Riemann Definition and 618 Threads

Georg Friedrich Bernhard Riemann (German: [ˈɡeːɔʁk ˈfʁiːdʁɪç ˈbɛʁnhaʁt ˈʁiːman] (listen); 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.
In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis.
His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory.
Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.

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  1. D

    Disproof of Riemann Hypothesis

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

    What does f(x)>g(x) mean for x in [a,b]?

    Homework Statement Prove or falsify the statement (see picture) The Attempt at a Solution I've got the answer already but I want to make sure I know is what is meant by f(x)>g(x) for x in [a,b]. Does it mean f(x) lies above g(x) throughout the entire interval?
  3. D

    What is x in the Riemann Hypothesis?

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

    Riemann Zeta Function and Pi in Infinite Series

    I was playing around with an infinite series recently and I noticed something peculiar, I was hoping somebody could clarify something for me. Suppose we have an infinite series of the form: \sum^_{n = 1}^{\infty} 1/n^\phi where \phi is some even natural number, it appears that it is always...
  5. P

    Orthonormal basis => vanishing Riemann curvature tensor

    Hey! If a (pseudo) Riemannian manifold has an orthonormal basis, does it mean that Riemann curvature tensor vanishes? Orthonormal basis means that the metric tensor is of the form (g_{\alpha\beta}) = \text{diag}(-1,+1,+1,+1) what causes Christoffel symbols to vanish and puts Riemann...
  6. F

    Trivial zeros of Zeta Riemann Function

    According to Wikipedia, the Zeta Riemann Function is defined as follows: \begin{equation} \zeta(z) = \sum_{k=1}^{\infty}\frac{1}{k^{z}}, \forall z \in \mathbb{C}, Re[Z] > 1. \end{equation} Well, the trivial zeros are the negative even numbers. Is that a consequence of the following...
  7. T

    Derivative of Riemann zeta function

    I'm trying to evaluate the derivative of the Riemann zeta function at the origin, \zeta'(0), starting from its integral representation \zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\frac{1}{e^t-1}. I don't want to use a symbolic algebra system like Mathematica or Maple. I am able to...
  8. D

    Quick question to clear up some confusion on Riemann tensor and contraction

    Let's say I want to calculate the Ricci tensor, R_{bd}, in terms of the contractions of the Riemann tensor, {R^a}_{bcd}. There are two definitions of the Riemann tensor I have, one where the a is lowered and one where it is not, as above. To change between the two all that I have ever seen...
  9. M

    Limit Sum Riemann: Solve Homework Problem

    Homework Statement Calculate the limit with Riemann. Homework Equations \displaystyle\lim_{n \to{+}\infty}{\displaystyle\frac{pi}{4}\cdot{} \displaystyle\sum_{k=0}^n{tan^2(\displaystyle\frac{k\cdot{} pi}{4n})\cdot{}\displaystyle\frac{1}{n}}} The Attempt at a Solution I don't know how to...
  10. K

    Riemann Sums: Finding the Limit as n Approaches Infinity

    Homework Statement Identify an=the summation from k=1 to n of (2n)/(4k2+1) as a Riemann sum of an appropriate function on an appropriate interval and find the limit as n approaches infinity of an. Homework Equations There is no interval givien so I assume its from 0 to 1. The...
  11. F

    Contraction in the Riemann Tensor

    Hi all, I'm trying to follow through some of my notes of a GR course. The notes are working towards a specific expression and the following line appears: R^{\alpha \beta}_{\gamma \delta ; \mu} + R^{\alpha \beta}_{\delta \mu ; \gamma} + R^{\alpha \beta}_{\mu \gamma ; \delta}=0 Which by...
  12. T

    General Relativity - Riemann Tensor and Killing Vector Identity

    Homework Statement I am trying to show that for a vector field Va which satisfies V_{a;b}+V_{b;a} that V_{a;b;c}=V_eR^e_{cba} using just the below identities. Homework Equations V_{a;b;c}-V_{a;c;b}=V_eR^e_{abc}(0) R^e_{abc}+R^e_{bca}+R^e_{cab}=0 (*) V_{a;b}+V_{b;a}=0 (**) The Attempt at a...
  13. S

    Regarding Riemann integration defination

    Regarding "Riemann integration defination" Hi, I did not understand the following: We have : Partition is always a "finite set". A function f is said to Riemann integrable if f is bounded and Limit ||P|| -> 0 L(f,P) = Limit||P|| -> 0 U(f,P) where L(f,P) and U(f,P) are...
  14. E

    Proving a function of bounded variation is Riemann Integrable

    Homework Statement If a function f is of bounded variation on [a,b], show it is Riemann integrable Homework Equations Have proven f to be bounded S(P) is the suprenum of the set of Riemann integrals of a partition (Let's say J) s(P) is the infinum of J S(P) - s(P) < e implies f...
  15. icystrike

    Are the Cauchy-Riemann Equations Ever Satisfied for f(z) = |z|?

