Related papers: Finite sums of arithmetic progressions

We describe an effective method for calculating certain infinite sums, generalizations of the classical Bernoulli polynomials. As shown by Edward Witten in his papers on two-dimensional gauge theories, the correlation functions of…

Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper…

The paper gives a unified and simple proof of both theorems and Cousin's theorem.

In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.

We prove several unique continuation results for biharmonic maps between Riemannian manifolds.

We give an exact coefficients formula of any infinite product of power series with constant term equal to $1$, by using structures from partitions of integers and permutation groups. This is an universal theorem for various of Binomial-type…

Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular…

We discuss in the context of finite extensions two classical theorems of Takahasi and Howson on subgroups of free groups. We provide bounds for the rank of the intersection of subgroups within classes of groups such as virtually free…

Often topological classes of one-dimensional dynamical systems are finite codimension smooth manifolds. We describe a method to prove this sort of statement that we believe can be applied in many settings. In this work we will implement it…

We prove and generalize some recent conjectures of Z.-W. Sun on infinite series whose summands involve products of harmonic numbers and several binomial coefficients. We evaluate various classes of infinite sums in closed form by…

In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.

We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite…

We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing those in a previous paper (McPhedran et al,…

In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involves Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups. As…

We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4. In general, for…

We present a self-contained development of the Weierstrass theory of those analytic functions (single-valued or multiform) which admit an algebraic addition theorem. We review the history of the theory and present detailed proofs of the…

We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for…

In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…

The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy and Littlewood's extended Goldbach's conjecture. We examine common features of other heuristics in additive prime number theory, such as…

We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.