What is Riemann: Definition and 613 Discussions

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. 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...
  2. J

    MHB Real Analysis - Prove the Riemann Integral Converges

    Just a couple questions. Problem 2: Just would like to know if this is the correct approach for this problem. Problem 3: I am just wondering if I can use Problem 2 to prove the first part of Problem 3? Because to me, they seem very similar. Problem 4: Would I use the MVT for integrals...
  3. J

    MHB Real Analysis - Riemann Integral Proof

    I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?
  4. davidge

    I Riemann tensor in 3d Cartesian coordinates

    Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get...
  5. TheSodesa

    Number of subdivisions in a Riemann integral (DFT)

    Homework Statement This is a combination of two questions, one being the continuation of the other 3) Calculate the DFT of the sequence of measurements \begin{equation*} \{ g \}_{k=0}^{5} = \{ 1,0,4,-1,0,0 \} \end{equation*} 4a) Draw the DFT calculated in question 3 on the complex plane. 4b)...
  6. MAGNIBORO

    I Solving Riemann Sum Problem: Integral of x^x

    Hi. I try to solve the integral $$\int_{0}^{1} x^{x} dx$$ Through sums of riemann But I came to the conclusion that the result is 0 that is wrong $$\int_{0}^{1} x^{x} dx = \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}$$ $$= \lim_{n\rightarrow...
  7. binbagsss

    Riemann Zeta Function shows non-trival zeros critical-strip symmetry

    1. Homework Statement I want to show that the non-trival zeros of the Riemann Zeta function all lie in the critical strip ## 0 < Re(s) < 1## and further to this that they are symmetric about the line ##Re(s)= 1/2 ## where ## \zeta(s) = \sum\limits^{\infty}_{n=1}n^{-s}## With the functional...
  8. Rectifier

    Understanding the Riemann Sum - Integral Connection

    The problem I want to calculate $$\sum^n_{k=1} \frac{4}{1+ \left(\frac{k}{n} \right)^2} \cdot \frac{1}{n}$$ when ##n \rightarrow \infty## The attempt ## \sum^n_{k=1} \underbrace{f(\epsilon)}_{height} \underbrace{(x_k-x_{k-1})}_{width} \rightarrow \int^b_a f(x) \ dx ##, when ##n \rightarrow...
  9. mhob

    A Contraction between Levi-Civita symbol and Riemann tensor

    How to proof that εμνρσ Rμνρσ =0 ? Thanks.
  10. binbagsss

    Mellin transform-- complete Riemann function

    Homework Statement Show that ## \int\limits^{\infty}_0 \sum\limits^{\infty}_{n=1}e^{-\pi n^{2} t} t^{s-1} dt = \sum\limits^{\infty}_{n=1}(\pi n^{2} )^{-s} \int\limits^{\infty}_{0} e^{-t} t^{s-1} dt ## Homework Equations ##\Phi (t)=\theta(it/2)=1+2\sum\limits^{\infty}_{n=1}e^{-\pi n^2 t}. ##...
  11. 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 >>
  12. tommyxu3

    I Ricci curvatures determine Riemann curvatures in 3-dimension

    Hello~ For usual Riemann curvature tensors defined: ##R^i_{qkl},## I read in the book of differential geometry that in 3-dimensional space, Ricci curvature tensors, ##R_{ql}=R^i_{qil}## can determine Riemann curvature tensors by the following relation...
  13. terryds

    Definite integral as Riemann sums

    Homework Statement Determine ##\int_{0}^{2}\sqrt{x}dx## using left riemann sums Homework Equations ##\int_{a}^{b}f(x)dx = \lim_{n\rightarrow \infty}\sum_{i=0}^{n-1}(\frac{b-a}{n})f(x_i)## The Attempt at a Solution [/B] ##\frac{b-a}{n}=\frac{2-0}{n}=\frac{2}{n}## ##\int_{0}^{2}\sqrt{x}dx =...
  14. binbagsss

    Riemann Zeta Function showing converges uniformly for s>1

    Homework Statement ## g(s) = \sum\limits^{\infty}_{n=1} 1/n^{-s}, ## Show that ##g(s)## converges uniformly for ## Re(s>1) ## Homework Equations Okay, so I think the right thing to look at is the Weistrass M test. This tells me that if I can find a ##M_{n}##, a real number, such that for...
  15. F

    I Definitions of the Riemann integral

    In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that...
  16. redtree

    A Riemann Tensor Equation: Simplifying the Riemann-Christoffel Tensor

    The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as: $$ R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n} $$ My question is that it seems that...
  17. A

    I Contour integration over Riemann surface

    Cauchy integral theorem states that the contour integration of a complex harmonic function along a closed simply connected path=0. What if this simply connected path is drawn over a Riemann surface of function like ##f(z)=\sqrt z##. Will that be possible in the first place? and will the...
  18. A

    I What are the independent components of the Riemann tensor

    What 20 index combinations yield Riemann tensor components (that are not identically zero) from which the rest of the tensor components can be determined?
  19. Victor Alencar

    A Geometrical interpretation of Ricci and Riemann tensors?

