If $n$ is a positive integer, write $F_n$ for the $n$th Fibonacci number, and $\omega(n)$ for the number of distinct prime divisors of $n$. We give a description of Fibonacci numbers satisfying $\omega(F_n) \leq 2$. Moreover, we prove that the inequality $\omega(F_n) \geq (\log n)^{\log 2 + o(1)}$ holds for almost all $n$. We conjecture that $\omega(F_n) \gg \log n$ for composite $n$, and give a heuristic argument in support of this conjecture.

We show an apparently new expression of Bernoulli numbers, simultaneously we give an expression of multiple zeta values $\zeta(2m, 2m, \dots, 2m)$.

For every rational function of degree more than one, there exists a transcendental meromorphic solution of the Schröder equation. By Yanagihara and Eremenko-Sodin, it is known that the Valiron, Nevanlinna and Picard exceptional sets of this solution are all same.

As an analogue of this result, we show that all the Valiron, Nevanlinna and Picard exceptional sets of iterations of a rational function of degree more than one are also same. As a corollary, the equidistribution theorem in complex dynamics follows.

The normalized $L^2$-norm of the traceless part of the Ricci curvature defines a Riemannian functional on the space of Riemannian metrics. In this paper, we will consider the critical Riemannian metrics with a flat conformal structure for this functional.

We study some aspects of noncommutative differential geometry on a finite Weyl group in the sense of S. Woronowicz, K. Bresser et al., and S. Majid. For any finite Weyl group $W$ we consider the subalgebra generated by flat connections in the left-invariant exterior differential algebra of $W$.

For root systems of type $A$ and $D$ we describe a set of relations between the flat connections, which conjecturally is a complete set.