We show that $\tau$-tilting finite simply connected algebras are representation-finite. Then, some related algebras are considered, including iterated tilted algebras, tubular algebras and so on. We also prove that the $\tau$-tilting finiteness of non-sincere algebras can be reduced to that of sincere algebras. This motivates us to give a complete list of $\tau$-tilting finite sincere simply connected algebras.

The compact simply connected Riemannian $4$-symmetric spaces were classified by J. A. Jiménez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma }$ of order four on $G$ and $H$ is a fixed points subgroup $G^{\gamma }$ of $G$. According to the classification by J. A. Jiménez, there exist seven compact simply connected Riemannian $4$-symmetric spaces $G/H$ in the case where $G$ is of type $E_8$. In the present article, we give the explicit form of automorphisms ${\tilde{\omega }}_4$, ${\tilde{\kappa }}_4$ and ${\tilde{\epsilon }}_4$ of order four on $E_8$ induced by the $C$-linear transformations ${\omega }_4$, ${\kappa }_4$ and ${\epsilon }_4$ of the 248-dimensional $C$-vector space ${\mathfrak{e}}_8^C$, respectively. Further, we determine the structure of these fixed points subgroups ${(E_8)}^{{\omega }_4}$, ${(E_8)}^{{\kappa }_4}$ and ${(E_8)}^{{\epsilon }_4}$ of $E_8$. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces $G/H$ above corresponding to the Lie algebras $\mathfrak{h}=\mathfrak{s}\mathfrak{u}(2)\oplus iR\oplus {\mathfrak{e}}_6$, $iR\oplus \mathfrak{s}\mathfrak{o}(14)$ and $\mathfrak{h}=\mathfrak{s}\mathfrak{u}(2)\oplus iR\oplus \mathfrak{s}\mathfrak{o}(12)$, where $\mathfrak{h}=\text{Lie}(H)$. With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in $E_8$.

Akiyama classified all cubic Pisot units with finite $\beta$-expansions. In this paper, we show a generalization of Akiyama’s theorem by using some reduction theorem. Moreover we give another application of this reduction theorem.

We consider a sum of the derivative of Dirichlet $L$-functions over the zeros of the Dirichlet $L$-function. We give an asymptotic formula for the sum.

We present new relationships between the work of H. Davenport and A. I. Popov. A new general formula involving the von Mangoldt function is presented, as well as expressions under the Riemann Hypothesis.