arXiv subjects
Publications and source records attributed to Douglas Azevedo.
In this paper, we propose extensions for the classical Kummer test, which is a very far-reaching criterion that provides sufficient and necessary conditions for convergence and divergence of series of positive terms. Furthermore, we present and discuss some interesting consequences and examples such as extensions of the Olivier's theorem and Raabe, Bertrand and Gauss's test.
In this paper, we use the infamous continuous and nowhere differentiable Weierstrass function as a prototype to define a Weierstrass fractal kernel. We investigate the properties of the reproducing kernel Hilbert space (RKHS) associated with this kernel by presenting an explicit characterization of this space. In particular, we show that this space has a dense subset composed of continuous but nowhere differentiable functions. Moreover, we present sharp estimates for the covering numbers of the unit ball of this space as a subset of the continuous functions.
For a certain function $J(s)$ we prove that the identity $$\frac{ζ(2s)}{ζ(s)}-\left(s-\frac{1}{2}\right)J(s)=\frac{ζ(2s+1)}{ζ(s+1/2)}, $$ holds in the half-plane Re$(s)>1/2$ and both sides of the equality are analytic in this half-plane.
We use the Prime Number Theorem to prove the existence of zero-free strips for the Riemann-zeta function. Precisely, we prove that there exists $δ>0$ for which if $0\leq r<δ$ then $ζ(s)\neq 0$ for Re$(s)>1-r$.
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this direct approach is a consequence of the the Kummer's characterization of summable sequences of positive terms. Some interesting consequences are then presented. In particular, we show how the twin-prime conjecture is related to our main result.
Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many values of $n$. As it is shown, the key ingredient to obtain this behaviour is a consequence of an extension of the Kummer's characterization of convergent series of positive terms.
In this note we present a method to bound gaps between primes via the divergence of the series of reciprocals of the prime numbers, a consequence of a version of the Bertrand's test for convergence of series of positive numbers and a suitable series of positive terms related to the logarithmic integral function. Using this proposed method we discuss the Polignac's conjecture and present some consequences.