    Homework Statement What does it mean by this: The cauchy riemann equations are never satisfied when x and y are different from zero and when x=y=0 . Looking at the example of f(z)= l z l = \sqrt{x^{2}+y^{2}} Homework Equations The Attempt at a Solution
  16. Z

    An identity about Gamma and Riemann function

    we know that \Gamma (s)= \int_{0}^{\infty}dxe^{-x}x^{s-1} however every factor of the Riemann Zeta can be obtained also from a Mellin transform \int_{0}^{\infty}dxf(x)x^{s-1} =(1-p^{-s})^{-1} where f(x) is the distribution \sum_{n=0}^{\infty}x \delta (x-p^{-n}) is there any...
  17. S

    Basic Complex Analysis: Cauchy Riemann

    Homework Statement Let f be a holomorphic function in the unit disc D1 whose real part is constant. Prove that the imaginary part is also constant. Homework Equations Cauchy Riemann equations The Attempt at a Solution Hi guys, I'm working through the basics again. I think here we...
  18. S

    Compare Riemann Integration & Definite Integration Accuracy

    Can some please draw a comparison between Riemann Integration and normal definite integration in terms of accuracy.
  19. quasar987

    (anti)-symmetries of the Riemann curvature tensor

    The Riemannian curvature tensor has the following symmetries: (a) Rijkl=-Rjikl (b) Rijkl=-Rijlk (c) Rijkl=Rklij (d) Rijkl+Rjkil+Rkijl=0 This is surely trivial, but I do not see how to prove that Rijkl=-Rjilk. :( Thanks.
  20. 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?
  21. L

    Riemann hypothesis and number theory

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

    Not riemann integrable => not lebesgue integrable ?

    "not riemann integrable" => "not lebesgue integrable"?? Hi, In general if a function is not Riemann integrable does this mean the function is also not Lebesgue integrable? Why or why not? I know that if the the function is Riemann integrable then its Lebesgue integrable, but I can't find...
  23. A

    Riemann vs. Lebesgue integral in QM

    When we talk about "Hilbert space" in (undergraduate) QM, we are typically talking about the space of square-integrable functions so that we can make sense out of \int_{-\infty}^{\infty} |\psi(\vec r,t)|^2 d^3x. But are we talking about Riemann-integrable functions or Lebesgue-integrable...
  24. I

    Proove interchange symmetry of the Riemann curvature tensor

    Homework Statement Proove that: R_{abcd} = R_{cdab} Homework EquationsThe Attempt at a Solution I'm not sure whether to expand the following equations any further (using the definitions for the christoffel symbols) and hope that I can re-label repeated indexes at a later stage or if there is...
  25. M

    Riemann Integrability, Linear Transformations

    Homework Statement If f,g are Riemann integrable on [a,b], then for c,d real numbers, (let I denote the integral from a to b) I (cf + dg) = c I (f) + d I (g) Homework Equations The Attempt at a Solution I have the proofs for c I(f) = I (cf) and I (f+g) = I (f)...
  26. M

    Approximating integral using riemann sums

    Homework Statement f: [0,1] -> Reals, f(x) = 3-x2 P={0,1/2,1} Find lower and upper Riemann sums, and approximate the definate integral using them and find the corresponding approximation error. Homework Equations The Attempt at a Solution So I tried finding the upper Riemann...
  27. 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...
  28. H

    Unclear on Riemann Zeta Function

    After reading about the Riemann Zeta Function on Wolfram Alpha (http://mathworld.wolfram.com/RiemannZetaFunction.html), it's still unclear to me how the Euler product formula is essentially equal to the limit of a p-series. Someone please enlighten me
  29. M

    Absolute value of riemann sums

    Homework Statement I'm trying to prove that Sp|f| - sp|f| \leq Spf - spf Where P is a partition of [a,b] and f is function that is riemann integrable. Homework Equations The Attempt at a Solution So I get to a point where M = supf(x) and m = inff(x) then |M|(b-a) - |m|(b-a)...
  30. C

    Show that the following function is Riemann integrable.

    Homework Statement Show that the function f: [0,1] -> R defined by: f(x) = 1, if x=1/k for some k f(x) = 0, else is Riemann integrable on [0,1] Homework Equations The Attempt at a Solution I attempted the problem using Cauchy's criterion but found that this function is...
  31. M

    Is f(x) = 1 if x is rational, 0 if x is irrational Riemann integrable on [0,1]?

