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. MevsEinstein

    B Does this help solve the Riemann Hypothesis?

    Hello PF! If ##\Re (s)## is the real part of ##s## and ##\Im (s)## is the imaginary part, then t is very easy to prove that $$\zeta (s) = \zeta ( \Re (s) ) \zeta ( \Im (s)i) - \displaystyle\sum_{n=1}^\infty \frac{1}{n^{\Re (s)}} [\displaystyle\sum_{k \in S, \mathbb{Z} \S = n}...
  2. fresh_42

    Insights The Extended Riemann Hypothesis and Ramanujan’s Sum

    [url="https://www.physicsforums.com/insights/the-extended-riemann-hypothesis-and-ramanujans-sum/"]Continue reading...
  3. The Riemann Hypothesis, Explained

    The Riemann Hypothesis, Explained

    The Riemann Hypothesis, Explained
  4. nomadreid

    Where is the usual sensationalism? (Riemann Hypothesis)

    That there is yet another mathematician (Dr Kumar Eswaran, Hyderabad) claiming to have solved the Riemann Hypothesis is not surprising. That the institute for which he works for (SNIST) is satisfied that the proof is correct is also not surprising. The eyebrows started to lightly ascend upon...
  5. S

    I Category Theory and the Riemann Hypothesis

    YouTube has been suggesting videos about category theory of late, and I have spent some time skimming through them, without really understanding where it's all going. A question came to mind, namely: It seems reasonably conceivable that group theory could perhaps supply a vital key to the...
  6. R

    I A thought about the Riemann hypothesis

    This is the Riemann Zeta function ##Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}##. It equals 0 only at the negative integers on the real axis and numbers of form ##1/2+x i##. The series can be expanded to this: $$\sum_{n=1}^{\infty} \frac{1}{n^s} = \sum_{n=1}^{\infty} \frac{1}{n^{1/2 + xi}} =...
  7. nomadreid

    I Connection of zeta to primes: less Euler than error or L-functions?

    Often I read that the Riemann Hypothesis (RH) is related to prime numbers because of the equivalence on Re(s)>1 of the zeta function and Eurler's product formula , but is it more accurate that the relevance of the RH to primes (or vice-versa) is either that the RH implies formulas for the...
  8. jedishrfu

    B Mathematicians Revive Abandoned Approach to the Riemann Hypothesis

    https://phys.org/news/2019-05-mathematicians-revive-abandoned-approach-riemann.html and more on the Reimann Hypothesis backstory from Numberphile:
  9. nomadreid

    I |Li(x) - pi(x)| goes to 0 under RH?

    Extremely quick question: According to http://mathworld.wolfram.com/PrimeNumberTheorem.html, the Riemann Hypothesis is equivalent to |Li(x)-π(x)|≤ c(√x)*ln(x) for some constant c. Am I correct that then c goes to 0 as x goes to infinity? Does any expression exist (yet) for c? Thanks.
  10. Spinnor

    B Get up to speed with the Riemann Hypothesis

    I found the following blog post from a Ph.D., "The Riemann Hypothesis, explained", while trying to get up to speed with the Riemann Hypothesis. It is at, https://medium.com/@JorgenVeisdal/the-riemann-hypothesis-explained-fa01c1f75d3f It seemed well written as I could understand it. Delete or...
  11. N

    A Has the Riemann hypothesis been proven?

    I am very baffled. I have heard through the grapevine that the Riemann hypothesis has been proven. My first reaction was of course to dismiss it as yet another failed attempt by someone who was not careful or by a crackpot, or some type of April's fool joke made a few months late.But what I...
  12. nomadreid

    I Video (analytic continuation) seems to mix 4-D & 2-D maps

    The question here is not asking for links to help understand analytic continuation or the Riemann hypothesis, but rather help in understand the bits of hand-waving in the following video’s explanations : https://www.youtube.com/watch?v=sD0NjbwqlYw (apparently narrated by the same person who does...
  13. Z

    MHB Solution to the Riemann Hypothesis in plain English

    The number line at x=1/2 is mediated by a concurrent incentive field whose shape can be extrapolated through the placement of prime numbers. Each prime number is a turning point in the n-dimensional movement of the imaginary number line, whose degree and angle can be determined through all the...
  14. Jason C

    Can Zeta ζ(½+it) be interpreted as a Wave function?

    In a recent article by BBM in Physical Review Letters highlights another approach to link QM to Zeta to Prove R.H. There approach proved unsuccessful. I want to ask professional Physicists if the following new approach have merit in connecting the Zeta function to QM? This new line of attack...
  15. M

    B Isn't the Riemann Hypothesis just a convention?

