Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [distribution-of-primes]

Use this tag for questions related to the branch of number theory studying distribution laws of prime numbers among natural numbers.

The central problems are to find the best expression as $x \to \infty$ for–

  • the number of prime numbers not exceeding $x$, and
  • the number of prime numbers not exceeding $x$ in an arithmetic progression.
122 questions
3
votes
1 answer

The Density of Primes

The below discussion comes from HM Edwards book on the Riemann Zeta Function. (2′)$$\sum_{p < x}1/p∼\log(\log)\,\,(→∞)$$, (...) Now $$\log(\log)= \ \int_{1}^{\log(x)} \frac{du}{u} \ = \int_{e}^{x} \frac{1}{v} \frac{dv}{\log v}$$ so (2') says that…
L. Tim
  • 231
2
votes
1 answer

Primes of form $6n-1$

we know that the probability that a given $n\in\mathbb{N}$ is a prime is $\frac{1}{\log n}$ and all primes except 2 and 3 are of form $6n\mp 1$. We can deduce that the probability that $6n-1$ is $\frac{1}{2\log n}$? Thank you.
Theory Nombre
  • 399
2
votes
1 answer

Is the twin of a known prime more likely to be prime?

If I have a large known prime, is the number 2 greater than that or 2 less than that more likely than chance to be prime? It seems that since finding new large primes is something of a newsworthy event (especially so if they are the new largest…
Ze'ev misses Monica
  • 182
1
vote
0 answers

prime number counting assuming RH: exact or not?

In https://www.quora.com/What-is-the-relationship-between-the-Riemann-Hypothesis-and-prime-numbers and https://en.wikipedia.org/wiki/Prime-counting_function#Exact_form talk of exact formulas for the prime number counting function (assuming the…
nomadreid
  • 352
1
vote
0 answers

Distribution of divisions of a circle

When viewing distributions divisions of a circle, an interesting behavior is displayed. Take n nested/circumscribed circles. Divide each circle in to n parts and plot a point for each division. There are patterns formed. The integers seem to form…
AbstractDissonance
  • 1
1
vote
1 answer

Is this new method to find primes?

By observation I found this method which can be proofed easily ,but not sure if it is mentioned somewhere before and that's what almost should be. This is the method: For every $p_n$ & $a$ that satisfy $p_n < p_n\#-a < p_{n+1}^2$ If $(a)$ is a prime…
Pentapolis
  • 601
0
votes
1 answer

Density of Primes in different sets

I am examining the density of Primes in other sets than the naturals. E.g. we want to have the density of Mersenne Primes. From the prime number theorem I know that in the naturals we have $$ \frac{\pi(n)}{\frac{n}{\log(n)}} \rightarrow 1. $$ But…
Lereu
  • 424