Related papers: On Riemann's Theorem About Conditionally Convergen…

We provide numerical procedures for possibly best evaluating the sum of positive series. Our procedures are based on the application of a generalized version of Kummer's test.

We describe how the reversion of a series is related to convolutional recurrence relations for the series, and we place this relationship in the context of Riordan arrays. As an example of the approach, we give new recurrence relations for…

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute…

Equivariant Riemann-Roch theorem for the complex variety under the action of complex linear reductive algebraic group.

We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.

The problem of convergence in law of normed sums of exchangeable random variables is examined. First, the problem is studied w.r.t. arrays of exchangeable random variables, and the special role played by mixtures of products of stable laws…

In this paper, we study the distribution of the sequence of integers $2^{\omega(n)}$ under the assumption of the strong Riemann hypothesis, where $\omega(n)$ denotes the number of distinct prime divisors of $n$. We provide an asymptotic…

For $\frac{1}{2}<x<1$, $y>0$, and $n\in\mathbb{N}$, let $\displaystyle\theta_n(x+iy)=\sum_{i=1}^n\frac{{\mbox{sgn}}\, q_i}{q_i^{x+iy}}$, where $Q=\{q_1,q_2,q_3,\cdots\}$ is the set of finite products of distinct odd primes, and…

On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

We show how to use extended word series in the reduction of continuous and discrete dynamical systems to normal form and in the computation of formal invariants of motion in Hamiltonian systems. The manipulations required involve complex…

We prove a new result on multiple summing operators and among other applications, we provide a new extension of Littlewood's $4/3$ inequality to $m$-linear forms.

We study the convergence sets of a class of alternating series. Among other things, our results establish the convergence of the series $\sum_n (-1)^n|\sin n|/n$.

We study the decomposition of real numbers into sums of L\"uroth sets, which are defined by numbers whose L\"uroth expansions have prescribed digit constraints. We establish several results on the congruence modulo 1 of sums of L\"uroth…

The Levy-Steinitz theorem characterizes the values that a conditionally convergent sequence in of real numbers can attain under permutations. We extend this analysis to sequences of countable sequences of real numbers, under pointwise…

In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (frame-like) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or…

The method of using rearrangements to give sufficient conditions for Fourier inequalities between weighted Lebesgue spaces is revisited. New results in the case q < p are established and a comparison between two known sufficient conditions…

The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also…

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.