Related papers: Note on Prime Gaps and Zero Spacings
We discuss recent advances on weak forms of the Prime $k$-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps between primes.
Gap balancing numbers are a certain generalization of balancing and cobalancing numbers that arise from studying the equation ${T(L)+T(B)=T(m)}$ where $T(i)$ is the $i$th triangular number. In this paper, we survey early results, attempt to…
A boundary behavior of mappings, which are closely related with Sobolev and Orlicz--Sobolev classes in the plane and in the space, is investigated. There are obtained theorems on boundary behavior of classes mentioned above.
We give the lower bound for the growth of the maximum value for a solution to the minimal surface equation with 0 boundary values over an unbounded simply connected domain.
We introduce a novel collection of uniqueness problems, with related sampling and interpolation issues. We call them deep zero problems, as they are concerned with local properties at a small number of given points.
In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erd\H{o}s-Rankin bound about large gaps between consecutive primes. The work answers in a strong…
In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…
Considered here is one quite general problem about description of extremal configurations maximizing the product of inner radii non-overlapping domains.
This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…
In this paper we establish bounds on the number of vertices for a few classes of convex sublattice-free lattice polygons. The bounds are essential for proving the formula for the critical number of vertices of a lattice polygon that ensures…
We survey the variants of Erd\H{o}s' distinct distances problem and the current best bounds for each of those.
We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic…
In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic…
Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed…
Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime…
This paper concerns the number of lattice points in a circle.
Using Duke's large sieve inequality for Hecke Gr{\"o}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application,…
In this article we summarize what is known about the initial-boundary value problem for general relativity and discuss present problems related to it.
In this work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically, we have inequalities for these indices relating them with other indices: the first Zagreb index, the forgotten index and the first variable…
We consider the edge- and vertex-isoperimetric probem on finite and infinite hexagonal grids: For a subset W of the hexagonal grid of given cardinality, we give a lower bound for the number of edges between W and its complement, and lower…