On the discrete convolution of the Liouville and Möbius functions

This paper investigates the discrete convolution of the Liouville function, deriving an explicit formula for its weighted averages to reveal new properties regarding its Dirichlet and power series as well as its generalizations to multiple factors.

The Gist

Imagine you are a chef trying to bake the perfect cake, but you don't have a recipe. Instead, you have a bag of ingredients that behave very strangely. Some ingredients are "good" (let's call them Liouville ingredients), and some are "bad" (let's call them Möbius ingredients).

The problem is that these ingredients are chaotic. If you pick one, you can't predict if the next one will be good or bad. Sometimes they cancel each other out; sometimes they amplify each other. Mathematicians have been trying to figure out what happens when you mix these ingredients together in specific ways.

This paper is like a new, high-tech cookbook that finally gives us a way to predict the flavor of the cake, even with these chaotic ingredients.

Here is the breakdown of what the authors (Marco, Alessandro, and Alessandro) did, using simple analogies:

1. The Chaotic Ingredients (The Liouville and Möbius Functions)

In math, there are two famous "chaotic" number functions:

• The Liouville Function ($\lambda$): Think of this as a coin flip for every number. If a number is made of an even number of prime factors (like $6 = 2 \times 3$), it's a "Heads" (+1). If it's an odd number of factors (like$2$), it's a "Tails" (-1).
• The Möbius Function ($\mu$): This is a similar coin flip, but if a number has a repeated factor (like $12 = 2 \times 2 \times 3$), it gets disqualified (0).

These functions jump around wildly. They don't follow a pattern you can easily see with your naked eye.

2. The Mixing Bowl (Discrete Convolution)

The authors are interested in what happens when you mix these ingredients. Specifically, they look at Sums.
Imagine you want to make a number $N$ by adding two smaller numbers together ($m_1 + m_2 = N$).

• You take the "flavor" of $m_1$ and the "flavor" of $m_2$.
• You multiply them together.
• You do this for every possible pair that adds up to $N$.

This is called a Discrete Convolution. It's like asking: "If I mix every possible pair of ingredients that sum to $N$, what is the total flavor of the mixture?"

In the world of prime numbers, there is a famous unsolved puzzle called Goldbach's Conjecture. It asks if every even number greater than 2 can be made by adding two prime numbers. This paper asks a similar question: "If we mix these chaotic Liouville ingredients, does the total flavor cancel out to zero, or does it leave a residue?"

3. The Magic Recipe (Explicit Formulas)

The authors' main achievement is creating a Magic Recipe (an "Explicit Formula").

Before this paper, if you wanted to know the total flavor of a mixture, you had to count every single pair one by one. That takes forever and is prone to error.
The authors found a shortcut. They realized that the "flavor" of the mixture is directly connected to the Riemann Zeta Function ($\zeta$).

Think of the Riemann Zeta Function as a Master Map of the universe of numbers. It has hidden "landmarks" called Zeros (specifically, the non-trivial zeros). These zeros are like the secret coordinates that determine how the chaotic ingredients behave.

The authors' formula says:

"To find the total flavor of the mixture, you don't need to count every pair. Instead, you just need to look at the coordinates of these secret landmarks (the Zeros) and plug them into this specific equation."

4. The "Weighted" Approach

The paper doesn't just look at simple sums; it looks at Weighted Averages.
Imagine you aren't just counting the ingredients; you are weighing them. Maybe the ingredients near the start of the list are heavier than the ones at the end.
The authors developed a flexible tool that works no matter how you weigh the ingredients, as long as the weights follow a few simple rules (like being smooth and not having sharp, jagged edges).

5. The Big Results

Using this new recipe, they discovered three cool things:

1. The Formula Works: They proved that if you assume the famous Riemann Hypothesis (a big guess that all the secret landmarks are in a straight line), you can write down a perfect formula for these sums.
2. The "Natural Boundary": They showed that while you can predict the flavor for most numbers, there is a "wall" at a certain point (the line $Re(s) = 1$) where the pattern becomes so chaotic that you can't predict anything beyond it. It's like a fog bank; you can see clearly up to a point, but then the map dissolves.
3. Scaling Up: They showed this recipe works not just for mixing two ingredients, but for mixing any number of ingredients (3, 4, 100, etc.).