    If the zeta function intersects the critical line when the real part is 1/2, then it will intersect some other line when some other real part is used. Isn't the Riemann Hypothesis just based on a particular convention for the critical line?
  16. S

    I Wrong proofs of Riemann hypothesis

    Hello! I read some stuff about the Riemann hypothesis and the formulation seems pretty clear. I also read that many proof of it (well basically all of them) are wrong. I was just wondering in which way are they wrong? (I haven't find a page with the wrong proofs, together with explanations of...
  17. F

    A Is this a correct proof of the Riemann Hypothesis?

    Found an article online detailing a proof of the Riemann Hypothesis: << link deleted by mentor - unacceptable source >>
  18. M

    A Has anything similar to the Riemann hypothesis ever solved

    Has anything similar to the Riemann hypothesis ever been solved? Specifically, has anyone proven that the real part of a result of some particular function always assumes a particular value?
  19. M

    A Is this a valid physical analogy for the Riemann Hypothesis?

    http://arxiv.org/abs/1202.2115 I know Arxiv isn't a real journal, but this caught my eye. Is this a meaningful physical interpretation of the Riemann hypothesis? From what I understand, the zeta function can be modeled as a wave, but attempting to solve for the real part requires infinite...
  20. fresh_42

    Exploring the Extended Riemann Hypothesis in Modern Physics

    I just thought about the critical concepts in mathematics and physics that arose in the last century: Goedel, Schroedinger, etc. My question is: Are there any physical theories that rely on the validity of the extended Riemann Hypothesis? I don't mean computer science, i.e. secure...
  21. mathbalarka

    MHB What is the Riemann Hypothesis?

    Throughout this note, I'll give a brief, explanatory, informal introduction to the Riemann Hypothesis (RH) explaining the statement of the conjecture, the difficulties of approaching it as well as some notable consequences of RH in the field of number theory. I hope the readers will enjoy the...
  22. G

    Some insights about the Riemann hypothesis needed

    The Riemann hypothesis states, whenever the Riemann zeta function hits 0, the real part of the input must be 0.5. Does any input with real part being 0.5 make the function hit 0? Also, assuming the hypothesis is true, would it suffice to prove that if the input's real part is not 0.5, then the...
  23. A

    How Riemann hypothesis would break internet security?

    I saw this in one of the episodes of Numb3rs - (a T.V. show that describes how math can be used to solve crimes) It basically said that if Riemann hypothesis is true then it could break all the internet security. I want to know how. I couldn't understand Riemann hypothesis from Wikipedia and...
  24. Z

    Riemann Hypothesis for dynamical systems

    what are teh differential equations associated to Riemann Hypothesis in this article ?? http://jp4.journaldephysique.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/jp4/abs/1998/06/jp4199808PR625/jp4199808PR625.html where could i find the article for free ? , have...
  25. K

    What would be the impact of solving the Riemann hypothesis?

    The Riemann hypothesis is arguably the most difficult and perplexing unsolved theorems in all of mathematics. There is currently a $1,000,000 prize for it's solution. It's been 153 years since it's inception in 1859 and no mathematician has ever been able to solve it, not even Bernhard Riemann...
  26. N

    How Many Mathematicians Are Secretly Trying to Solve the Riemann Hypothesis?

    I once had a math professor who said that although they might not admit it, most mathematicians try to solve the Riemann hypothesis in their spare time. How true is this? Who is trying to solve the Riemann hypothesis? I don't just mean "are you?" (although feel free to speak up if you are)...
  27. Sherlock1

    MHB Riemann hypothesis finally gets resolved

    Happy April fool's day! (Rofl)
  28. S

    Exploring the Riemann Hypothesis and Analytic Continuation

    I don't know anything of complex analysis or analytic number theory or analytic continuation. But i read about zeta function and riemann hypothesis over wikipedia, clay institute's website and few other sources. I started with original zeta function...
  29. O

    Riemann Hypothesis and Goldbach Conjecture Proof?

    Hey guys, I saw these just showed up on arXiv, published by some unknown who claims to have invented his own number system and is not affiliated with any academic institutions. What do you make of this? http://arxiv.org/abs/1110.3465 http://arxiv.org/abs/1110.2952
  30. Z

    Riemann Hypothesis equivalence

    let be the function \sum_{\rho} (\rho )^{-1} =Z and let be the sum S= \sum_{\gamma}\frac{1}{1/4+ \gamma ^{2}} here 'gamma' runs over the imaginary part of the Riemann Zeros then is the Riemann Hypothesis equivalent to the assertion that S=2Z ??
  31. D

    Disproof of Riemann Hypothesis

    I think I have managed to disprove the Riemann Hypothesis. Hope I have not made a calculation error.
  32. D

    What is x in the Riemann Hypothesis?

    Th Riemann Hypothesis states that every 0 lies on x=0.5. What is x here?
  33. F

    How would the world change if the Poincaré conjecture was proven?

