Let $n = p_{1} p_{2} \cdots p_{r}$ be a product of $r$ prime numbers which are not necessarily different. We define then an arithmetic function $\mu_{m}(n)$ by $$\mu_{m}(n) = \rho^{r} \quad (\rho = e^{2\pi i/m}),$$ where $m$ is a natural number. We further define the function $L(s, \mu_{m})$ by the Dirichlet series $$L(s, \mu_{m}) = \sum_{n=1}^{\infty} \frac{\mu_{m}(n)}{n^{s}} = \prod_{p} \Bigl( 1-\frac{\rho}{p^{s}} \Bigr)^{-1} \quad (\Re s > 1), $$ and will show that $L(s, \mu_{m})$, $(m \geq 3)$, has an infinitely many valued analytic continuation into the half plane $\Re s > 1/2$.

We establish that almost all natural numbers not congruent to $5$ modulo $9$ are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

The commutators of convolution operators are considered. By localization and Fourier transform estimates, a sufficient condition such that these commutators are bounded on $L^2(\mathbb{R}^n)$ is given. As applications, some new results about the $L^2(\mathbb{R}^n)$ boundedness for the commutators of homogeneous singular integral operators are established.

We introduce a notion of kernel systems on finite groups: roughly speaking, a kernel system on the finite group $G$ consists in the data of a pseudo-Frobenius kernel in each maximal solvable subgroup of $G$, subject to certain natural conditions. In particular, each finite $CA$-group can be equipped with a canonical kernel system. We succeed in determining all finite groups with kernel system that also possess a Hall $p'$-subgroup for some prime factor $p$ of their order; this generalizes a previous result of ours (Communications in Algebra 18(3), 1990, pp. 833-838). Remarkable is the fact that we make no a priori abelianness hypothesis on the Sylow subgroups.

We consider a positively graded noetherian domain $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ for which $R_{0}$ is essentially of finite type over a perfect field $K$ of positive characteristic and we assume that the generic fibre of the natural morphism $\pi : Y = \rm{Proj}(R) \to Y_{0} = \rm{Spec}(R_{0})$ is geometrically connected, geometrically normal and of dimension $> 1$. Then we give bounds on the "ranks" of the $n$-th homogeneous part $H^{2}_{R_{+}} (R)_{n}$ of the second local cohomology module of $R$ with respect to $R_{+} := \bigoplus_{m > 0} R_{m}$ for $n < 0$. If $Y$ is in addition normal, we shall see that the $R_{0}$-modules $H^{2}_{R_{+}} (R)_{n}$ are torsion-free for all $n < 0$ and in this case our bounds on the ranks furnish a vanishing result. From these results we get bounds on the first cohomology of ample invertible sheaves in positive characteristic.

Let $\widehat {\Gamma}$ be a free abelian group of finite rank, let $\Gamma$ be a sub-semigroup of $\widehat {\Gamma}$ satisfying certain finiteness conditions, and let $\mathfrak{L}=\bigoplus_{(\alpha, a) \in \Gamma \times \mathbb{Z}_2} \mathfrak{L}_{(\alpha, a)}$ be a ($\Gamma\times\mathbb{Z}_{2}$)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of $\mathfrak{L}$, we derive a new recursive formula for the dimensions of homogeneous subspaces of $\mathfrak{L}$. When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson's root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.

We study the existence of $L^2$ holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of $L^2$ holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.

There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra's construction of independent cyclotomic units.

In this paper, firstly, we show the Bondi-mass type estimate of solutions of Calabi flow on closed $4$-manifolds. Secondly, in our applications, we obtain the long time existence on closed $4$-manifolds. In particular, we are able to show the asymptotic convergence of a subsequence of solutions of the Calabi flow on closed Einstein $4$-manifolds.

It is shown, that the so-called Blaschke condition characterizes in any bounded smooth convex domain of finite type exactly the divisors which are zero sets of functions of the Nevanlinna class on the domain. The main tool is a non-isotropic $L^1$ estimate for solutions of the Cauchy-Riemann equations on such domains, which are obtained by estimating suitable kernels of Berndtsson-Andersson type.