Lemmas on the Number of Divisors, on Prime Numbers and Trigonometric Sums

2005 ◽  
pp. 23-35
Keyword(s):  
Prime Numbers ◽  
Trigonometric Sums ◽  
Number Of Divisors

Estimates of Certain Simple Trigonometric Sums with Prime Numbers

1985 ◽  
pp. 133-159
Author(s):  
Ivan Matveevič Vinogradov
Keyword(s):  
Prime Numbers ◽  
Trigonometric Sums

A remark on a theorem of the Goldbach-Waring type

2004 ◽  
Vol 41 (3) ◽  
pp. 309-324
Author(s):  
C. Bauer
Keyword(s):  
Prime Numbers

Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but ≪ x23027/23040+ε even integers N ≤ x can be written as N = p21 + p32 + p43 + p45.

The First 50 Million Prime Numbers

1977 ◽  
Vol 1 (S2) ◽  
pp. 7-19 ◽  
Author(s):  
Don Zagier
Keyword(s):  
Prime Numbers

Density of summable subsequences of a sequence and its applications

2020 ◽  
Vol 70 (3) ◽  
pp. 657-666
Author(s):  
Bingzhe Hou ◽  
Yue Xin ◽  
Aihua Zhang
Keyword(s):  
Prime Numbers ◽  
Linear Operators ◽  
Upper Density ◽  
Asymptotic Densities

AbstractLet x = $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

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