The Takeaway

In simple terms, this paper is a breakthrough in understanding how chaotic number patterns interact.

• Old way: Count, count, count, and hope for the best.
• New way: Use the "Master Map" (Riemann Zeta) and the "Secret Landmarks" (Zeros) to calculate the result instantly.

It's like going from manually counting every grain of sand on a beach to having a satellite that can tell you exactly how much sand is there based on the shape of the coastline. It doesn't solve the ultimate mystery of why primes are the way they are, but it gives us a powerful new telescope to see them more clearly.

Technical Summary

Here is a detailed technical summary of the paper "On the Discrete Convolution of the Liouville and Möbius Functions" by Marco Cantarini, Alessandro Gambini, and Alessandro Zaccagnini.

1. Problem Statement

The paper investigates the arithmetic properties of the discrete convolution of the Liouville function $\lambda(n)$ and the Möbius function $\mu(n)$. Specifically, it focuses on the sum:
$$S(N) := \sum_{m_1 + m_2 = N} \lambda(m_1)\lambda(m_2)$$
and its generalization to $d$ factors:
$$S_d(N) := \sum_{m_1 + \dots + m_d = N} \lambda(m_1)\dots\lambda(m_d)$$

This problem is analogous to the Goldbach conjecture, which concerns the convolution of the von Mangoldt function $\Lambda(n)$. While the Goldbach problem seeks to prove $R(N) > 0$ for even $N$, the Liouville/Möbius convolution problem asks for the asymptotic behavior of $S(N)$. Previous work (Corrádi & Kátai, Mangerel) has established that $S(N) = o(N)$ under certain hypotheses or for large $N$.

The primary goal of this paper is to derive explicit formulas for weighted averages of these convolution sums. These formulas aim to provide a deeper understanding of the distribution of these sums, their connection to the non-trivial zeros of the Riemann zeta function $\zeta(s)$, and the analytic properties of their associated Dirichlet and power series.

2. Methodology

The authors employ a combination of analytic number theory techniques, specifically:

• Explicit Formulas via Perron's Formula: The core strategy involves applying Perron's formula to the Dirichlet series associated with the Liouville function, $\frac{\zeta(2s)}{\zeta(s)}$.
• **Extension to $0 < x < 1$:** A crucial technical innovation is the derivation of explicit formulas for the summatory function$L(x) = \sum_{n \le x} \lambda(n)$that are valid for **all**$x > 0$, not just$x \ge 1$. This is necessary because the convolution integral$\int_0^x L(y)L(x-y)dy$involves values of$L$ near 0.
• Laplace Convolution: The authors study the Laplace convolution of $L(x)$ with itself, denoted $C_\lambda(x) = \int_0^x L(y)L(x-y)dy$. They show that this integral corresponds to the Cesàro average of order 1 of $S(n)$.
• General Weighted Averages: Utilizing a general framework established in a previous work [6], the authors derive a formula for weighted sums of the form $\sum \lambda(n)\lambda(m)f(\frac{n+m}{\eta})$. This involves integrating the explicit formula of $L(x)$ against a weight function $f$.
• Assumptions: The main results rely on two standard conjectures in analytic number theory:
  1. Riemann Hypothesis (RH): All non-trivial zeros of $\zeta(s)$ lie on the critical line $\text{Re}(s) = 1/2$.
  2. Simplicity of Zeros (SZ): All non-trivial zeros are simple (multiplicity 1).
  3. Conjecture (SZ): A bound on the sum of reciprocals of the derivative of $\zeta$ at the zeros, $\sum_{0<\gamma<T} \frac{1}{|\zeta'(\rho)|} \ll T(\log T)^{1/2}$.

