A new expression for the product of the two Dirichlet series is given. From this new expression we derive many results. Among other things we give here another proof of Wilton's expression for the product of two Riemann's zeta functions through our new expression. Other results including mean value theorems will be treated elsewhere.

In this paper it is discussed for which meromorphic functions $f$ the homogeneous differential $f(z) f''(z) - a(f'(z))^2$ has only finitely many zeros. It is shown that any transcendental meromorphic functions $f(z)$ have the form $R(z) \exp(P(z))$ for a rational function $R$ and a polynomial $P$ with the property if $a \ne 1, (n \pm 1) / n$, $n \in N$. This result settles one conjecture proposed by W. Bergweiler.

Let $a$ be a positive integer which is not a perfect $h$-th power with $h \geq 2$, and $Q_a(x; k,l)$ be the set of primes $p \leq x$ such that the residual order of $a$ in $\mathbf{Z} / p\mathbf{Z}^{\times}$ is congruent to $l \bmod{k}$. It seems that no one has ever considered the density of $Q_a(x; k,l)$ for $l \ne 0$ when $k \geq 3$. In this article, the natural densities of $Q_a(x; 4,l)$ ($l = 0, 1, 2, 3$) are considered. When $l = 0, 2$, calculations of $\sharp Q_a(x; 4,l)$ are simple, and we can get these natural densities unconditionally. On the contrary, the distribution properties of $Q_a(x; 4,l)$ for $l = 1, 3$ are rather complicated. Under the assumption of Generalized Riemann Hypothesis, we determine completely the natural densities of $\sharp Q_a(x; 4,l)$ for $l = 1, 3$.

In this paper, we announce that every right-angled Coxeter group determines its Coxeter system up to isomorphism. This implies that the Dranishnikov's rigidity conjecture is the case for right-angled Coxeter groups, i.e., every right-angled Coxeter group determines its boundary up to homeomorphism.

We show that the Nash structural sheaf over the real spectrum of a commutative ring is determined only by the underlying space. We also calculate the stalks and global sections of it in the restricted case. As an application, we show some basic properties of `separated' morphisms.

The object of this paper is to study topological Euler numbers in a semi-stable degeneration of surfaces by using the semi-stable minimal model program. As its application, we find some restrictions of singularities in a semi-stable degeneration of surfaces with general fiber a minimal $\kappa = 0$ surface.

We prove the existence of $3$-dimensional flips by using the crepant descent method. Our proof depends on the existence of good members in the anticanonical linear system and uses explicit computations of blowing ups of terminal singularities.