    I do not get the conceptual difference between Riemann and Ricci tensors. It's obvious for me that Riemann have more information that Ricci, but what information? The Riemann tensor contains all the informations about your space. Riemann tensor appears when you compare the change of the sabe...
  20. 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?
  21. 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...
  22. mertcan

    A Riemann tensor and covariant derivative

    hi, I tried to take the covariant derivative of riemann tensor using christoffel symbols, but it is such a long equation that I have always been mixing up something. So, Could you share the entire solution, pdf file, or links with me? ((( I know this is the long way to derive the einstein...
  23. A

    B Can Riemann zeta function be written as ##f(s)=u(s)+iv(s)##?

    I don't recall that I have seen Riemann zeta function put in the form of ##f(s)=u(s)+iv(s)##.
  24. W

    Riemann tensor given the space/metric

    Homework Statement Given two spaces described by ##ds^2 = (1+u^2)du^2 + (1+4v^2)dv^2 + 2(2v-u)dudv## ##ds^2 = (1+u^2)du^2 + (1+2v^2)dv^2 + 2(2v-u)dudv## Calculate the Riemann tensor Homework Equations Given the metric and expanding it ##~~~g_{τμ} = η_{τμ} + B_{τμ,λσ}x^λx^σ + ...## We have...
  25. Nemo1

    MHB Solving Limits and Riemann Sums: Tips from Nemo

    Hi Community, I have the following question: I have done basic solving of limits and also of Riemann sums but never had to do them in the same question. Would I be correct in saying that I need to solve for the Riemann sum first then take the limit of the integral? Cheers Nemo
  26. jamalkoiyess

    Study Reimann Geometry: High School+Astronomy

    So I was reading now about the new geometries and I wanted to know if I can study the Reimann Geometry knowing that I finished high school or if I could just know about it but not about the formulas. I am so interested in the subject because it is used in astronomy.
  27. nomadreid

    I Riemann zeta: regularization and universality

    I am not sure which is the appropriate rubric to put this under, so I am putting it in General Math. If anyone wants to move it, that is fine. Two questions, unrelated except both have to do with the Riemann zeta function (and are not about the Riemann Hypothesis). First, in...
  28. T

    MHB Find a formula for the Riemann sum and take the limit of the sum as n->infinite

    For the function given below find a formula for the Riemann sum obtained by dividing the interval [1,5] into n equal subintervals and using the right-hand endpoint for each c subscript k. Then take a limit of thissum as n-> infinite to calculate the area under the curve over [1,5]. Below you...
  29. S

    I Does an inverse Riemann exist?

    I already googled this but I did not find a definite answer. Is there such a thing as a 'inverse riemann'? Specifically, where you invert the start number to be at the top of the riemann symbol and then decrement down to the end value which is on the bottom of the riemann symbol?
  30. D

    Covariant derivative of Killing vector and Riemann Tensor

    I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...
  31. S

    A Is the Riemann Curvature Tensor a Mathematical Tool or Physically Significant?

    Can someone explain mathematically why do we say Riemann Curvature Tensor has all the information about curvature of Space Thank You
  32. Kevin McHugh

    I Reducing Riemann Components to 20 - Step-by-Step Guide

    I used the expression Rabcd=-Rbacd=-Rabdc=Rcdab to reduce the number of components. I also used if a=b=0 the R=0 and if and c=d=0 then R=0. This reduced the number of components to 64. How do I get them down to 21? I know I need another equality to reduce it to 20. <<Mentor note: Fixed...
  33. Crush1986

    Question about Riemann Surface Problem

    Homework Statement f(z)=z^\frac{3}{2} find the branch points, branch cuts, and Riemann sheet structure. Homework Equations none The Attempt at a Solution So, I converted this to complex exponential form r^\frac{3}{2} e^\frac{i*3*\Theta}{2} From here I mapped around a circle that was...
  34. C

