On the Distribution of Integers with Restricted Prime Factors I

Alexander P. Mangerel

paper

On the Distribution of Integers with Restricted Prime Factors I

arXiv:1511.08038

Abstract

Let $E_{0}, \dots, E_{n}$ be a partition of the set of prime numbers, and define $E_{j} (x) := \sum_{p \leq x p \in E _{j}} \frac{1}{p}$ . Define $π (x; E, k)$ to be the number of integers $n \leq x$ with $k_{j}$ prime factors in $E_{j}$ for each $j$ . Basic probabilistic heuristics suggest that $x^{- 1} π (x; E, k)$ , modelled as the distribution function of a random variable, should satisfy a joint Poisson law with parameter vector $(E_{0} (x), \dots, E_{n} (x))$ , as $x \to \infty$ . We prove an asymptotic formula for $π (x; E, k)$ which contradicts these heuristics in the case that for each $j$ , $E_{j} (x)^{2} \leq k_{j} \leq lo g^{\frac{2}{3} - ε} x$ for each $j$ under mild hypotheses. As a particular application, we prove an asymptotic formula regarding integers with prime factors from specific arithmetic progressions, which generalizes a result due to Delange.

38 pages

On the Distribution of Integers with Restricted Prime Factors I

Abstract

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