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. 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 ^^;
  2. W

    Does Time Variation Necessarily Imply Full Spacetime Metric?

    Background: Math: An affine parameter provides a metric along a geodesic but not a metric of the space, for example between geodesics. A connection provides an affine parameter, and a non-trivial connection gives rise to Riemann curvature. Given the existence of a connection with Riemann...
  3. M

    What is the Relationship Between Riemann Space and Relativity?

    Homework Statement Hello …….. I have a question about a statement mentioned in the book “Introduction to tensor calculus and continuum mechanics” . it is :- Where the space (Vn) is Riemann space . Is this statement really true ? Homework Equations The Attempt at a...
  4. G

    If supremum=infimum, is f Riemann integrable?

    This is my first time posting & I am not familiar with how to get all the correct math symbols or how to use Latex, so I did the best I could. Homework Statement Suppose f is bounded on [a,b] and there is a partition P* of [a,b] for which S(f,P*)=S(f,P*). Is f Riemann integrable on...
  5. 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...
  6. P

    I am taking issue with the Riemann Integral

    Basically from what I understand the integral of a function, say ∫x^2dx from say 0 to 1, can be represented as the supremum/infimum of the function values within each of a countably infinite number of vanishingly small intervals in the domain created by a countably infinite number of partition...
  7. J

    Riemann lebesgue lemma. wikipedia. 2010-06-26

    http://en.wikipedia.org/wiki/Riemann-Lebesgue_lemma Have I made a mistake when it looks to me that the Wikipedia proof on Riemann-Lebesgue lemma looks like nonsense? How are you supposed to use dominated convergence theorem there?
  8. A

    Upper and lower Riemann sums

    Let f be a Riemann integrable function defined on an interval [a,b], and let P = \{a = x_0 < x_1 < \ldots < x_n = b\} be a partition of [a,b]. I don't understand why the definition of (say) the upper Riemann sum of f associated with P is always given as U(f,P) = \sum_{i=1}^n M_i (x_i -...
  9. A

    How can I convert the Riemann Sum into an Integral?

    If I have a function c(x,Δx) that gives the area between x and x + Δx of a function. The area under the function can be given by: Sum from j = 0 to n-1 of c(b/n j,c/b) As n tends to infinity and b is the upper limit of integration. How can I convert this from a sum into a integral? I'm not...
  10. A

    Riemann tensor: indipendent components

    Hi, thanks for the attention and excuse for my bad english. I'm studying general relativity and I have a doubt about the number of indipendent component of the riemann curvature tensor. We have two kind of riemann tensor: type (3,1) Rikml type (4,0) Rrkml There are also some symmetry...
  11. R

    Riemann integrable sequences of functions

    Let f_n : [0,1] → [0,1] be a sequence of Riemann integrable functions, and f : [0, 1] → [0, 1] be a function so that for each k there is N_k so that supremum_(1/k<x≤1) of |f_n(x) − f(x)| < 1/k , for n ≥ N_k . Prove that f is Riemann integrable and ∫ f(x) dx = lim_n→∞ ∫ f_n(x) dx I am really...
  12. K

    Integrability of f on (c,d) from (a,b): Proof

    Homework Statement If a<c<d<b and f is integrable on (a,b), show that f is integrable on (c,d) Homework Equations The Attempt at a Solution I know that f is integrable on (a,b) iff for all e>0 there exists step functions g and h such that g \leq f1(a,b) \leq h and I(g-h) <e (...
  13. I

    Is g(f(x)) Riemann Integrable if g(x) is piecewise continuous?

    Homework Statement We have a corollary that if f(x) is in the set of Riemann Integrable functions and g(x) is continuous, then g(f(x)) is also a riemann integrable function Show that if g(x) is piecewise continuous then this is not true Homework Equations Hint: take f to be a ruler...
  14. L

    Are Individual Functions Riemann Integrable if Their Sum is?

    suppose f and g are bounded functions on [a,b] such that f+g is in R[a,b] Then, does it follow that f and g are also in R[a,b]? i wanto to prove whether it is or not
  15. J

    3D printing of Riemann Surfaces

    Hello. Does anyone know of a group that has used 3D printing techniques such as laser sintering to create Riemann surfaces of some simple functions? For example, just \sqrt{z}? Actually I would be interested in more complex function and preferable color-code various components of the surface...
  16. T

    Help With Understanding Riemann Problem for Traffic Flow Modeling

    I'm having a lot of trouble getting my head around this topic.. I am currently trying to create a numerical model of traffic flow, which has worked out to be a nonlinear hyperbolic PDE. If anyone could explain the concept of the Riemann problem to me in lamen terms, it would be greatly...
  17. Ranku

    Riemann tensor and flat spacetime

    When Riemann tensor = 0, spacetime is flat. Is the geometry of this flat spacetime that of special relativity?
  18. stevmg

    Riemann sum and anti-derivative

    How do you mathematically equate a Riemann sum as area under the curve to an anti-derivative? How do you prove that, theoreticlly, the one is equalent to the other? Assuming the function is continuous between points a and b, there is always a Riemann sum and thus the function is integrable...
  19. G

    Riemann prime distribution for dummies?

