Related papers: On a sequence involving the prime numbers

We estimate the local laws of the distribution of the middle prime factor of an integer, defined according to multiplicity or not. An asymptotic estimate with effective remainder is provided for a wide range of values. In particular this…

We provide an elementary proof of an asymptotic formula for prime counting functions. As a minor application we give a new reduction of the proof of Chebotar\"ev's density theorem to the cyclic case.

We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.

We briefly describe some well-known means and their properties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, deriving from the harmonic mean, motivate the definition of a new kind of mean that we…

We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.

We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only…

We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations,…

In this note we prove an inequality involving primes and the product of consecutive primes.

By involving some exponential sums related to $\Lambda(n)$ in arithmetic progression, we can obtain some new results for von Mangoldt function over {\bf nonhomogeneous} Beatty sequences in arithmetic progressions, which improve some recent…

We consider a certain equidistributed sequence of rational numbers constructed from the primes. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known…

We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.

There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the…

In this note, we find a new inequality involving primes and deduce several Bonse-type inequalities.

Let $a,b>0$ be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form $ax^2+by^3$ with $x \leq X^{1/2}$ and $y \leq X^{1/3}$. The proof…

We illustrate the interplay between certain discrete and continuous problems, by presenting a method for the study of the asymptotics of a divergent sequence, through consideration of the asymptotics of its continuous analogue

Sequences diverge either because they head off to infinity or because they oscillate. Part 1 constructs a non-Archimedean framework of infinite numbers that is large enough to contain asymptotic limit points for non-oscillating sequences…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the…

The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has…

Let $(u_n)_{n \geq 0}$ be a nondegenerate linear recurrence of integers, and let $\mathcal{A}$ be the set of positive integers $n$ such that $u_n$ and $n$ are relatively prime. We prove that $\mathcal{A}$ has an asymptotic density, and that…

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…