Let $K = \mathbf{Q}(\sqrt{m})$ be a real quadratic field, $\mathcal{O}_K$ its ring of integers and $G = \operatorname{Gal}(K/\mathbf{Q})$. For $\gamma \in H^1(G, \mathcal{O}_K^{\times})$, we associate a module $M_c/P_c$ for $\gamma = [c]$. It is known that $M_c/P_c \approx \mathbf{Z}/\Delta_m \mathbf{Z}$ where $\Delta_m = 1$ or 2 and we will determine $\Delta_m$.

We study the duplication formulas of triple sine and cosine functions from various expressions point of view.

We introduce and study two kinds of zeta functions $\zeta(u; G,X)$ and $Z(u; G,X)$ as well as moments $m(k; G,X)$ attached to a given finite group action $G \curvearrowright X$. We show that zeta functions determine the moments, and moments determine the multiple transitivity of the action. In the symmetric group case we give an explicit formula of moments and calculate zeta functions of the infinite symmetric group $\mathfrak{S}_{\infty}$.

Let $\widetilde{\mathfrak{H}}^{(g)}$ be the graded ring generated by the $r$-th higher weight enumerators of all codes of any length, $1 \leq r \leq g$. In this note we will prove that $\widetilde{\mathfrak{H}}^{(g)}$ is not finitely generated.

Let $M$ be a positive integer with $M > 4$, and let $\varphi$ denote Euler's totient function. If a positive integer $n$ satisfies the Diophantine equation (*) $M \varphi(n) = n - 1$, then the number of prime factors of $n$ is much bigger than $M$. Moreover, the set of all squarefree integers which do not fulfil (*) contains ``nice'' subsets.