Grander than the Riemann Hypothesis

In summary, there have been many attempts made to produce a formula or function that generates all prime numbers, but none have been successful. Examples of closed-form solutions have been given, but they do not generate all primes. It has been suggested that such a formula could potentially solve the Riemann hypothesis, but this has not been proven. There is a possibility that a "nice" form of the prime-counting function could help prove the Riemann hypothesis, but this is currently only speculation.
  • #1
tgt
522
2
...But it may not exist yet.

Has any mathematician thought about producing a formula or function which spits out all the prime numbers? i.e 1->2, 2->3, 3->3, 4->5, 5->7, 6->11 etc.

Any attempts been made?

What the closest that people have thought?
 
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  • #2
You mean a closed-form solution, I assume? As in, piecewise with a finite number of piecewise parts?

I think it's safe to assume that nothing of the sort yet exists... if it did, I believe it would answer the Riemann hypothesis. No?

I also believe I read something somewhere about a closed-form polynomial never being able to generate "only" prime numbers.

See:

http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
 
  • #3
tgt said:
Has any mathematician thought about producing a formula or function which spits out all the prime numbers? i.e 1->2, 2->3, 3->3, 4->5, 5->7, 6->11 etc.

Yes. There are many of these. Ribenboim (2004) lists many examples on pp. 131 to 155, as does Guy in UPNT A17 (pp. 58-65 in the third edition). See also Prime Formulas on MathWorld.

Hardy & Wright mention this in Chapter 1 (p. 6 in the latest printing) and give several examples.

csprof2000 said:
You mean a closed-form solution, I assume? As in, piecewise with a finite number of piecewise parts?

Willans, Wormell, Mináč, and Gandhi all give examples of closed-form solutions.

csprof2000 said:
I think it's safe to assume that nothing of the sort yet exists... if it did, I believe it would answer the Riemann hypothesis. No?

It would not solve the Riemann hypothesis.

csprof2000 said:
I also believe I read something somewhere about a closed-form polynomial never being able to generate "only" prime numbers.

See:

http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

A one or two-variable polynomial can't produce all the primes (and only primes). But a 26-variable one can; Jones, Sato, Wada and Wiens gives an explicit example after Matijasevič showed it was possible.
 
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  • #4
I found this one really cool:
http://mathworld.wolfram.com/PrimeDiophantineEquations.html
How on Earth do you find such a formula :p
I guess it is not a coincident that the number of variables in this diophantine equation is the same as the number of variables in the polynomial that CRGreatHouse referred to.
 
  • #5
Kurret said:
I guess it is not a coincident that the number of variables in this diophantine equation is the same as the number of variables in the polynomial that CRGreatHouse referred to.

Yes, that was the result I was referring to. Matijasevič showed that is was possible by showing that recursively enumerable sets are precisely Diophantine sets, and Jones, Sato, Wada and Wiens made the polynomial you mention.
 
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  • #6
CRGreathouse said:
It would not solve the Riemann hypothesis.

It could solve it couldn't it? If we have such a bijection [tex]f : \mathbb{N} \to \mathbb{P}[/tex] wouldn't [tex]f^{-1}(p)=\pi(p)[/tex]?
 
  • #7
*-<|:-D=<-< said:
It could solve it couldn't it? If we have such a bijection [tex]f : \mathbb{N} \to \mathbb{P}[/tex] wouldn't [tex]f^{-1}(p)=\pi(p)[/tex]?

How would that help?
 
  • #8
If you had a closed form for [tex]f^{-1}(p)[/tex] it might be easier extract the necessary information to show that [tex]f^{-1}(x) = Li(x) + \mathcal{O}(\sqrt{x}\log(x))[/tex], of course it is only speculation as it would depend entirely of the form of [tex]f(n)[/tex].
 
  • #9
OK, so let
[tex]F(j)=\left\lfloor{\cos^2\pi\frac{(j-1)!+1}{j}\right\rfloor.[/tex]

Then
[tex]\pi(x)=\sum_{j=2}^xF(j).[/tex]

Alternately, let
[tex]F(j)=\frac{\sin^2\pi\frac{(j-1)!^2}{j}}{\sin^2\frac\pi j}[/tex]

Now you have two closed-form formulas for the prime-counting function. Does this help you prove the RH?
 
  • #10
Well, no ( :þ ), because I can't show that [tex]\sum_{j=2}^x F(j) - Li(x)[/tex] is [tex]\mathcal{O}(\sqrt{x} \log x)[/tex].

What I was trying to say that if we have a "nice" form of that function it might be easier to show that it is. I'm not saying it would solve RH automatically, I'm only saying it might do it, depending of the form.
 

1. What is the Riemann Hypothesis?

The Riemann Hypothesis is one of the most famous unsolved problems in mathematics. It was proposed by mathematician Bernhard Riemann in the 19th century and states that all non-trivial zeros of the Riemann zeta function lie on a certain line in the complex plane. It has far-reaching implications in number theory and has been a subject of intense research for over a century.

2. How is "Grander than the Riemann Hypothesis" different from the Riemann Hypothesis?

"Grander than the Riemann Hypothesis" is a term coined by mathematician John Derbyshire in his book of the same name. It refers to the idea that there may be a grand overarching theory in mathematics that encompasses the Riemann Hypothesis and other unsolved problems. This theory is yet to be discovered and may involve connections between various branches of mathematics.

3. Is there any evidence for "Grander than the Riemann Hypothesis"?

There is currently no concrete evidence for "Grander than the Riemann Hypothesis" as it is a speculative concept. However, many mathematicians believe that there may be connections between different areas of mathematics that have yet to be discovered, and this could potentially lead to a grand unifying theory.

4. Why is "Grander than the Riemann Hypothesis" important?

"Grander than the Riemann Hypothesis" is important because it reflects the ongoing pursuit of understanding the fundamental nature of mathematics. It challenges mathematicians to think beyond individual problems and look for deeper connections between different areas of mathematics. It also highlights the vast potential for future discoveries in mathematics.

5. How do scientists approach the study of "Grander than the Riemann Hypothesis"?

Scientists approach the study of "Grander than the Riemann Hypothesis" by exploring connections between different areas of mathematics and investigating potential underlying principles or patterns. This involves collaboration and interdisciplinary research, as well as analyzing existing theories and searching for new ones. It is a constantly evolving and challenging field of study.

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