Related papers: Greedy Sums and Dirichlet Series

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…

A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples of positive integers satisfying the property that their sum is equal to their…

We prove that a countable direct sum of chains has either one, countably many or else continuum many isomorphism classes of siblings. This proves Thomass\'e's conjecture for such structures. Further, we show that a direct sum of chains of…

In this article, we present a family of numerical approaches to solve high-dimensional linear non-symmetric problems. The principle of these methods is to approximate a function which depends on a large number of variates by a sum of tensor…

In this paper, we introduce a new concept namely degree polynomial for vertices of a simple graph. This notion leads to a concept namely degree polynomial sequence which is stronger than the concept of degree sequence. After obtaining the…

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.

In the first part, we consider generalized quadratic Gauss sums as finite analogues of the Jacobi theta function, and the reciprocity law for Gauss sums as their transformation formula. We attach finite Dirichlet series to Gauss sums using…

We prove a discretized Product Theorem for general simple Lie groups, in the spirit of Bourgain's Discretized Sum-Product Theorem.

A recent paper of Furdui and Valean proves some results about sums of products of "tails" of the series for the Riemann zeta function. We show how such results can be proved with weaker hypotheses using multiple zeta values, and also show…

With an artificial (p', n')-system it has been proved that even numbers > (p(x))^2 are the sum of two p > p(x).

Sums of the singular series constants that appear in the Hardy--Littlewood $k$-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over…

We show that deciding the equality of two Dedekind sums $S(c,b)$, $S(d,b)$ is equivalent to deciding whether a Dedekind sum defined by $b, c, d$ takes a certain value. By means of this result we construct infinite sequences of pairwise…

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…

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…

For any positive integers $h\ge 2$ and $g\ge 1$, we present a greedy algorithm that provides an infinite $B_h[g]$ sequence with $a_n\le 2gn^{h+(h-1)/g}.$

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

Erd\H{o}s and Graham found it conceivable that the best $n$-term Egyptian underapproximation of almost every positive number for sufficiently large $n$ gets constructed in a greedy manner, i.e., from the best $(n-1)$-term Egyptian…

In this article the infinite product of bicomplex numbers is defined and the convergence and divergence of this product are discussed.

Motivated by the change-making problem, we extend the notion of greediness to sets of positive integers not containing the element $1$, and from there to numerical semigroups. We provide an algorithm to determine if a given set (not…

We establish the existence of a quantum lift of the greedy basis.