On a family of arithmetic series related to the Möbius function

This paper generalizes a recent result by Alladi and Johnson by proving that for any set of prime numbers with a natural density, the weighted sum of the Möbius and distinct prime factor functions over integers whose smallest prime factor lies in that set converges to zero, while also providing an effective estimate for the rate of this convergence.

The Gist

Here is an explanation of G´erald Tenenbaum's paper, "On a family of arithmetic series related to the M¨obius function," translated into simple, everyday language using analogies.

The Big Picture: A Cosmic Balancing Act

Imagine the world of numbers as a massive, chaotic marketplace. In this market, every number is a product made of "prime ingredients" (like 2, 3, 5, 7, etc.).

The paper investigates a very specific question: If we look at all the numbers in this market, but only count the ones whose smallest ingredient comes from a specific group, do the "positive" and "negative" values cancel each other out perfectly?

To understand this, we need three characters:

1. The Mobius Function ($\mu$): Think of this as a scorecard.

   • If a number is made of an even number of distinct prime ingredients, it gets a +1 score.
   • If it's made of an odd number, it gets a -1 score.
   • If it has a repeated ingredient (like $12 = 2 \times 2 \times 3$), it gets a 0 (it's disqualified).
   • Why does this matter? In number theory, these +1s and -1s usually cancel each other out perfectly, like a tug-of-war where both teams are equally strong.

2. The "Smallest Prime" Rule ($P^-(n)$): Imagine every product in the market has a "tag" showing its smallest ingredient.

   • For the number 12 ($2 \times 2 \times 3$), the tag says 2.
   • For the number 15 ($3 \times 5$), the tag says 3.
   • For the number 35 ($5 \times 7$), the tag says 5.

3. The "Distinct Factors" Counter ($\omega$): This is just a counter that counts how many different ingredients are in the product.

   • 12 has ingredients {2, 3}, so the count is 2.
   • 35 has ingredients {5, 7}, so the count is 2.

The Problem: The Alladi-Johnson Discovery

A few years ago, mathematicians Alladi and Johnson discovered something strange. They looked at a specific group of numbers: those where the "smallest ingredient tag" was a specific number (like numbers where the smallest prime is 3, or 7, or 11).

They found that if you add up the scores ($\mu$) multiplied by the ingredient counts ($\omega$) for all these numbers, the total sum is exactly zero.

It's like saying: "If I only look at products tagged with '3', the positive scores and negative scores cancel out perfectly."

The New Discovery: Does it work for any group?

Tenenbaum asks: "Does this cancellation happen for any group of prime tags we choose, or just for specific patterns (like arithmetic progressions)?"

He proves that yes, it works for almost any group, as long as that group is "fairly distributed" in the long run.

The "Fairness" Analogy

Imagine you have a bag of marbles (the prime numbers). You want to pick a subset of marbles to form a "VIP Club."

• The Rule: The VIP Club must be "fair." This means that as you look at larger and larger numbers, the VIP Club shouldn't suddenly become huge or suddenly disappear. It should maintain a steady, predictable density.
• The Result: Tenenbaum proves that if your VIP Club is "fair" (mathematically, it has a "natural density"), then the sum of the scores for all numbers whose smallest prime tag is in the VIP Club will eventually settle down to zero.

The "Rate of Convergence": How fast does it happen?

The paper doesn't just say "it equals zero." It asks, "How close are we to zero right now?"

Think of it like a pendulum swinging. Eventually, it stops in the middle (zero). But how much is it swinging left or right at any given moment?

• Tenenbaum provides a formula to estimate how big the "swing" is.
• The "swing" depends on how "fair" your VIP Club is. If the club is very regular, the swing is tiny. If the club is a bit wobbly, the swing is a bit bigger, but it still gets smaller as you look at bigger numbers.

The "Counter-Example": When the Rules Break

The paper also shows what happens if you break the "fairness" rule.

