# 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.

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### $H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?

I'm aware of the Gram series which is equivalent to $R(x)$ (Riemann prime counting function): $$R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n}).$$ Over the interval $x=2$ to $x=10^4$ the average ...
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### Primes of the form 2...21

I was wondering what properties could have these numbers: $21, 221, 2221, 22221, ...$ At glance I thought this set would have infinitely many primes. Immediately I went to Python and I realized that ...
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### Why number 6 is the most frequent gap when subtracting all consecutive primes(the smaller from the larger)?

Using JavaScript, i felt like collecting all the distances between primes and see what pattern they may have. here is what i got: i generated all primes up to a 1000000, and made an object that counts ...
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### Why is the Riemann explicit formula for primes (via the Riemann Hypothesis) any better than existing formulae?

Many people, such as here, don't consider Willan's formula for primes, and other such formulas given here, as meaningful formulae for computing primes. My understanding of the main criticisms are that ...
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### Collinear primes $(n,p_n)$

This is about prime numbers such that at least three points $(i,p_i)$, $(j,p_j)$ and $(k,p_k)$ are on the same straight line. Conjectures: For any pair $(i,p_i),\, i>1$, there are two different ...
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### Is there really no way to generate infinitely many primes?

Is there really no way to generate infinitely many primes? A previous answer for someone asking about the Infinite generation of primes, says: There is no exact way to generate primes continuously. ...
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### A sequence is defined by the nth term: $n^3 -21n^2 +99n +121$. What primes does the sequence contain if continued to infinity? [closed]

A sequence is defined by the nth term: $$n^3 -21n^2 +99n +121$$ What primes does the sequence contain if continued to infinity? (Question given by maths teacher in stretch and challenge workshop)
1 vote
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### angular distribution of primes in a number field

For my master's thesis, I need to work on a problem related to the "angular" distribution of primes in a Number field. For the sake of simplicity, let's take a real quadratic number field K. ...
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### Any deducible pattern in this sequence? Is it related to primes?

Here's a part of a sequence that I want to understand better: $S=\{0,0,1,1,2,2,3,2,4,3,4,3,5,4,5,4,6,5,7,5,8,5,7,5,8,7,...\}.$ The sequence comes from finding distinct values for $f_n(x)$ where,  ...
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### How to calculate how many primes there are of $N$ bits
Is there a way to calculate how many primes are exactly $N$ bits in length, without generating them? I know that you can calculate how many primes are below $N$, but not how/if you can calculate ...