The main purpose of this paper is, using the estimates for character sums and the analytic method, to study the $2k$-th power mean of the inversion of Dirichlet $L$-functions with the weight of general quadratic Gauss sums, and give two interesting asymptotic formulas.

In this paper we study the map properties of the homogeneous fractional integral operator $T_{\Omega,\alpha}$ on $L^p({\Bbb R}^n)$ for $n/\alpha\le p\le\infty.$ We prove that if $\Omega$ satisfies some smoothness conditions on $S^{n-1},$ then $T_{\Omega,\alpha}$ is bounded from $L^{n/\alpha}({\Bbb R}^n)$ to $BMO({\Bbb R}^n),$ and from $L^p({\Bbb R}^n)\, (n/\alpha<p\le\infty)$ to a class of the Campanato spaces ${\cal L}_{l,\lambda} ({\Bbb R}^n),$ respectively. As the corollary of the results above, we show that when $\Omega$ satisfies some smoothness conditions on $S^{n-1},$ the homogeneous fractional integral operator $T_{\Omega,\alpha}$ is also bounded from $H^p({\Bbb R}^n)\, (n/(n+\alpha)\le p\le1)$ to $L^q({\Bbb R}^n)$ for $1/q=1/p-\alpha/n.$ The results are the extensions of Stein-Weiss (for $p=1$) and Taibleson-Weiss's (for $n/(n+\alpha)\le p<1$) results on the boundedness of the Riesz potential operator $I_\alpha$ on the Hardy spaces $H^p({\Bbb R}^n).$

In this paper, we proved a result that if two meromorphic functions $f(z)$ and $g(z)$ share five small functions $a_j(z) \ \ (j=1,\ldots,5)$ in the sense of $\overline{E}_{k)}(a_j,f)=\overline{E}_{k)}(a_j,g)$, $(j=1,\ldots,5)$ ($k\geq 22$), then we have $f(z)\equiv g(z)$.

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

It is proved that the central values of symmetric square $L$-functions of normalized Hecke eigenforms for the full modular group on average satisfy an analogue of the Lindelöf hypothesis in weight aspect, under the assumption that these values are non-negative.

Let $C \subset \mathbb{P}^{g-1}$ be a canonical curve of genus $g$. In this article we study the problem of extendability of $C$, that is when there is a surface $S\subset \mathbb{P}^g$ different from a cone and having $C$ as hyperplane section. Using the work of Epema we give a bound on the number of moduli of extendable canonical curves. This for example implies that a family of large dimension of curves that are cover of another curve has general member nonextendable. Using a theorem of Wahl we prove the surjectivity of the Wahl map for the general k-gonal curve of genus $g$ when $k = 5, g \geq 15$ or $k = 6, g \geq 13$ or $k \geq 7, g \geq 12$.

We formulate a generalization of K. Takeuchi's method to classify smooth Fano $3$-folds and use it to give a list of numerical possibilities of $\mathbb{Q}$-Fano $3$-folds $X$ with Pic $X = \mathbb{Z} (-2K_{X})$ and $h^{0} (-K_{X}) \geq 4$ containing index $2$ points $P$ such that $(X, P) \simeq (\{xy +z^{2}+u^{a}=0\} / \mathbb{Z}_{2}(1, 1, 1, 0), o)$ for some $a \in \mathbb{N}$. In particular we prove that then $(-K_{X})^{3} \leq 15$ and $h^{0} (-K_{X}) \leq 10$. Moreover we show that such an $X$ is birational to a simpler Mori fiber space.

In the previous paper, we obtained a list of numerical possibilities of $\mathbb{Q}$-Fano $3$-folds $X$ with Pic $X = \mathbb{Z} (-2K_{X})$ and $h^{0} (-K_{X}) \geq 4$ containing index $2$ points $P$ such that $(X, P) \simeq (\{ xy+z^{2}+u^{a} = 0 \} / \mathbb{Z}_{2}(1, 1, 1, 0), o)$ for some $a \in \mathbb{N}$. Moreover we showed that such an $X$ is birational to a simpler Mori fiber space. In this paper, we prove their existence except for a few cases by constructing a Mori fiber space with desired properties and reconstructing $X$ from it.

This paper is concerned with a finitely generated module $M$ over a(commutative Noetherian) local ring $R$. In the case when $R$ is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of $M$ with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when $R$ is universally catenary and such that all its formal fibres are Cohen-Macaulay. These formulae involve certain subsets of the spectrum of $R$ called the pseudo-supports of $M$; these pseudo-supports are closed in the Zariski topology when $R$ is universally catenary and has the property that all its formal fibres are Cohen-Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.