3. Key Contributions and Results

A. Explicit Formula for the Convolution Sum

The paper establishes a truncated explicit formula for the weighted average of $S(n)$. Under RH and SZ, for a suitable weight function $f$ and parameter $\eta$:
$$\sum_{n \le \eta b} \sum_{m \le \eta b - n} \lambda(n)\lambda(m) f\left(\frac{m+n}{\eta}\right) = \text{Main Terms} + \text{Oscillatory Terms} + \text{Error}$$
The formula consists of:

1. Main Term: A term involving $\frac{\pi \eta}{8 \zeta(1/2)^2} \int f''(w) w^2 dw$.
2. Oscillatory Terms (Single Zeros): A sum over non-trivial zeros $\rho$ of $\zeta(s)$, involving terms like $\frac{\zeta(2\rho)}{\zeta'(\rho)} \Gamma(\rho) \eta^{\rho+1/2}$.
3. Oscillatory Terms (Double Zeros): A double sum over pairs of zeros $\rho_1, \rho_2$, involving products of $\zeta(2\rho_i)$ and Gamma functions.
4. Error Term: Controlled by $O_\epsilon(\eta^{1/2+\epsilon})$ and integrals involving the second derivative of the weight function.

B. Analytic Continuation of Dirichlet Series

A significant theoretical result is the analytic continuation of the Dirichlet series associated with $S(n)$:
$$D(s) = \sum_{n=1}^\infty \frac{S(n)}{n^s}$$

• Result: Assuming RH and SZ, $D(s)$ admits an analytic continuation to the half-plane $\text{Re}(s) > 1$.
• Natural Boundary: If the imaginary parts of the non-trivial zeros of $\zeta(s)$ are linearly independent over $\mathbb{Q}$, the line $\text{Re}(s) = 1$ acts as a natural boundary for the series. This implies the series cannot be analytically continued beyond this line due to the density of singularities generated by the sums of the zeros ($\gamma_1 + \gamma_2$).

C. Power Series and Generalizations

• Power Series: The authors derive an explicit formula for the power series $\sum S(n) e^{-ny}$, showing its asymptotic behavior as $y \to 0^+$.
• Möbius Function: Parallel results are derived for the convolution of the Möbius function, $S^*(n) = \sum \mu(m_1)\mu(m_2)$. Since $\Lambda(n)$ can be expressed via $\mu(n)$, these results provide indirect information about the Goldbach-type sums involving $\Lambda(n)$.
• Arbitrary Number of Factors ($d$-term): The methodology is generalized to the $d$-fold convolution $S_d(N)$. The authors provide explicit formulas for the Laplace convolution of $d$ copies of $L(x)$, showing that the main term scales with $x^d$ and the oscillatory terms involve sums over $d$ zeros (though the paper explicitly details the structure for $d=2$ and generalizes the integral representation for arbitrary $d$).

4. Significance

• Bridging Additive Problems: The paper connects the study of the Liouville/Möbius convolution (often viewed through the lens of the Chowla conjecture) with classical additive problems like Goldbach's conjecture. It demonstrates that the techniques used for $\Lambda(n)$ can be adapted to $\lambda(n)$ and $\mu(n)$.
• Explicit Formulas for Weighted Averages: By providing explicit formulas for general weights $f$, the paper offers a flexible tool for analyzing the distribution of these sums in various contexts, not just simple averages.
• Boundary Behavior: The identification of $\text{Re}(s)=1$ as a natural boundary for the Dirichlet series of $S(n)$ (under linear independence assumptions) is a strong structural result, highlighting the deep connection between the additive properties of $\lambda(n)$ and the distribution of the zeros of $\zeta(s)$.
• Technical Extension: The extension of explicit formulas for $L(x)$ to the range $0 < x < 1$is a necessary technical step that allows for the rigorous treatment of the convolution integral starting from 0, a detail often overlooked in standard treatments which focus on$x \ge 1$.

In summary, this paper provides a rigorous analytic framework for understanding the discrete convolution of multiplicative functions, yielding precise asymptotic formulas and deep insights into the analytic structure of the associated generating functions, contingent on the Riemann Hypothesis.