    How would the world benefit from the Riemann hypothesis being solved? Mathematicians have been trying to solve this for over 100 years, but have been unable to due to it's mind-boggling complexity and difficulty. What would the world benefit if this theorem was to be solved?
  34. L

    Riemann hypothesis and number theory

    Would the field of the number theory collapse or flourish if the Riemann Hypothesis is proved as true?
  35. R

    The clue to the proof of Riemann hypothesis

    This could be the way to proof. remember, this is not a proof. today I found a clue to solution to Riemann hypothesis: Let it be Riemann zeta function :ζ(s) The proof that all the non trivial zeroes lie on the critical strip when s = 1/2 + it let us suppose there are other zeroes...
  36. F

    Elliott-Halberstam conjecture and the Riemann Hypothesis

    I was wondering if one of the consequences of the Elliott-Halberstam conjecture would imply the Riemann Hypothesis (RH) or the Generalized Riemann Hypothesis (GRH)? Or at least if there is a connection between the Elliott-Halberstam conjecture and RH or GRH? I ask because the...
  37. marcus

    Riemann Hypothesis and Quantum Mechanics

    Maybe someone will find something interesting in this paper. They have a reference to some 1995 work by Alain Connes. I didn't have time to look into this very much. Maybe it's amusing and maybe not: http://arxiv.org/pdf/1012.4665v1
  38. C

    Fermi estimate of the week: Riemann hypothesis

    For those not familiar with the term Fermi estimate/problem/question see here: http://www.vendian.org/envelope/dir0/fermi_questions.html http://en.wikipedia.org/wiki/Fermi_problem My question: Between the time that Riemann posed his famous question (in 1859) and now, how many hours have...
  39. C

    Very simple (Dis)proof of Riemann hypothesis, Goldbach, Polignac, Legendre conjecture

    (Dis)proof of Riemann hypothesis,Goldbach,Polignac,Legendre conjecture I'm just an amateur and not goot at english ^^;
  40. Z

    A proposed Hamiltonian operator for Riemann Hypothesis

    HERE http://vixra.org/pdf/1007.0005v1.pdf is my proposed proof of an operator whose Eigenvalues would be the Imaginary part of the zeros for the Riemann Hypothesis the ideas are the following* for semiclassical WKB evaluation of energies the number of levels N(E) is related to the integral of...
  41. C

    Trace formula in noncommutative geometry and the Riemann hypothesis

    Does anyone know where to find this paper? Formule de trace en géométrie non-commutative et hypothèse de Riemann = Trace formula in noncommutative geometry and the Riemann hypothesis http://cat.inist.fr/?aModele=afficheN&cpsidt=2561461 The purchase link is broken there.. it gets stuck...
  42. Z

    About this strategy to prove Riemann Hypothesis

    http://arxiv1.library.cornell.edu/PS_cache/math/pdf/0102/0102031v10.pdf and http://arxiv1.library.cornell.edu/PS_cache/math/pdf/0102/0102031v1.pdf what do you think ? Author defines 2 operators D_{+} and D_{-} so they satisfy the properties D_{+} = D^{*}_{-} D_{-} = D^{*}_{+}...
  43. N

    Navigating Research on Riemann Hypothesis

    Let me start off by saying I have not yet had a formal course in Number Thoery and have only read briefly on the subject...hence the question: How close (in terms that would be understood by someone in my position) is the math community to proving the Riemann Hypothesis? I'm assuming there...
  44. 2

    Can someone explain zeros and zeta function for Riemann Hypothesis? (Yr13)

    Hi, I'm Yr 13 and just wanted to do some further reading/exploring. So i understand that the zeta function is something to do with summing up like this: 1/ (1^s) + 1/(2^s) etc etc Now, I just want to know what are non-trivial zeros and trivial zeros? I just want to be able to understand this...
  45. I

    Accurate Proof and varification for Riemann Hypothesis

    Accurate Proof verification of Riemann’s Hypothesis Riemann Hypothesis states that \int \frac{1}{ln (x)} has a root at \frac{1}{2} when s=2 The time series expansion of Log function is, [tex] \ln(x) = \frac {[x-1}{[x-2}+ \frac{1){3} \frac{x-3}{x-4} + \frac{1}{5}\frac{x-5}{x-6}+……...
  46. H

    Russian Dolls Matryoshka Approach to Riemann Hypothesis

    Well we know what matryoshka dolls are? Those nested dolls one inside another. I am a mere laymen and amateur that's why I am using descriptive terms instead of math rigor. So what should the approach be: If RH nest Hilbert-Polya conjecture, then what things nest HP conjecture? And ad...
  47. L

    The Riemann Hypothesis for High School Students

    Hi All, I would like to present what I believe to be a simple way to convey the essence of the Riemann Hypothesis to High School students. I hope you like it, and reply with suggestions for further improvements. Note for teachers: the rationale behind the graphs lays with the geometric...
  48. T

    Grander than the Riemann Hypothesis

    ...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?