We discover a general optimal inequality for graph hypersurfaces in affine $(n + 1)$-space $\mathbf{R}^{n+1}$ involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.

Let $k > 1$ be the fundamental discriminant, and let $\chi(n)$, $\varepsilon$ and $h$ be the real primitive character modulo $k$, the fundamental unit and the class number of the real quadratic field $\mathbf{Q}(\sqrt{k} )$, respectively. And let $[x]$ denote the greatest integer not greater than $x$.

In [3], M.-G. Leu showed $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^k \chi(n) / n \big] + 1$ for all $k$, and $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^{[k/2]} \chi(n) / n \big]$ in the case $k \neq m^2 + 4$ with $m \in \mathbf{Z}$.

In this paper we will show $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^{[k/2]} \chi(n) / n \big]$ for all fundamental discriminants $k > 1$.

Let $R_{\psi}$ be a twisting operator for a quadratic primitive character $\psi$ and $\tilde{T}(n^2)$ the $n^2$-th Hecke operator of half-integral weight. When $\psi$ has an odd conductor, we already found trace identities between twisted Hecke operators $R_{\psi} \tilde{T}(n^2)$ of half-integral weight and certain Hecke operators of integral weight for almost all cases (cf. [U1--3]). In this paper, the restriction is removed and we give similar trace identities for every quadratic primitive character $\psi$, including the case that $\psi$ has an even conductor.

In this paper, we deal with the problem of uniqueness of meromorphic functions concerning one question of Gross, and obtain some results that are improvements of that of former authors. Moreover, the example shows that the result is sharp.

An asymptotic formula is given for the number of cyclic quartic fields with discriminant $\leq x$ and divisible by a given prime.

Let $S$ be a sphere in $\mathbf{R}^n$ whose center is not in $\mathbf{Q}^n$. We pose the following problem on $S$. \[ \text{``What is the closure of $S \cap \mathbf{Q}^n$ with respect to the Euclidean topology?''} \] In this paper we give a simple solution for this problem in the special case that the center $a = (a_i) \in \mathbf{R}^n$ of $S$ satisfies \[ \left\{ \sum_{i=1}^n r_i (a_i - b_i); \ r_1, \dots, r_n \in \mathbf{Q} \right\} = K \] for some $b = (b_i) \in S \cap \mathbf{Q}^n$ and some Galois extension $K$ of $\mathbf{Q}$. Our solution represents the closure of $S \cap \mathbf{Q}^n$ for such $S$ in terms of the Galois group of $K$ over $\mathbf{Q}$.