In this note, we study the $q$-Dirichlet series $Z_q(s)$ which was evaluated at non-positive integers by Satoh. We consider the values of $Z_q(s)$ at positive integers. By letting $q \to 1$, we get the Euler formulas for $\zeta(2k)$ and the recent formulas for $\zeta(2k+1)$ given by Cvijović-Klinowski. We also consider the relation between $Z_q(s)$ and Jackson's $q$-$\Gamma$-function.

We investigate an easy way to compute the Iwasawa $\lambda$-invariant of imaginary quadratic fields and give some explicit examples.

Let $X$ be a complete algebraic variety over $\mathbf{C}$. We consider a log variety $(X, \Delta)$ that is weakly Kawamata log terminal. We assume that $K_X + \Delta$ is a $\mathbf{Q}$-Cartier $\mathbf{Q}$-divisor and that every irreducible component of $\lfloor \Delta \rfloor$ is $\mathbf{Q}$-Cartier. A nef and big $\mathbf{Q}$-Cartier $\mathbf{Q}$-divisor $H$ on $X$ is called nef and log big on $(X, \Delta)$ if $H \vert_B$ is nef and big for every center $B$ of non-``Kawamata log terminal'' singularities for $(X, \Delta)$. We prove that, if $L$ is a nef Cartier divisor such that $aL - (K_X + \Delta)$ is nef and log big on $(X, \Delta)$ for some $a \in \mathbf{N}$, then the complete linear system $|mL|$ is base point free for $m \gg 0$.