    Calculating Covariant Riemann Tensor with Diag Metric gab

    Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...
  35. 1

    Understanding dx in Integration

    Hello, I am currently in my first year of college, and I already took calculus in high school. I was able to solve all the problems, but I feel like I didn't understand everything conceptually. When integrating dy/dx=x you get, ∫x dx=1/2x2. But what exactly happened to the dx, why did it...
  36. N

    Upper and lower bound Riemann sums

    Homework Statement Find the upper, lower and midpoint sums for $$\displaystyle\int_{-3}^{3} (12-x^{2})dx$$ $$\rho = \Big\{-3,-1,3\Big\}$$ The Attempt at a Solution For the upper: (12-(-1)^2)(-1-(-3)) + (12-(-1))(3-(-1)) =74 For the lower: (12-(-3)^2)(-1-(-3))+(12-3)(3-(-1)) =42 For midpoint...
  37. 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...
  38. L

    Riemann's proof of the existence of definite integrals

    Hello, Since it was mentioned in my textbook, I've been trying to find Riemann's proof of the existence of definite integrals (that is, the proof of the theorem stating that all continuous functions are integrable). If anyone knows where to find it or could point me in the right direction, I...
  39. ShayanJ

    Contracting Riemann tensor with itself

    In chapter 8 of Padmanabhan's "Gravitation: Foundations and Frontiers" titiled Black Holes, where he wants to explain that the horizon singularity of the Schwarzschild metric is only a coordinate singularity, he does this by trying to find a scalar built from Riemann tensor and show that its...
  40. N

    What are the definitions and properties of Riemann sums?

    Just want to see if I actually understand what these all mean. Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...
  41. L

    Functional equation Riemann Zeta function

    There are two forms of Riemann functional equation. One is more symmetric and follows from the other and the duplication theorem of the Gamma function. At least, that's been claimed here...
  42. D

    Proving Non-linear Wave Equation for Riemann Tensor

    Hello, I am working through Hughston and Tod "An introduction to General Relativity" and have gotten stuck on their exercise [7.7] which asks to prove the following non- linear wave equation for the Riemann tensor in an empty space: ∇e∇eRabcd = 2Raedf Rbecf − 2Raecf Rbedf − Rabef Rcdef I have...
  43. D

    Deriving Riemann Tensor Comp. in General Frame

    How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...
  44. RJLiberator

    The Cauchy Riemann Equations for this Function

    Homework Statement Verify that each of the following functions is entire: f(z)=(z^2-2)e^(-x)e^(-iy) Homework Equations The Cauchy Riemann equations u(x,y) = ______ and v(x,y) = ______ u_y=-v_x u_x=v_y The Attempt at a Solution So, I've done a few of these problems and understand that to...
  45. V

    MHB Riemann sums: linear integrand

    ∫(4−7x)dx ====> Integral is from 1 to 8 Similar question but for some reason I can't get the answer after following all the steps
  46. S

    M: Solve Riemann Sum Problem Homework

    Homework Statement [/B] Hello, thank you in advance for your help. I am calculating a Riemann sum with right hand endpoints. I hit a small snag, and I appreciate your help in getting me straight.Homework Equations f(x) = x2+ 1, over the interval [0,1]. This is problem number such-and-such from...
  47. S

    MHB Riemann Integral: Two Questions

    We only have the epsilon-delta definition to work with for these. Prove that f is integrable and verify the value. On [0,1] f(x)=1 if x=1/2 else 0. \int_{0}^{1} \,f =0 Prove: If f is integrable on [0,1] then \lim_{{n}\to{\infty}}\ \frac{1}{n} \sum_{k=1}^{n} f(\frac{k}{n}) = \int_{0}^{1} \,f .
  48. F

    Mathematica Is anyone familiar with using Atlas 2 for mathematica?

    Is anyone familiar with Atlas 2 for mathematica to calculate the Riemann Tensor, Ricci Tensor, and scalar I have a metric that I need to calculate these things for. Can anyone help? I'm not too up on Mathematica either.
  49. Enrico Fermi

    Do not understand concept of Riemann geometry.

    Hello I do not fully grasp the concept of Riemann geometry.Please can you use mathematical descriptions but explain them because I am only a 10 year old but of course I know significantly the theories of dimensions. Thank you.
  50. S

    Solving Riemann Sum for "Deformation of Water by Magnetic Field

    I'm not sure if this is the right place to post this in, but I'm trying to recreate the "Deformation of water by a magnetic field" experiment by Chen et al. The PDF version of the paper can be accessed via Google (for some reason it won't let me provide a direct link). On the 2nd page of the...
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