• Imagine you create a VIP Club that is unfair. You pick primes in a weird pattern: "I'll take all primes between 1 million and 2 million, but ignore the next 100 million, then take the next 1 billion, then ignore the next 10 billion..."
• In this case, the cancellation fails. The sum doesn't go to zero; it gets stuck at a negative number (around -0.69, or $-\log 2$).
• The Lesson: The magic cancellation only works if the group of primes you choose is "well-behaved" and doesn't have huge gaps or sudden spikes.

The Mathematical Magic: How did he prove it?

Tenenbaum uses a toolkit of advanced mathematical techniques, which he translates into a story of splitting the problem:

1. The Small Primes (The "Noise"): He first looks at numbers with very small prime tags. He proves that these contribute very little to the total sum (they are like background noise).
2. The Large Primes (The "Signal"): He then looks at numbers with larger prime tags. Here, he uses a powerful method called the Selberg-Delange method (think of it as a high-powered telescope) to analyze the distribution of these numbers.
3. The Saddle-Point: He uses a technique called "saddle-point estimation." Imagine walking over a mountain pass. The path of least resistance (the saddle point) tells you exactly how the numbers behave. This allows him to calculate the exact rate at which the sum approaches zero.

Why does this matter?

This paper is a "friendly token" to two giants of number theory, George Andrews and Bruce Berndt. It extends a recent discovery to a much broader universe of possibilities.

In simple terms:
Tenenbaum showed that the universe of numbers has a deep, hidden symmetry. No matter which "fair" group of prime numbers you pick to focus on, the chaotic +1s and -1s of the Mobius function will always find a way to balance each other out to zero. It's a beautiful confirmation that even in the randomness of prime numbers, there is a strict order waiting to be found.

The Takeaway:
If you pick a group of primes that isn't "crazy" (it has a natural density), the sum of the Mobius scores for numbers starting with those primes will always vanish. The universe balances itself.

Technical Summary

Here is a detailed technical summary of the paper "On a family of arithmetic series related to the Möbius function" by G´erald Tenenbaum.

1. Problem Statement

The paper investigates the convergence behavior of specific arithmetic series involving the Möbius function ($\mu$) and the number of distinct prime factors function ($\omega$), conditioned on the smallest prime factor of the integers, denoted as $P^-(n)$.

Specifically, the author addresses the following question: Given a set of prime numbers $\mathcal{P}$, does the infinite series
$$\sum_{P^-(n) \in \mathcal{P}} \frac{\mu(n)\omega(n)}{n}$$
converge to zero?

This builds upon a recent result by Alladi and Johnson, who proved that this sum vanishes when $\mathcal{P}$ is an arithmetic progression (i.e., primes $p \equiv \ell \pmod k$). Tenenbaum seeks to determine the extent to which this result depends on the structure of $\mathcal{P}$, specifically whether it holds for any set of primes possessing a natural density.

2. Methodology

The proof employs advanced techniques from analytic number theory, combining combinatorial identities with complex analysis. The methodology is structured in two main parts:

A. Decomposition and Combinatorial Identities

The sum is split into two parts based on a parameter $y$ (where $2 \le y \le \sqrt{x}$):

1. Small primes ($P^-(n) \le y$): The contribution from primes in $\mathcal{P} \cap [2, y]$.
2. Large primes ($P^-(n) > y$): The contribution from primes in $\mathcal{P} \cap (y, \infty)$.

The author utilizes a duality identity derived via Möbius inversion. By defining a characteristic function $\chi_{\mathcal{P}}(n)$ and an auxiliary function $g_y(n)$, the author establishes a bound on the contribution of small primes. A key lemma (Lemma 2.1) relates the sum over small primes to the largest prime factor $P^+(n)$, showing that the contribution from small primes is negligible ($O(1/u)$) regardless of the specific set $\mathcal{P}$, provided $y$ is chosen appropriately.

B. Analytic Techniques (Selberg-Delange and Perron's Formula)

For the contribution of large primes, the author uses the Selberg-Delange method.

