Related papers: The distribution of prime numbers
These are lecture notes written at the University of Zurich during spring 2014 and spring 2015. The first part of the notes gives an introduction to probability theory. It explains the notion of random events and random variables,…
We obtain an upper bound for the distribution of primes in the form $n^4 + k$ up to $x$, averaged over $k$ with small square-full part. As a corollary, we show that for almost all $k$, there is an expected amount of primes in the form $n^4…
In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d <= 5. In analogy with the classical Erdos-Kac theorem, we prove for S_d-extensions that the number of such primes is normally…
The Riesz distribution for real normed division algebras is derived in this work. Then two versions of these distributions are proposed and some of their properties are studied.
The probability of finding a prime multiplet, i.e., a sequence of primes $p$ and $p+a_i$, $i=1... m$, being all primes where $p$ is some prime less than the integer $n$ is naively $1/log(n)^{m+1}$. It is shown that, in reality, it is…
These lecture notes are based on lectures given in 2019 Saint-Flour Probability School.
This is a survey article outlining what is known about absolute primes.
In this paper, we will give some estimation for the average error of the prime number theorem.
Two constructed prime number subsets (called prime brother & sisters and prime cousins) lead to a third one (called isolated primes) so that all three disjoint subsets together generate the prime number set. It should be suggested how the…
The aim of this paper is to prove Cotlar's ergodic theorem modeled on the set of primes.
This text is a survey of the general theory of stochastic processes, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which were developed by the French probability school…
Extending the classical Dirichlet's density theorem on coprime pairs, in this paper we describe completely the probability distribution of the number of coprime pairs in random squares of fixed side length in the lattice $\mathbb{N}^2$. The…
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
This is a pedagogical article cited in the foregoing research note, quant-ph/9911050
In recent work by Botkin, Dawsey, Hemmer, Just and the present author, a deterministic model of prime number distribution is developed based on properties of integer partitions that gives almost exact estimates for $\pi(n)$, the number of…
We prove asymptotic results for the singular series associated to the distribution of three primes. Assuming a quantitative version of Hardy and Littlewood's conjecture on prime 3-tuples, we deduce an asymptotic formula related to the joint…
If species abundance distributions are dominated by the simple processes of individuals in a community giving birth and death independently, the result is a log series distribution. I calculate this in a number of different ways, using both…
Recently we have introduced a novel characterisation of the distribution of twin primes that consists of three essential elements. These are: that the twins are most naturally viewed as a subsequence of the primes themselves, that the…
An unconditional inequality of the totient function is contributed to the literature. This result is associated with various problems about the distribution of prime numbers.
This paper is my contribution to the planned publication Recent Perspectives in Random Matrix Theory (Cambridge University Press). Addressed is the problem of computing spacing distributions in the bulk for the three symmetry classes…