7 problems

Let $a$ and $b$ be positive integers with $a\mid b$, and let $S$ be a finite gcd-closed set of positive integers. For each $x\in S$, let $G_S(x)$ denote the set of greatest-type di…

Let $S=\{x_1,\ldots,x_n\}$ be an odd-GCD-closed set, and let $f$ be a completely multiplicative function, meaning $f(ab)=f(a)f(b)$ for positive integers $a,b$. If $f$ is strictly m…

Let $S=\{x_1,\ldots,x_n\}$ be an odd-GCD-closed set, meaning that $S$ consists of odd positive integers and is closed under greatest common divisors. For a real exponent $\alpha$,…

A positive integer $x$ is singular if there exists a GCD-closed set $S=\{x_1,\ldots,x_n\}$ with $1\leq x_1<\cdots<x_n=x$ such that $\Psi_{S,\frac{1}{N}}(x)=0$. It is a primitive si…

Let $S$ be a GCD-closed set of odd positive integers, meaning that every element of $S$ is odd and the greatest common divisor of any two elements of $S$ belongs to $S$. Let…

Let $S=\{x_1,\ldots,x_n\}$ be a GCD-closed set of odd positive integers, and let $[[x_i,x_j]^r]$ be its power LCM matrix with nonzero exponent $r$. Hong's conjecture. Every such po…

Let $S=\{x_1,x_2,\ldots,x_n\}$ be a GCD-closed set of positive integers, let $r\neq0$, and let $[[x_i,x_j]^r]$ denote its power LCM matrix. Shen's conjecture. The matrix…