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

We introduce the Limiter, a universal extension of the real numbers and of the limit functional that assigns a canonical limit in an enlarged space to every real sequence. Motivated by generalized summation methods such as Borel summation…

Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…

In this article, we investigate conditional large values of quadratic Dirichlet character sums with multiplicative coefficients. We prove some Omega results under the assumption of the generalized Riemann hypothesis.

Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.

We extend Salem's Integral equation to the non-homogenous form, and offer the associated criteria for the Riemann Hypothesis. Explicit solutions for the non-homogenous case are given, which in turn give further insight into Salem's criteria…

In this paper we use a dynamical approach to prove some new divergence theorems on complete non-compact Riemannian manifolds.

A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…

It is shown that, given any finite set of pairs of random events in a Boolean algebra which are correlated with respect to a fixed probability measure on the algebra, the algebra can be extended in such a way that the extension contains…

A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.

We rewrite Riemann Zeta function as a sum over the primes. Each term of the sum is a product that depends only on the summation index (a prime) and the primes following it.

We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation".…

The properties of the Riemann extensions of nonriemannian spaces defined by the first order systems of differential equations are considered.

We give a simple and general central limit theorem for a triangular array of m-dependent variables. The result requires only a Lindeberg condition and avoids unnecessary extra conditions that have been used earlier. The result applies also…

We prove an extension of the Bourgain-Sarnak-Ziegler theorem and then apply it to bound certain polynomial exponential sums with modular coefficients.

The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…

One of the approaches to the Riemann Hypothesis is the Nyman-Beurling criterion. Cotangent sums play a significant role. Here we investigate the values of these cotangent sums for various shifts of the argument.

Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banach space and let $\tau$ be a permutation of natural numbers. We study the set $\operatorname{LIM}(\sum_{n=1}^\infty x_{\tau(n)})$ of all limit points of a sequence…

The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide…

As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity…

We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.