zeros."

It's like finding that the height of a mountain is actually determined by the vibrations of invisible strings deep underground. This connects the simple act of stacking blocks to the deepest mysteries of prime numbers.

Summary

• The Problem: Can we balance two types of number towers based on prime factors?
• The Method: They used a mathematical "lens" (Circle Method) to separate the easy parts from the hard parts.
• The Breakthrough: They proved the hard parts (chaos) are small enough to ignore and the easy parts (order) are perfectly balanced.
• The Surprise: The solution is deeply connected to the mysterious "ghosts" (zeros) of the Riemann Zeta function.

In short, the authors showed that in the infinite world of numbers, even and odd arrangements of these special blocks eventually cancel each other out, leaving a perfect, balanced symmetry that is secretly governed by the most famous unsolved problem in mathematics.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the asymptotic behavior of weighted partitions of a positive integer $n$. Specifically, it addresses a question regarding the parity of "admissible" partitions defined by the prime factorization of their parts.

• Admissible Partitions: A partition is admissible if every part is a square-free integer. Parts with an even number of prime factors are termed "blue" ($B$), and those with an odd number are "red" ($R$). Furthermore, in an admissible partition, each red part can appear at most once.
• The Core Question: Let $E(n)$ be the number of even admissible partitions (even number of red parts) and $O(n)$ be the number of odd admissible partitions. The authors ask: Is $E(n) \sim O(n)$ as $n \to \infty$?
• Mathematical Formulation: This problem is modeled using a generating function with weights derived from the Möbius function $\mu(n)$. The generating function is:
  $$\Psi_w(z) = \sum_{n=1}^\infty p_w(n) z^n = \prod_{m=1}^\infty (1 - z^m)^{-w(m)}$$
  For the specific case of interest, the weight is $w(n) = \mu(n)$. The difference between even and odd partitions is given by $p_\mu(n) = E(n) - O(n)$. Proving $E(n) \sim O(n)$ is equivalent to showing that $p_\mu(n)$ grows significantly slower than the total number of partitions, specifically that $\log p_\mu(n) = o(\sqrt{n})$.

The authors generalize this to Möbius convolutions $w(n) = \mu_k(n) = (\mu * \mu * \dots * \mu)(n)$ (convolved $k$ times), where the Dirichlet series is $1/\zeta(s)^k$.

2. Methodology

The paper employs the Hardy-Littlewood Circle Method, a standard technique in analytic number theory for estimating coefficients of generating functions. The strategy involves:

1. Generating Function Analysis: Expressing the partition function $p_{\mu_k}(n)$ via Cauchy's integral formula:
   $$p_{\mu_k}(n) = \rho^{-n} \int_{-1/2}^{1/2} \exp(\Phi_{\mu_k}(\rho e(\theta)) - 2\pi i n \theta) \, d\theta$$
   where $\Phi_{\mu_k}(z) = \sum_{j=1}^\infty \sum_{n=1}^\infty \frac{\mu_k(n)}{j} z^{jn}$.
2. Arc Decomposition: The integration interval is split into Major Arcs ($\mathcal{M}$) and Minor Arcs ($\mathcal{m}$).
   • Major Arcs: Neighborhoods of rational numbers $a/q$ with small denominators. Here, the behavior is governed by the poles of the Riemann zeta-function.
   • Minor Arcs: The remainder of the interval. Here, the goal is to show the contribution is negligible compared to the main term.
3. Exponential Sum Bounds: The estimation of the minor arcs relies heavily on bounding exponential sums twisted by Möbius convolutions:
   $$S_{\mu_k}(X, \alpha) = \sum_{n \le X} \mu_k(n) e(n\alpha)$$
   The authors develop new bounds for these sums using Type I and Type II estimates (based on work by Koukoulopoulos) and the Dirichlet hyperbola method.
4. Explicit Formula Derivation: For the Major Arcs, the authors perform a contour integration of the Dirichlet series associated with $\Phi_{\mu_k}$, shifting the path to the left to capture residues from the poles of $\zeta(s)$. This yields an explicit formula involving the non-trivial zeros of the Riemann zeta-function.

