Articles of prime gaps

A question about an asymptotic formula

I’ve been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann’s hypothesis is true, but I was unable to find a journal reference for this. Does anybody know of any journal reference or any other source where I can find this conditional result?

Corollaries of Green-Tao Theorem?

there is already a good thread which discusses some corollaries of the Green-Tao Theorem, here: Constructing arithmetic progressions The question I was wondering about is of a similar flavor but isn’t mentioned, namely, does there exist a $c \in \mathbb{R}$ such that given any $k \in \mathbb{N}$, can you always find a progression of primes […]

Does $\sum_{n=1}^\infty \frac{1}{p_ng_n}$ diverge?

I know of Euler’s proof that the sum of the reciprocals of the primes diverges. But what if we multiply the primes by it’s following prime gap. In other words, is $$\sum_{n=1}^\infty \frac{1}{p_ng_n} = \infty$$ true or false?

Analog of Cramer's conjecture for primes in a residue class

Let $q$ and $r$ be fixed coprime positive integers, $$ 1 \le r < q, \qquad \gcd(q,r)=1. $$ Suppose that two prime numbers $p$ and $p’$, with $p<p’$, satisfy $$ p \equiv p’ \equiv r \ ({\rm mod}\ q), \tag{1} $$ and no other primes between $p$ and $p’$ satisfy $(1)$. Then we have the […]

Geometric mean of prime gaps?

The arithmetic mean of prime gaps around $x$ is $\ln(x)$. What is the geometric mean of prime gaps around $x$ ? Does that strongly depend on the conjectures about the smallest and largest gap such as Cramer’s conjecture or the twin prime conjecture ?

Are there arbitrarily large gaps between consecutive primes?

This question already has an answer here: I'm trying to find the longest consecutive set of composite numbers 5 answers

Improved Betrand's postulate

I want to show that $2p_{n-2} \geq p_{n}-1$… Bertand’s postulate shows us that $4p_{n-2}\geq p_{n}$ but can we improve on this? any ideas?

The square of n+1-th prime is less than the product of the first n primes.

I wanted to prove the following question in an elementary way not using Bertrand postulate or analytic estimates like $x/\log x$. The question is $$ p_{n+1}^2<p_1p_2\cdots p_n,\qquad(n\geq4) $$ I made the following argument. Does anyone have some opinion or simpler ideas to complete. We consider two cases: Case 1: $N=p_1p_2\cdots p_n-1$ is composite: then there […]

Asymptotic Distribution of Prime Gaps in Residue Classes

Define $\pi_{n,a}(x)$ as the number of primes $p$ less than $x$ such that $p\equiv a\bmod n$ for coprime $n,a$. This function can be asymptotically approximated by $$\pi_{n,a}(x)=\frac{\operatorname{Li}(x)}{\varphi(n)}$$ This allows for the conclusion that, as $x$ tends to infinity, $\pi_{4,1}(x)\sim\pi_{4,3}(x)$. In other words, there are as many primes congruent to $1 \bmod 4$ as there are […]

On the difference between consecutive primes

Let $(p_n)$ be the sequence of prime numbers and $g_n = p_{n+1} – p_n$ Question: Is it known that $g_n \le n$? Remark: it’s known that $g_n < p_n^{\theta}$ with $\theta = 0.525$ for $n$ sufficiently large (see here), and that $p_n < n(\ln n + \ln\ln n )$ for $n \ge 6$ (see here). […]