5 problems

Let $S^{(1)}(x)$, $S^{(2)}(x)$, $C^{(1)}(x)$, and $C^{(2)}(x)$ be the subgroup-counting functions collected in $\mathcal{F}[x]$, and let … where $f(x)\in\mathcal{F}[x]$. Mean-squar…

Let $S^{(1)}(x)$, $S^{(2)}(x)$, $C^{(1)}(x)$, and $C^{(2)}(x)$ be the subgroup-counting functions collected in $\mathcal{F}[x]$, and let … where $f(x)\in\mathcal{F}[x]$ and the…

Let $R$ be a finite group, and let $T\leq R$ be a subgroup. Define … For a non-zero integer $n$, let $\lambda(n)$ denote the number of prime divisors of $n$, counted with multiplic…

The subgroup-count inequality. The inequality

Let $S_{\mathfrak H(q)}(z)$ denote the generating function for the subgroup numbers of the Hecke group $\mathfrak H(q)$. For a positive integer $h$, define … Let $\alpha$ be a posi…