[Paper Review] The probability that random positive integers are k-wise relatively prime

Jerry Hu|arXiv (Cornell University)|Aug 7, 2012

Analytic Number Theory Research4 citations

TL;DR

This paper derives an exact asymptotic formula for the probability that s randomly chosen positive integers are k-wise relatively prime, generalizing the classical result that two random integers are relatively prime with probability $6/\pi^2$. Using recursive structure and multiplicative arithmetic functions, the authors establish that this probability is $A_{s,k} = \prod_p \left(1 - \frac{1}{p}\right)^{s-k+1} \sum_{m=0}^{k-1} \binom{s}{m} \left(1 - \frac{1}{p}\right)^{k-1-m} \frac{1}{p^m}$, which reduces to $1/\zeta(s)$ in the pairwise case.

ABSTRACT

An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.

Motivation & Objective

• To generalize the classical result that two random integers are coprime with probability $6/\pi^2$ to the case of k-wise relative primality among s integers.
• To derive an exact asymptotic formula for the natural density of s-tuples that are k-wise relatively prime.
• To extend previous work on pairwise relative primality by Tóth (2002) to the broader k-wise setting using recursive and multiplicative number-theoretic techniques.
• To characterize the density of s-tuples that are k-wise relatively prime and also relatively prime to given auxiliary integers $u_1, \dots, u_{k-1}$.

Proposed method

• The paper defines $Q_{s,k}^{(u)}(n)$ as the count of s-tuples in $[1,n]^s$ that are k-wise relatively prime and satisfy relative primality conditions with respect to a $(k-1)$-tuple $u = (u_1, \dots, u_{k-1})$.
• It establishes a recursive relation: $Q_{s+1,k}^{(u)}(n) = \sum_{\substack{j=1 \\ (j,u_1)=1}}^n Q_{s,k}^{(j*u)}(n)$, where $j*u$ is a transformed $(k-1)$-tuple ensuring compatibility with k-wise conditions.
• The authors introduce multiplicative arithmetic functions $f_{s,k,i}(u_i)$ to encode the relative primality constraints with $u_i$, derived via Möbius inversion and prime-wise product decomposition.
• They prove the asymptotic formula $Q_{s,k}^{(u)}(n) = A_{s,k} \prod_{i=1}^{k-1} f_{s,k,i}(u_i) n^s + O(\theta(u_1) n^{s-1} \log^{\delta(s,k)} n)$, where $\delta(s,k) = \max \left\{ \binom{s-1}{i} \mid i=1,\dots,k-1 \right\}$.
• The key identity involves expressing the density as an Euler product over primes: $A_{s,k} = \prod_p \left(1 - \frac{1}{p}\right)^{s-k+1} \sum_{m=0}^{k-1} \binom{s}{m} \left(1 - \frac{1}{p}\right)^{k-1-m} \frac{1}{p^m}$.
• The proof uses induction, Möbius inversion, and estimates on divisor sums involving $\tau_k(n)$, relying on known asymptotics for sums of multiplicative functions.

Experimental results

Research questions

• RQ1 What is the exact asymptotic probability that s randomly selected positive integers are k-wise relatively prime, for general $s \geq 1$ and $k \geq 2$?
• RQ2 How does this probability change when the s-tuple is also required to be relatively prime to a given set of $k-1$ auxiliary integers $u_1, \dots, u_{k-1}$?
• RQ3 Can the recursive structure of k-wise relative primality be captured via a multiplicative function framework that allows for exact density computation?
• RQ4 What is the error term in the asymptotic density estimate, and how does it depend on the size and structure of the auxiliary integers $u_i$?
• RQ5 How does the k-wise coprimality probability relate to the Riemann zeta function and Euler products, and how does it generalize the $1/\zeta(s)$ result for pairwise coprimality?

Key findings

• The exact natural density of s-tuples of positive integers that are k-wise relatively prime is given by $A_{s,k} = \prod_p \left(1 - \frac{1}{p}\right)^{s-k+1} \sum_{m=0}^{k-1} \binom{s}{m} \left(1 - \frac{1}{p}\right)^{k-1-m} \frac{1}{p^m}$.
• When the s-tuple must also be i-wise relatively prime to a given $u_i$ for $i=1,\dots,k-1$, the density becomes $A_{s,k} \prod_{i=1}^{k-1} f_{s,k,i}(u_i)$, where $f_{s,k,i}(u_i)$ is a multiplicative arithmetic function defined via prime factors of $u_i$.
• The error term in the asymptotic count $Q_{s,k}^{(u)}(n)$ is bounded by $O(\theta(u_1) n^{s-1} \log^{\delta(s,k)} n)$, where $\theta(u_1)$ is the number of squarefree divisors of $u_1$, and $\delta(s,k) = \max_{1 \leq i \leq k-1} \binom{s-1}{i}$.
• The recursive structure $Q_{s+1,k}^{(u)}(n) = \sum_{\substack{j=1 \\ (j,u_1)=1}}^n Q_{s,k}^{(j*u)}(n)$ holds, with $j*u$ defined via a transformation involving GCDs and LCMs to preserve k-wise conditions.
• The function $f_{s,k,i}(u_i)$ satisfies the identity $\frac{f_{s,k,i}(u_i)}{f_{s,k,i+1}(u_i)} = \sum_{d|u_i} \frac{\mu(d) \binom{s}{i}^{\omega(d)}}{\alpha_{s,k,i}(d)}$, linking it to Möbius inversion and prime-power decomposition.
• The formula reduces to the known result $1/\zeta(s)$ when $k=2$, recovering the classical probability that s integers are pairwise relatively prime.