3 problems

For $N\geq 2$, let $t$ be the number of positive divisors of $N$, let $g_1,\ldots,g_r$ be the generators specified in the surrounding setup, and let $\sigma_{N,l}$ be the associate…

Let $N\geq 2$ be an integer, and write $\beta(N)=\beta(N,N)$. Formula conjecture. 1. If $N\not\equiv 1,9\pmod{24}$, then $\beta(N)$ is periodic in $N$ with period $24$, with values…

Let $N\geq 2$ and $1<l\mid N$. The homomorphism $\sigma_{N,l}$ has image $\operatorname{im}(\sigma_{N,l})=\beta(N,l)\mathbb{Z}$ for a unique positive integer $\beta(N,l)$. Dependen…