What is Riemann hypothesis: Definition and 74 Discussions

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.
The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:

The real part of every nontrivial zero of the Riemann zeta function is 1/2.
Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit.

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  1. E

    Riemann Hypothesis: Exploring Unsolved Mysteries

    I have two questions: Why hasn't the hypothesis been proved yet? Is it because we don't know why re(s) has to be 1/2 and thus can't prove it, or is it because we know why re(s) has to be 1/2 but we just don't know how to prove it. Why exactly does re(s) have to be 1/2? \zeta...
  2. I

    Riemann Hypothesis: Question on Critical Line

    I have a question concerning the Riemann Hypothesis, a conjecture about the distribution of zeros of the Riemann-zeta function. the trivial zeros (s=-2, s= -4, s=-6) arent much of a concern as the NON-trivial zeros, where any real part of the non-trivial zero is = 1/2. What i am having...
  3. C

    Lagarias’ equivalence to the Riemann hypothesis

    Lagarias’ equivalence to the Riemann hypothesis should be discussed, i.e., if hn := n-th harmonic number := 1/1 + 1/2 + · · · + 1/n, and σn := divisor function of n := sum of positive divisors of n, then if n > 1, hn + ehn ln hn > σn. There is a $1,000,000 prize for the proof of this at...
  4. C

    Proof of the Riemann Hypothesis

    http://arxiv.org/abs/0806.0892"
  5. H

    Prove Riemann Hypothesis: High School Student Guide

    i am a high school student i want to prove the riemann hypothesis but i do not how to start:confused:
  6. K

    Understanding the Trivial Zeros of the Riemann Hypothesis

    Can somebody explain me about the trivial zeros? Why \zeta(-2) = \zeta(-4) = \zeta(-6) = 0 = \zeta(k) So \zeta(k) \sum_{n=1}^{ \infty} \frac{1}{n^k} = 0 ?
  7. N

    Is this the end of the Riemann Hypothesis?

    http://www.arsmathematica.net/archives/2007/05/22/latest-on-latest-paper-on-arxiv/
  8. W

    Explain Riemann Hypothesis: Imaginary Numbers & Interval

    Can someone please explain to what exactly the Riemann Hypothesis is? My friend said it is something to do with imaginary numbers and how they behave in a certain interval- just wondering.
  9. benorin

    An elementary problem equivalent to the Riemann hypothesis

    Let H_{n}=\sum_{k=1}^{n}\frac{1}{k} be the nth harmonic number, then the Riemann hypothesis is equivalent to proving that for each n\geq 1, \sum_{d|n}d\leq H_{n}+\mbox{exp}(H_{n})\log H_{n} where equality holds iff n=1. The paper that this came from is here: An Elementary Problem...
  10. B

    Exploring the Riemann Hypothesis and Its Implications for Prime Numbers

    let \zeta(z)=\sum_{n \in \mathbb{N}} n^{-z} ~ {{a+ib}}>1 then, \zeta(z)=0 iff z=-2n where n is a natural number. pi(x)=\int_0^\infty\frac{dx}{\xS[x+1]} gamma(x+) where S[x+1]= \sum_{n \in \mathbb{N}} n^-{x+1} I have discovered that pi(x)=\int_a^b\frac{dx}/logx = 1/log b+ 2/log b...
  11. W

    Anybody tried solving Riemann Hypothesis?

    Just wondering if any of the posters seriously tried solving Riemann Hypothesis. And if yes, then what kinds of problems did you run into?
  12. -Job-

    What is the Riemann Hypothesis and why is it so difficult to solve?

    I know this is one of the famous unsolved problems still hanging around. Could someone give me the "gist" of it, and what the implications are if it is solved one way or the other? I looked it up on Wikipedia but that didn't help me much. Has anyone any idea why it is so hard to solve (i imagine...
  13. E

    Could Riemann Hypothesis be completely false?

    Heilbronn proved that the Epstein Zeta function did not satisfy RH...but the Zeta function \zeta(s) can be put in a form of an Epstein function but a factor k..let be the functional equation for Epstein functions: \pi^{-s}\Gamma(s)Z_{Q^{-1}}(s)=|Q|^{1/2}\pi^{s-n/2}\Gamma(n/2-s)Z_{Q}(n/2-s)...
  14. E

    A proof of Riemann hypothesis (but of course the snobbish journals don,t want

    to publish it because i,m not a famous teacher,mathematician from a snob and pedant univesity of Usa of England...this is the way science improves..only by publishing works from famous mathematician..:mad: :mad: :mad: :mad: :mad: o fcourse if i were Louis de Branges or Alain Connes or other...
  15. Jameson

    How do non-trivial zeros of the Riemann Zeta Function occur?

    I've tried my best to understand the Riemann Zeta Function on my own, but I appeal to the knowledge of you guys to help me understand more. For s >1 , the Riemann Zeta Fuction is defined as: \zeta(s)=\sum_{n=1}^{\infty}n^{-s} I have no problem with this. That series obviously converges...
  16. E

    Are Riemann hypothesis and Goldbach conjecture related?

    this is a question i have i mean are RH and Goldbach conjecture related? i mean in the sense that proving RH would imply Goldbach conjecture and viceversa: RIemann hypothesis: (RH) \zeta(s)=0 then s=1/2+it Goldbach conjecture,let be n a positive integer then: 2n=p1+p2 ...
  17. E

    An strategy to prove Riemann hypothesis

    Let be the Hamitonian of a particle with mass m in the form: H=\frac{-\hbar^{2}}{2m}D^{2}\phi(x)+V(x)\phi(x) then the RH is equivalent to prove that exist a real potential V(x) of the Hamiltonian so that the values E_n H\phi=E_{n}\phi satisfy the equation \zeta(1/2+iE_{n})=0 that is...
  18. W

    Riemann Hypothesis effect on the world

    What would be the effects on the world and to the individual or individuals if the Riemann Hypothesis was solved?
  19. P

    What is the connection between the Riemann hypothesis and prime numbers?

    Hi, When I hear about the Riemann hypothesis, it seems like the first thing I hear about it is its importance to the distribution of prime numbers. However, looking online this seems to be a very difficult thing to explain. I understand that the Riemann Hypothesis asserts that the zeroes of...
  20. S

    Riemann Hypothesis: Explaining \zeta(s) & Diagrams

    hello all after doing a bit of research on the riemann hypothesis I came along this paragraph, in which I don't understand, especially the first sentence , how would one be able to show that? It can be shown that \zeta (s) = 0 when s is a negative even integer. The famous Riemann...
  21. I

    What progress has been made on the Riemann Hypothesis since 2004?

    http://news.uns.purdue.edu/UNS/html4ever/2004/040608.DeBranges.Riemann.html Any news since then? There are links to the papers themselves on the bottom of the page. But I can't understand much, I'm afraid.
  22. C

    De Branges and the Riemann Hypothesis

    In June of this year the mathematician Louis de Branges published in Internet a proposed "proof" of the Riemann Hypothesis. The page is: http://www.math.purdue.edu/~branges/riemannzeta.pdf Years ago De Branges proved the Bieberbach Conjecture. He has tried several times to proof the RH...
  23. I

    Riemann Hypothesis: Will it Ever be Proven?

    Just an opinion question, do you think that the riemann hypothesis will ever be proven? If so, how long do you think it will take?
  24. T

    Riemann Hypothesis: What Is It? Online Resources

    What is the Riemann Hypothesis? Where can I find good online literature upon the subject?:smile:
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