• Generating Functions: The sum is interpreted as the derivative at $z=1$ of a Dirichlet series $A(x, y; z) = \sum \frac{\mu(n)z^{\omega(n)}}{n^s}$ restricted to $P^-(n) > y$.
• Perron's Formula: The partial sums are expressed as contour integrals involving the generating function $H(s, y; z)$.
• Zero-Free Regions: The analysis relies on the Korobov-Vinogradov zero-free region for the Riemann zeta function to estimate the error terms.
• Dickman Function: The asymptotic behavior is linked to the Dickman function $\varrho(v)$ via the Laplace transform, which arises naturally when evaluating the contour integrals near the singularity at $s=1$.
• Density Hypothesis: The proof assumes the set $\mathcal{P}$ has a natural density $\delta$, formalized by the condition:
  $$\varepsilon_{\mathcal{P}}(t) := \frac{1}{t} \sum_{\substack{p \le t \\ p \in \mathcal{P}}} \log p - \delta = o(1)$$

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The paper proves that if a set of primes $\mathcal{P}$ has a natural density $\delta$ (satisfying the condition above), then the infinite series converges to zero:
$$\sum_{P^-(n) \in \mathcal{P}} \frac{\mu(n)\omega(n)}{n} = 0$$
Crucially, the author provides an effective estimate for the rate of convergence. For $x \ge 3$ and $e^{(\log_2 x)^b} \le y \le \sqrt{x}$ (with $b > 5/3$), the partial sum satisfies:
$$\sum_{\substack{n \le x \\ P^-(n) \in \mathcal{P}}} \frac{\mu(n)\omega(n)}{n} \ll \varepsilon^*_{\mathcal{P}}(y) \log u + \frac{1}{u}$$
where $u = (\log x)/\log y$ and $\varepsilon^*_{\mathcal{P}}(y) = \sup_{t > y} |\varepsilon_{\mathcal{P}}(t)|$.

Counter-Example Construction

The paper demonstrates that the condition of having a natural density is necessary. If $\mathcal{P}$ is constructed by selecting intervals $[\sqrt{x_j}, x_j]$ for a rapidly increasing sequence $\{x_j\}$, the sum does not converge to zero. Specifically:
$$\liminf_{x \to \infty} \sum_{\substack{n \le x \\ P^-(n) \in \mathcal{P}}} \frac{\mu(n)\omega(n)}{n} \le -\log 2$$
This highlights that the cancellation properties of the Möbius function rely heavily on the uniform distribution of the primes in the set.

Special Cases

The author provides refined asymptotic formulas for specific cases:

1. All Primes: $\sum_{n \le x} \frac{\mu(n)\omega(n)}{n} \sim -\frac{1}{\log x}$.
2. Singleton Set: For a fixed prime $p$, $\sum_{P^-(n)=p} \frac{\mu(n)\omega(n)}{n} \sim \frac{\zeta(1, p)}{p \log x}$.
   The paper notes that one cannot heuristically reconstruct the sum over all primes by simply summing the results over singleton sets, illustrating the non-additive nature of these asymptotic behaviors.

4. Significance

• Generalization: This work significantly generalizes the result of Alladi and Johnson. It moves beyond arithmetic progressions to any set of primes with a natural density, showing that the "duality" between small and large prime factors is a robust phenomenon in analytic number theory.
• Effective Bounds: Unlike many asymptotic results in this field, Tenenbaum provides explicit error terms dependent on the deviation of the prime set's density from its limit. This allows for quantitative assessments of convergence rates.
• Methodological Insight: The paper demonstrates the power of combining Möbius inversion duality with the Selberg-Delange method and saddle-point estimates. It clarifies the precise role of the distribution of primes (via the error term $\varepsilon_{\mathcal{P}}$) in the cancellation of oscillatory arithmetic functions.
• Counter-Intuitive Behavior: The construction of the counter-example serves as a cautionary tale in analytic number theory, showing that without a regular distribution of primes (natural density), the expected cancellation in sums involving $\mu(n)$ can fail dramatically.

In summary, Tenenbaum establishes that the vanishing of the series $\sum \mu(n)\omega(n)/n$ conditioned on $P^-(n)$ is a direct consequence of the natural density of the prime set, providing a rigorous framework and effective bounds for this class of arithmetic series.