We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical 4-folds.

A complex reflection determines an $L$-function which is a generalization of the Artin-Mazur zeta function associated with an element of the symmetric group. The present paper shows that the $L$-function is the Ruelle zeta function associated with a weighted $\mathbf{Z}$-dynamical system.

In this note, we shall define the balancing Wieferich prime which is an analogue of the famous Wieferich primes. We prove that, under the $abc$ conjecture for the number field $\mathbf{Q}(\sqrt{2})$, there are infinitely many balancing non-Wieferich primes. In particular, under the assumption of the $abc$ conjecture for the number field $\mathbf{Q}(\sqrt{2})$ there are at least $O(\log x/{\log \log x})$ such primes $p \equiv 1(\mathrm{mod}\ k)$ for any fixed integer $k> 2$.

Recently, Andrews, Chan, Kim and Osburn introduced a $q$-series $h(q)$ for the study of the first positive rank and crank moments for overpartitions. They conjectured that for all integers $m \geq 3$, \begin{equation*} \frac{1}{(q)_{∞}} (h(q) - m h(q^{m})) \end{equation*} has positive power series coefficients for all powers of $q$. Byungchan Kim, Eunmi Kim and Jeehyeon Seo provided a combinatorial interpretation and proved it is asymptotically true. In this note, we show this conjecture is true if $m$ is any positive power of 2, and we show that in order to prove this conjecture, it is only to prove it for all primes $m$. Moreover we give a stronger conjecture. Our method is completely different from that of Kim et al.