Chebyshev upper estimates for Beurling's generalized prime numbers

Let N be the counting function of a Beurling generalized number system and let pi be the counting function of its primes. We show that the L-1-condition integral(infinity)(1)vertical bar N(x) - ax/x vertical bar dx/x < infinity and the asymptotic behavior N(x) = ax + O (x/log x), for some a > 0, suffice for a Chebyshev upper estimate pi(x) log x/x <= B < infinity

The structure of quasiasymptotics of Schwartz distributions

In this article complete characterizations of quasiasymptotic behaviors of Schwartz distributions are presented by means of structural theorems. The cases at infinity and the origin are both analyzed. Special attention is paid to the quasiasymptotic of degree -1 and it is shown how the structural theorem can be used to study Ces\`{a}ro and Abel summability of trigonometric series and integrals. Further properties of quasiasymptotics at infinity are discussed, the author presents a condition over test functions which allows one to evaluate them at the quasiasymptotic, these test functions are in bigger spaces than $\mathcal{S}$. An extension of the structural theorems for quasiasymptotics is given, the author studies a structural characterization of the behavior $f(\lambda x)=O(\rho(\lambda))$ in $\mathcal{D'}$, where $\rho$ is a regularly varying function

Structural theorems for quasiasymptotics of distributions at infinity

Complete structural theorems for quasiasymptotics of distributions are presented in this article. For this, asymptotically homogeneous functions and associate asymptotically homogeneous functions at infinity with respect to a slowly varying function are employed. The proposed analysis, based on the concept of asymptotically and associate asymptotically homogeneous functions, allows to obtain easier proofs of the structural theorems for quasiasymptotics at infinity in the so far only known case: when the degree of the quasiasymptotic is not a negative integer. Furthermore, new structural theorems for the case of negative integral degrees are obtained by this method

On Diamond's L1L^1 criterion for asymptotic density of Beurling generalized integers

We give a short proof of the $L^{1}$ criterion for Beurling generalized integers to have a positive asymptotic density. We actually prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)=\sum_{n_{k}\leq x} \mu(n_k)/n_k=o(1)$, with $\mu$ the Beurling analog of the Moebius function.Comment: 13 page

Tauberian class estimates for vector-valued distributions

We study Tauberian properties of regularizing transforms of vector-valued tempered distributions, that is, transforms of the form $M^{\mathbf{f}}_{\varphi}(x,y)=(\mathbf{f}\ast\varphi_{y})(x)$, where the kernel $\varphi$ is a test function and $\varphi_{y}(\cdot)=y^{-n}\varphi(\cdot/y)$. We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. Our goal is to characterize spaces of Banach space valued tempered distributions in terms of so-called class estimates for the transform $M^{\mathbf{f}}_{\varphi}(x,y)$. Our results generalize and improve earlier Tauberian theorems of Drozhzhinov and Zav'yalov [Sb. Math. 194 (2003), 1599-1646]. Special attention is paid to find the optimal class of kernels $\varphi$ for which these Tauberian results hold.Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:1012.509

On General Prime Number Theorems with Remainder

We show that for Beurling generalized numbers the prime number theorem in remainder form \pi(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n\in\mathbb{N} is equivalent to (for some $a>0$) N(x) = ax + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n \in \mathbb{N}, where $N$ and $\pi$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number theorem with remainders in the Ces\`aro sense.Comment: 15 page

Solution to the first Cousin problem for vector-valued quasianalytic functions

We study spaces of vector-valued quasianalytic functions and solve the first Cousin problem in these spaces.Comment: 23 page