    Homework Statement Let A={1/n, n =natural number} f: [0,1] -> Reals f(x) = {1, x in E, 0 otherwise Prove f is riemann integrable on [0,1] Homework Equations The Attempt at a Solution Not quite sure, but I think supf = 1 and inf f= 0 no matter what partition you take, then...
  32. A

    Left endpoint approximation & Riemann Sums (Sigma)

    1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...
  33. R

    Cauchy riemann equations and constant functions

    Homework Statement Let f(z) be analytic on the set H. Let the modulus of f(z) be constant. Does f need be constant also? Explain. Homework Equations Cauchy riemann equations Hint: Prove If f and f* are both analytic on D, then f is constant. The Attempt at a Solution I think f need be...
  34. 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...
  35. O

    Calculate Riemann tensor according to veilbein

    Homework Statement How to use veilbein to calculate Riemann tensor, Ricci tensor and Ricci scalar? (please give me the details) de^a+\omega_{~b}^a\wedge e^b=0, R_{~b}^a=d\omega_{~b}^a+\omega_{~c}^a\wedge\omega_{~b}^c. The metric is...
  36. T

    Convergence of sum involving Mobius function (with Riemann Zeta)

    Hello everyone. I was hoping someone could clarify this "heuristic" argument I found online. First, what is the analytic function they speak of and is its derivative difficult to compute? Second, does this look like a legit argument? : If you take the derivative w.r.t s of both sides of sum...
  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. A

    Cauchy Riemann Equations (basic doubt)

    Lets say we have a function of a complex variable z , f(z). I read that for the function to be differentiable at a point z0 , the CR equations are a necessary condition but not a sufficient condition. Can someone give me an example where the CR equations hold but the function is not...
  39. 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...
  40. H

    Riemann Integrable <-> Continuous almost everywhere?

    Riemann Integrable <--> Continuous almost everywhere? I ran across a statement somewhere in the forums saying that a function is Riemann-integrable iff it is continuous almost everywhere, i.e. if its set of discontinuities has measure 0. Is that right? What about the case of a function...
  41. A

    What Definite Integral Does This Riemann Sum Represent?

    Homework Statement Rn=\sum(i*e^(-2i/n))/n^2, i=1 Identify this Riemann sum corresponding to a certain definite integral. Homework Equations The Attempt at a Solution I got till 1/n^2 [1/e^(2/n)+2/e^(4/n)+3/e^(6/n)...n/e^2] and that's it. To my understanding I should be...
  42. K

    Proof using Riemann Integral definition

    Homework Statement Suppose that f:[a, b] → ℝ is a function that is zero for all x ∈ [a, b] except for the values x_1,x_2,…,x_k. Find ∫[a b](f(x)dx) and prove your result. Homework Equations Definition of a Riemann integrable function...
  43. C

    A little renewed Disproof of Riemann hypothesis

    hmm...
  44. A

    General metric with zero riemann tensor

    A metric consistent with interval: \mathrm{d}s^2=-\mathrm{d}\tau^2+\frac{4\tau^2}{(1-\rho^2)^2}\left(\mathrm{d}\rho^2+\rho^2\mathrm{d}\theta^2+\rho^2\sin(\theta)^2\mathrm{d}\varphi^2\right) get zero for riemann's tensor, therefor must be isomorphic with minkowski tensor. But I don't find thus...
  45. T

    Math Real Analysis Problem, Riemann Sum Integral?

    Part 1. Homework Statement The problem literally states... " The Integral. limit of n-> infinity of n*[1/(n+1)^2 + 1/(n+2)^2 + 1/(n+3)^2 + 1/(2n)^2] = 1/2 " According to the teacher, the answer is 1/2. I don't know why or how to get there. Part 2. The attempt at a solution...
  46. M

    Suggestion for a good book on Riemann Surfaces - your personal experiences

    Hello everyone - I'm a third year student at Cambridge university, and I've recently started taking a course on Riemann surfaces along with a number of other pure courses this year. The problem is, the lecturer of the course is of a rather sub-par standard - whilst I don't doubt he's probably...
  47. B

    What is the different between Minkowski Space an Semi Riemann Space?

    Last summer I took Semi Riemann Geometry lesson. Almost all the definitions in Semi Riemann geometry are with the same Minkowski geometry. I don't understand what is the different between Minkowski Space an Semi Riemann Space.
  48. V

    Proof that f(x) = 1/sqrt(x) is Riemann Integrable

    Homework Statement The problem given is: Show that the function f(x) = 1/sqrt(x) is integrable on the compact interval [0,1]. Homework Equations We are only allowed to use theorems, definitions, and properties that have been covered in class or are in the book. The ones I...
  49. L

    What are the Functions of Riemann Invariants?

    Can someone tell me what a riemann invariant is in general terms please? i can't find it anywhere! thanks, lav
  50. D

    No. of field equations and components or Riemann tensor?

    no. of field equations and components or Riemann tensor?? Someone was trying to explain to me about curvature in space. From what I got from what they were saying doesn't make sense to me. I'm not sure I understand what the number of components, N, of R\alpha,\beta,\gamma,\delta when compared...
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