Is Prime Number Density Uniformly Distributed Among Integers?

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Discussion Overview

The discussion revolves around the concept of prime number density and whether it is uniformly distributed among integers. Participants explore the nature of prime distribution, potential relationships, and existing research on the topic, including references to the prime number theorem and the Riemann zeta function.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions the uniformity of prime distribution among integers, noting that primes do not appear uniformly when listing numbers from 1 to 50.
  • Another participant suggests looking into the prime number theorem as a potential resource for understanding prime distribution.
  • A participant expresses curiosity about whether there exists a relationship that quantifies the number of primes within a certain bound and questions the existence of complex patterns in their distribution.
  • Some participants humorously reference their own experiences with mathematical proofs, indicating a light-hearted approach to the discussion.
  • One participant asserts that significant progress in understanding prime distribution requires more than empirical studies, implying the complexity of the topic.
  • A later reply mentions a connection between the Fourier transform of the distribution of zeros of the zeta function and the distribution of primes and prime powers.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of prime number density or its distribution. Multiple competing views and questions remain unresolved throughout the discussion.

Contextual Notes

Participants express uncertainty regarding the existence of specific relationships or patterns in prime distribution, and there are references to complex mathematical concepts that are not fully elaborated upon.

Winzer
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Before I went to bed I had an idea about integers. Is there such thing as a prime number density? I just listed 1 through 50 and found that primes aren't uniformly distributed(that I noticed). Now by typical density definition the density should be the number of primes as a function of some bound over the space. Has anyone done work on this?
 
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Look for prime number theorem.
 
Winzer said:
Has anyone done work on this?

The margin is too small to even begin to list them.
 
matt grime said:
The margin is too small to even begin to list them.
But does there exist a relationship that tells us exactly how many primes are within a certain bound?
Is there some complex pattern?
 
Funny you should say that I just found a remarkable proof of Fermats last theorem but my margin was too small to write it down.Now I have forgotten it.Damm.
 
That's exactly what I said when I sent in my paper to the Clay institute: the margin was too small but the proofs of all seven so called unsolvables are trivial--Do I get my money now?. They didn't take it to well.
 
It doesn't matter who you are, it'll be safe to say that many people smarter than you have spent the equivalent of many lifetimes of full-time study looking at the distribution of prime numbers.

Suffice to say that any progress in this area isn't going to come about from empirical study of their distribution.
 
Actually, the Fourier transform of the distribution of zeros of the zeta at +1/2 is equal to the distribution of primes and prime powers.
 

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