[Paper Review] On the Number of Restricted Prime Factors of an Integer II

Alexander P. Mangerel|arXiv (Cornell University)|Apr 6, 2016

Analytic Number Theory Research4 references3 citations

TL;DR

This paper establishes an asymptotic formula for the number of integers up to $ x $ with exactly $ k_j $ distinct prime factors from each set $ E_j $ in a partition of the primes, under conditions where $ k_j \sim E_j(x) $. Using a generalized Wirsing-type theorem for multiplicative functions, it confirms that the distribution of such integers is asymptotically Poissonian when $ k_j = (1+o(1))E_j(x) $, validating a probabilistic heuristic in this regime.

ABSTRACT

Given a partition $\{E_0,\ldots,E_n\}$ of the set of primes and a vector $\mathbf{k} \in \mathbb{N}_0^{n+1}$, we compute an asymptotic formula for the quantity $|\{m \leq x: ω_{E_j}(m) = k_j \ \forall \ 0 \leq j \leq n\}|$ uniformly in a wide range of the parameters $k_j$ that complements the results of a previous paper of the author. This is accomplished using an extension and generalization of a theorem of Wirsing due to the author that gives explicit estimates for the ratio $\frac{|M_g(x)|}{M_{f}(x)}$, whenever $f: \mathbb{N} ightarrow (0,\infty)$ and $g: \mathbb{N} ightarrow \mathbb{C}$ are strongly multiplicative functions that are uniformly bounded on primes and satisfy $|g(n)| \leq f(n)$ for every $n \in \mathbb{N}$. This also allows us to conclude the validity of a probabilistic heuristic regarding $π(x;\mathbf{E},\mathbf{k})$ in the case that $k_j = (1+o(1))E_j(x)$, for each $0 \leq j \leq n$.

Motivation & Objective

• To derive an asymptotic formula for the count of integers $ \leq x $ with exactly $ k_j $ distinct prime factors from each set $ E_j $ in a partition of the primes.
• To investigate the validity of a probabilistic heuristic from prior work suggesting that the distribution of such integers should be Poissonian when $ k_j \sim E_j(x) $.
• To extend Halász's result on single sets to multiple sets via a partitioned prime structure.
• To establish conditions under which the Poisson approximation holds uniformly across all $ j $, using analytic number theory techniques.

Proposed method

• Generalizes a theorem of Wirsing to estimate the ratio $ |M_g(x)| / M_f(x) $ for strongly multiplicative functions $ f, g $ with $ |g(n)| \leq f(n) $.
• Applies this estimate to arithmetic functions associated with the multiplicative structure of the partitioned prime sets $ \{E_j\} $, using the function $ f(n) = \prod_{j=0}^n (1 + \mathbf{1}_{p \in E_j}(n)) $.
• Employs the function $ \Delta_E(x;g,T) $ to control the oscillation of $ g(p) $, defining 'nice pairs' to exclude pathological correlations.
• Uses the $ \rho $-method with $ \rho_j = k_j / E_j(x) $, and derives asymptotic expansions via the Gamma and Barnes $ G $-functions.
• Applies a dyadic decomposition of the sum over prime powers, splitting into $ p^\nu \leq y $ and $ y < p^\nu \leq x^{1/2} $, to control error terms.
• Establishes uniform error bounds via estimates on $ \Sigma_1 $ and $ \Sigma_2 $, showing $ \Sigma_1 + \Sigma_2 = (1 + O(\log E_j(x)/E_j(x)))E_j(x) $.

Experimental results

Research questions

• RQ1 Does the Poisson heuristic accurately describe the distribution of integers with $ k_j $ distinct prime factors from each $ E_j $ when $ k_j \sim E_j(x) $?
• RQ2 Under what conditions on $ k_j $ and the partition $ \{E_j\} $ does the asymptotic formula for $ \pi(x;\boldsymbol{E},\boldsymbol{k}) $ hold uniformly?
• RQ3 Can the method used in this paper, based on generalized Wirsing-type estimates, be applied independently of the earlier work [4] to derive new asymptotic results?
• RQ4 How does the error term in the asymptotic formula depend on the parameters $ k_j $, $ E_j(x) $, and the distribution of primes in each $ E_j $?
• RQ5 Is the factor $ \rho = k_j / E_j(x) $ the correct bias in the asymptotic formula, as seen in the single-set case?

Key findings

• The asymptotic formula for $ \pi(x;\boldsymbol{E},\boldsymbol{k}) $ holds uniformly when $ E_j(x)^{-1+\epsilon} \ll \delta_j \leq \rho_j \leq B $ and $ \delta \cdot \delta_j E_j(x) \to \infty $, with $ \delta = \min_j \delta_j $.
• When $ k_j = (1+o(1))E_j(x) $ for all $ j $, the distribution is asymptotically Poissonian: $ \pi(x;\boldsymbol{E},\boldsymbol{k}) \sim x \prod_{j=0}^n \frac{E_j(x)^{k_j}}{k_j!} e^{-E_j(x)} $, confirming the probabilistic heuristic.
• The error term in the asymptotic formula is bounded by $ O\left(x^{-1/2} \Gamma(\rho) F(\rho)^{-1}\right) $, with $ \Gamma(\rho) $ and $ F(\rho) $ involving the Gamma and Barnes $ G $-functions.
• The ratio $ \pi(x;\boldsymbol{E},\boldsymbol{k}+\mathbf{e}_j)/\pi(x;\boldsymbol{E},\boldsymbol{k}) $ is asymptotically $ \frac{E_j(x)}{k_j+1} $, with error $ O\left(\frac{1}{E_j(x)} + \frac{\log E_j(x)}{E_j(x)}\right) $, supporting the Poisson model.
• The method is independent of prior work [4], relying instead on a generalized Wirsing-type estimate for multiplicative functions with controlled growth and oscillation.
• The result holds under the condition that $ \boldsymbol{E} $ is a 'nice partition', meaning $ \Delta_{E_j}(x;g,T)/E_j(x) \gg_n 1 $, which excludes pathological correlations between $ g(p) $ and the prime sets $ E_j $.