[Paper Review] Exponentially $S$-numbers

Vladimir Shevelev|arXiv (Cornell University)|Oct 18, 2015

Analytic Number Theory Research3 references3 citations

TL;DR

This paper establishes a general asymptotic formula for the counting function of exponentially $S$-numbers—positive integers whose prime exponents all belong to a given sequence $S \in \mathbf{S}$—proving they have a natural density $h(E(S))$ when $s(1) = 1$. The main result provides an explicit density formula and an error term $O\left(\sqrt{x}\log x \cdot e^{c\frac{\sqrt{\log x}}{\log\log x}}\right)$, valid uniformly across all such sequences, with $c \approx 7.443$. The work generalizes prior results on square-free and powerful numbers and extends to a broader class of exponential $S$-number systems.

ABSTRACT

Let $\mathbf{S}$ be the set of all finite or infinite increasing sequences of positive integers. For a sequence $S=\{s(n)\}, n\geq1,$ from $\mathbf{S},$ let us call a positive number $N$ an exponentially $S$-number $(N\in E(S)),$ if all exponents in its prime power factorization are in $S.$ Let us accept that $1\in E(S).$ We prove that, for every sequence $S\in \mathbf{S}$ with $s(1)=1,$ the exponentially $S$-numbers have a density $h=h(E(S))$ such that $$\sum_{i\leq x,\enskip i\in E(S)} 1 = h(E(S))x+R(x), where R(x) does not depend on $S$ and $h(E(S))=\prod_{p}(1+\sum_{i\geq2}\frac{u(i)-u(i-1)}{p^i}),$ where $u(n)$ is the characteristic function of $S.$

Motivation & Objective

• To establish the existence and explicit computation of the natural density $h(E(S))$ for exponentially $S$-numbers, defined as positive integers whose prime exponents all lie in a given sequence $S \in \mathbf{S}$.
• To derive a uniform error term in the asymptotic counting function that does not depend on the specific choice of $S$, provided $s(1) = 1$.
• To generalize prior results on square-free numbers ($S = \{1\}$) and exponentially square-free numbers ($S = B$) to arbitrary sequences $S$ starting with 1.
• To investigate the structure of the density function $h(E(S))$ and its dependence on the characteristic function $u(n)$ of $S$, leading to an Euler product formula.
• To extend the framework to a more general setting where each prime is assigned its own exponent sequence $S_n$, yielding a new class of exponentially $\mathbf{A}$-numbers.

Proposed method

• Define exponentially $S$-numbers as integers whose prime exponents are all in a fixed sequence $S \in \mathbf{S}$, with $1 \in E(S)$ by convention.
• Use a decomposition of $E(S)$ into disjoint classes $E(S)^{(a)}$ indexed by powerful numbers $a \in E(S) \cap E(\Upsilon)$, where $\Upsilon = \{2,3,4,\dots\}$.
• Apply Lemma 1 on the counting function $b_r(x)$ of square-free numbers coprime to $r$, with a remainder term bounded by $O\left(\sqrt{x} \cdot e^{c\frac{\sqrt{\log r}}{\log\log r}}\right)$ for large $r$.
• Express the total count $\sum_{i \leq x, i \in E(S)} 1$ as a sum over $a \in E(S) \cap E(\Upsilon)$ of $b_{r(a)}(x/a)$, leading to an asymptotic expansion involving the density of square-free numbers.
• Derive the density formula $h(E(S)) = \prod_p \left(1 + \sum_{i \geq 2} \frac{u(i) - u(i-1)}{p^i}\right)$ using multiplicative number theory and properties of Euler products.
• Generalize the result to the case of a sequence $\mathbf{A} = \{S_n\}$, where each prime $p_n$ has its own exponent sequence $S_n$, yielding a new density formula involving $\prod_n \left(1 + \sum_{i \geq 2} \frac{u_n(i) - u_n(i-1)}{p_n^i}\right)$.

Experimental results

Research questions

• RQ1 What is the natural density of the set of exponentially $S$-numbers when $S$ is any increasing sequence of positive integers with $s(1) = 1$?
• RQ2 Can a uniform error term be established for the counting function of exponentially $S$-numbers across all such sequences $S$?
• RQ3 How does the density $h(E(S))$ depend on the characteristic function $u(n)$ of the sequence $S$?
• RQ4 Is the set of all possible densities $\{h(E(S))\}$ dense in the interval $[6/\pi^2, 1]$?
• RQ5 What is the asymptotic behavior of the counting function for exponentially $\mathbf{A}$-numbers, where each prime has its own exponent sequence $S_n$?

Key findings

• For every sequence $S \in \mathbf{S}$ with $s(1) = 1$, the set of exponentially $S$-numbers has a positive natural density $h(E(S))$, while $h(E(S)) = 0$ if $s(1) > 1$.
• The density is given explicitly by the Euler product: $h(E(S)) = \prod_p \left(1 + \sum_{i \geq 2} \frac{u(i) - u(i-1)}{p^i}\right)$, where $u(n)$ is the characteristic function of $S$.
• The counting function satisfies $\sum_{i \leq x, i \in E(S)} 1 = h(E(S))x + O\left(\sqrt{x} \log x \cdot e^{c \frac{\sqrt{\log x}}{\log \log x}}\right)$ with $c = 4\sqrt{2.4 / \log 2} \approx 7.443$.
• The error term is uniform across all sequences $S$ with $s(1) = 1$, and does not require the Riemann Hypothesis, unlike prior results for specific cases.
• The set $\{h(E(S))\}$ is not dense in $[6/\pi^2, 1]$; there exists a positive gap between $\prod_p (1 - \frac{p-1}{p^3})$ and $\prod_p (1 - \frac{1}{p^3})$, showing that not all densities in the interval are achievable.
• The generalization to exponentially $\mathbf{A}$-numbers, where each prime $p_n$ is assigned a sequence $S_n$, yields a density formula $h(E(\mathbf{A})) = \prod_{n \geq 1} \left(1 + \sum_{i \geq 2} \frac{u_n(i) - u_n(i-1)}{p_n^i}\right)$, with the same error term structure.