Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities

1. Problem Statement and Context

The paper addresses the asymptotic behavior of sums involving the Möbius function $\mu(n)$, the number of distinct prime factors $\omega(n)$, and the smallest prime factor $p_1(n)$ of an integer $n$, specifically when $p_1(n)$ lies in a specific arithmetic progression.

• Historical Background: In 1977, Alladi established a "duality" connecting the distribution of the smallest prime factor $p_1(n)$ and the largest prime factor $P_1(n)$. He proved that if the largest prime factors are uniformly distributed in reduced residue classes modulo $\ell$ (a consequence of the Prime Number Theorem for Arithmetic Progressions, PNTAP), then specific weighted sums involving $\mu(n)$ and $p_1(n)$ vanish or converge to specific densities.
• Recent Developments: In 2024, Alladi and Jason Johnson extended this to the second-order duality ($k=2$), proving that $\sum \mu(n)\omega(n)/n = 0$ and related results for arithmetic progressions.
• The Gap: The paper aims to generalize these results to all higher orders $k \geq 3$. Specifically, it seeks to prove that for any $k \geq 3$, the sum $\sum_{p_1(n) \equiv j \pmod \ell} \frac{\mu(n)\omega(n)^{k-1}}{n}$ vanishes, and to establish a corresponding density theorem involving binomial coefficients of $\omega(n)$.

2. Methodology

The authors employ a combination of analytic number theory, combinatorial identities, and asymptotic analysis. The methodology is structured in several key steps:

A. Higher-Order Duality Identities

The core engine of the proof is the generalization of Alladi's 1977 duality identities. For a function $f$ defined on primes and an integer $k \geq 1$:
$$\sum_{1<d|n} \mu(d) \binom{\omega(d)-1}{k-1} f(p_1(d)) = (-1)^k f(P_k(n))$$
where $P_k(n)$ is the $k$-th largest prime factor of $n$. This identity allows the authors to transform sums involving the smallest prime factor (which are hard to estimate directly) into sums involving the $k$-th largest prime factor (which are easier to estimate using distribution theorems).

B. Asymptotic Estimates for $\mu(n)\omega(n)^k$

The authors first establish strong quantitative bounds for the partial sums of $\mu(n)\omega(n)^k$.

• They utilize the hyperbola method to split sums.
• They rely on the Selberg-Delange method (or elementary equivalents using strong forms of the PNT) to derive asymptotic expansions for $\pi_k(x)$ (the count of integers with exactly $k$ prime factors).
• They prove that for $k \geq 2$, $\sum_{n \leq x} \mu(n)\omega(n)^k = o(x)$ and $\sum_{n \leq x} \frac{\mu(n)\omega(n)^k}{n} = o(1)$.

C. Uniform Distribution of $k$-th Largest Prime Factors

A critical component is proving that the sequence of $k$-th largest prime factors $\{P_k(n)\}$ is uniformly distributed in the reduced residue classes modulo $\ell$.

• Lemma 5.1: The authors show that the set of integers where the definitions of $P_k(n)$ (strict vs. weak inequalities) differ is negligible ($o(x)$).
• Theorem 5.3: They prove that for any fixed $k \geq 1$ and modulus $\ell$, the number of integers $n \leq x$ such that $P_k(n) \equiv j \pmod \ell$ is:
  $$N_k(x; \ell, j) = \frac{x}{\phi(\ell)} + O\left(\frac{x(\log \log x)^k}{\log x}\right)$$
  This result relies on Tenenbaum's estimates for $\Psi_k(x, T)$ (integers with $k$-th largest prime factor $\leq T$) and careful error analysis using the strong form of the PNT.

D. The "Cancellation" Mechanism

To prove the main results for $k \geq 3$, the authors use the duality identity to convert the sum over $p_1(n)$ into a sum over $P_k(n)$.

• They express $\omega(n)^{k-1}$ (or binomial coefficients) as linear combinations of falling factorials using Stirling numbers of the second kind.
• This decomposition leads to a linear combination of sums involving $P_1(n), P_2(n), \dots, P_k(n)$.
• Because $P_1, \dots, P_k$ are all uniformly distributed with the same density $1/\phi(\ell)$, the main terms in the resulting sums cancel out perfectly (e.g., coefficients sum to zero), leaving only the error terms.

3. Key Contributions and Results

A. Vanishing Sums for Higher Orders

The paper proves that for any integer $k \geq 2$:
$$\sum_{n=2}^{\infty} \frac{\mu(n)\omega(n)^k}{n} = 0$$
Furthermore, for integers $j, \ell$ with $(j, \ell)=1$ and $k \geq 3$:
$$\sum_{\substack{n=2 \\ p_1(n) \equiv j \pmod \ell}}^{\infty} \frac{\mu(n)\omega(n)^{k-1}}{n} = 0$$
This generalizes the $k=1$ (Alladi 1977) and $k=2$ (Alladi-Johnson 2024) results.

B. Density-Type Theorems

The authors establish a precise density result involving binomial coefficients. For $k \geq 3$:
$$(-1)^k \sum_{\substack{n=2 \\ p_1(n) \equiv j \pmod \ell}}^{\infty} \frac{\mu(n) \binom{\omega(n)-1}{k-1}}{n} = \frac{1}{\phi(\ell)}$$
This result connects the arithmetic density of the smallest prime factor in a progression to the combinatorial structure of the number of prime factors.

C. Quantitative Bounds

Unlike previous qualitative results, this paper provides strong quantitative estimates (error terms). For example, for the partial sum up to $x$:
$$\sum_{\substack{n \leq x \\ p_1(n) \equiv j \pmod \ell}} \frac{\mu(n)\omega(n)^{k-1}}{n} \ll \frac{(\log \log x)^{k+1/2}}{\sqrt{\log x}}$$
The error terms degrade as $k$ increases but remain sufficiently small to ensure convergence to zero.

D. Technical Lemmas

• Lemma 6.4: A novel combinatorial lemma proving a cancellation identity involving Stirling numbers of the second kind: $\sum_{j=1}^k (-1)^{j-1}(j-1)! S_{k,j} = 0$. This is crucial for the cancellation of main terms in the higher-order proofs.

4. Significance and Impact

1. Unification of Duality: The paper completes the picture of Alladi's duality theory by extending it from the first and second orders to arbitrary orders $k$. It demonstrates that the uniform distribution of the $k$-th largest prime factor is a universal property that drives the vanishing of these Möbius-weighted sums.
2. Methodological Advancement: The authors successfully navigate the increasing complexity of error terms as $k$ grows. They demonstrate that despite the error terms becoming larger (in terms of powers of $\log \log x$), the structural cancellation provided by the duality identities is robust enough to yield the desired asymptotic results.
3. Foundation for Algebraic Extensions: The concluding remarks highlight that these results serve as a blueprint for extending duality to algebraic number fields and function fields. The paper explicitly notes that subsequent work (by Sengupta, Sahoo, and Jha) has already begun applying these higher-order duality identities to Galois extensions and function fields, confirming the paper's role as a foundational step in modern analytic number theory.
4. Connection to PNTAP: The work reinforces the deep connection between the distribution of prime factors (specifically the $k$-th largest) and the Prime Number Theorem for Arithmetic Progressions, showing that the "randomness" of prime factors extends to higher-order statistics.

In summary, Alladi and Sengupta have successfully generalized a profound number-theoretic duality, providing rigorous proofs for all higher orders and establishing a framework for future research in algebraic settings.