3. Key Contributions and Results

A. Asymptotic Bounds for Weighted Partitions

The primary result regarding partitions is Theorem 1.2, which establishes a sharp upper bound for the logarithm of the partition function weighted by the convolution $\mu * \mu$:
$$\log p_{\mu * \mu}(n) = O_B\left( \frac{\sqrt{n}}{(\log n)^B} \right)$$
for any fixed $B > 0$.

• Significance: Since the total number of partitions $p(n)$ grows as $\exp(\pi \sqrt{2n/3})$, this result implies that $p_{\mu * \mu}(n)$ is exponentially smaller than $p(n)$.
• Corollary (Theorem 1.1): As a direct consequence, $E(n) - O(n) = p_\mu(n)$ is negligible compared to $E(n) + O(n)$. Thus, the number of even admissible partitions is asymptotic to the number of odd admissible partitions as $n \to \infty$.

B. Bounds on Exponential Sums

To achieve the partition bounds, the authors prove new estimates for exponential sums twisted by Möbius convolutions.

• Theorem 1.4: Provides a bound for $S_\mu(X, \alpha)$ involving the denominator $q$ of the rational approximation of $\alpha$.
• Theorem 1.5: Generalizes this to $S_{\mu_k}(X, \alpha)$ for any $k \ge 2$. The bound depends on sequences $a_k, b_k, c_k$ defined recursively, yielding:
  $$S_{\mu_k}(X, \alpha) \ll_\epsilon X^{a_k+\epsilon} + \frac{X (\log X)^{k^2}}{q^{b_k}} + X^{c_k} q^{1-c_k} (\log X)^{k^2}$$
• Theorems 1.6 & 1.7: Extend these results to convolutions involving the characteristic functions of square-full and square-free integers.

C. Explicit Formula for the Generating Function

Theorem 1.3 provides an explicit formula for $\Phi_{\mu * \mu}(\rho e(\theta))$ on the major arcs. It expresses the function as a sum of:

1. A main term involving $\log X$ and $\log(2\pi)$.
2. A sum over the non-trivial zeros $\rho$ of the Riemann zeta-function:
   $$\sum_{|\text{Im}(\rho)| < T_\nu} f(X, \theta, \rho)$$
3. A sum over the trivial zeros (negative even integers).
   This connects the partition problem directly to the distribution of the zeros of $\zeta(s)$, similar to the classical explicit formula for the Chebyshev function $\psi(x)$.

4. Technical Innovations

• Handling Negative Weights: Unlike previous works on weighted partitions (e.g., by Meinardus or Debryune/Tenenbaum) which often assume non-negative weights, this paper deals with weights ($\mu_k$) that oscillate between positive and negative values. This prevents the existence of a standard "main term" in the asymptotic expansion, requiring the authors to prove an upper bound rather than an asymptotic equivalence for the difference $E(n) - O(n)$.
• Refined Exponential Sum Estimates: The authors improve upon standard Type I/II estimates by carefully handling the support of the variables in the Dirichlet convolution and utilizing specific properties of the Möbius function in arithmetic progressions (via Lemma 2.6, a consequence of the Siegel-Walfisz theorem).
• Unconditional Zeta Bounds: The proof of the explicit formula (Theorem 1.3) is conducted unconditionally. The authors use a result by Ramachandra, Sankaranarayanan, and Inoue (Lemma 2.8) to bound $1/\zeta(s)$ in the critical strip, avoiding the need to assume the Riemann Hypothesis.

5. Significance

• Resolution of a Parity Question: The paper definitively answers the question of whether even and odd admissible partitions are equidistributed, showing they are asymptotically equal.
• Bridging Partition Theory and Zeta Zeros: It provides a concrete example of how the zeros of the Riemann zeta-function influence the asymptotic behavior of partition functions with arithmetic weights.
• Methodological Advancement: The techniques developed for bounding exponential sums with Möbius convolutions are of independent interest and can be applied to other problems involving arithmetic functions in additive number theory. The paper demonstrates that even when weights are highly oscillatory (like $\mu$), the Circle Method can still yield strong upper bounds, provided the exponential sums are sufficiently controlled.

In summary, this work successfully applies advanced analytic number theory techniques to solve a specific combinatorial problem, revealing deep connections between partition functions, the Möbius function, and the Riemann zeta-function.