    I saw a documentary recently that talked about the distribution of prime numbers and their similarity to vibrations in a sphere of quartz when struck by metal ball bearings. I tried to look up Riemann online and was overloaded with advanced math. Is there a resource where I can find out more...
  20. S

    Proof of Cauchy Criterion for Riemann Integrals

    Homework Statement Some proofs I've looked at vary, but they generally follow the format show here: http://en.wikibooks.org/wiki/Real_Analysis/Riemann_integration#Theorem_.28Cauchy_Criterion.29 This isn't a question about an exercise, but rather a request for a clarification or a way of...
  21. A

    Riemann tensor in normal coordinates

    This is essentially a "homework question", but I'm not looking for an explicit solution so I have posted it here. 1. Homework Statement Find a simplified expression for the Riemann tensor in terms of the connection in normal coordinates. 2. Homework Equations Riemann tensor =...
  22. A

    Riemann tensor in normal coordinates (General Relativity)

    Homework Statement Find a simplified expression for the Riemann tensor in terms of the connection in normal coordinates. Homework Equations Riemann tensor = (derivative of connection term) - (derivative of connection term) - (connection term)(connection term) - (connection...
  23. K

    Proving the Hint: Riemann Integrability of Product Functions

    Homework Statement Homework Equations The Attempt at a Solution Right now, I'm still trying to understand why the hint is true. This is what I've got so far... Let ||f||∞= sup{|f(x)|: x E [a,b]} M_i(f,P) = sup{f(x): x_{i - 1} ≤ x ≤ x_i} m_i(f,P) = inf{f(x): x_{i - 1} ≤ x ≤ x_i}...
  24. 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...
  25. I

    Riemann surface of four closed string interaction (Zwiebach section26.5)

    There are three kinds of light-cone string diagrams for four closed string interactions. As displayed by fig. 26.10, 26.11 and 26.12 of section 26.5 of Zwiebach's book. For each light-cone string diagram, it is characterized by two parameters, the time difference of the two interaction...
  26. E

    Proof of Inf. Riemann Zeta Function Zeros at re(s)=1/2

    Does anybody know where I can find the proof that an infinite number of zeros of the riemann zeta function exist when re(s) = 1/2?
  27. D

    Riemann Zeta Function Z(z)

    I was wondering how do you calculate the Riemann value, of a Riemann Zeta Function, for example the riemann zeta function for n = 0, is -1/2, which envolves a bernoulli number (what is a bernoulli number and what roll does it play in the Riemann Zeta Function), can anyone explain that to me...
  28. S

    Riemann integration in R^n

    Homework Statement a) Let A in R^n be compact, and let f: A -> R be continuous and also non-negative. Show that if there exists some a in A with f(a) > 0, then \int_{A}f > 0 b) Now let A in R^n be a closed rectangle, and suppose f: A -> R be bounded and integrable. Show that if f(x) > 0 for all...
  29. P

    Riemann Stieltjes Integral help

    Let f = x, for 0<=x<=1 1, for 1<x alpha = x^2, for 0<=x<=1 1, for 1<x Find Integral (f) d(alpha) -- from 0 to 23 pls help!
  30. S

    What is Riemann zeta function.

    Could anyone tell me what is the Riemann zeta function. On Wikipedia , the definition has been given for values with real part > 1 , as : Sum ( 1 / ( n^-s) ) as n varies from 1 to infinity. but what is the definition for other values of s ? It is mentioned that the zeta function is the...
  31. M

    Trivial zeros of the Riemann zeta function

    Clearly I am missing something obvious here, but how is it that negative even numbers are zeros of the Riemann zeta function? For example: \zeta (-2)=1+\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+...=1+4+9+.. Which is clearly not zero. What is it that I am doing wrong?
  32. A

    Square of the Riemann zeta-function in terms of the divisor summatory function.

    Hi, The divisor summatory function, D(x), can be obtained from \zeta^{2}(s) by D(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty}\zeta^{2}(w)\frac{x^{w}}{w}dw and I was trying to express \zeta^{2}(s) in terms of D(x) but I didnt succeed, could someone help?
  33. 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^{*}_{+}...
  34. J

    No use of the tangent in Riemann sum?

    Where is the use of the "tangents at every point on the curve" in the Riemann sum? Riemann sum allows us to find the area of under the curve, and this involves only the height of each rectangle (i.e. the function f(x) at each x), and the width (i.e. the x), and the two are multiplied together...
  35. N

    Is There a Topological or Geometrical Approach to the Riemann Hypothesis?

    Hello dear forum members I wanted to know where are the research on the Riemann hypothesis , the latest advances ,who are the currently leading experts and is now known that mathematics it requires for its resolution
  36. W

    Struggling with Describing a Riemann Surface for log(z2 - za)?

    I am having trouble describing the Riemann surface of log(z) + log(z-a)
  37. R

    Surface Area of Sphere as a Riemann Sum

    Homework Statement How do you solve the surface area of a sphere using Riemann Sums?Homework Equations The Attempt at a Solution I started out with 2 * (lim n->∞ [ (i=1 to n) ∑ [ 2*pi*(√(r^2 - (i/rn)^2))*(r/n) ] ]) where the summation is the surface area of the cylinders (or discs) inside a...
  38. M

    Calculating Riemann Tensor for S^2 with Pull-Back Metric from Euclidean Space

    Find the Riemann tensor of the 2-sphere of radius r S^{2}_{r}={(x,y,z) \in\Re^{3}|x^{2} + y^{2} + z^{2} = r^{2}} with metric g obtained as the pull-back of the Euclidean metric gR3 by the inclusion map S^{2} \hookrightarrow\Re^{3}. Any help would be appreciated. Thanks
  39. A

    Riemann integrable functions continuous except on a set of measure zero?

    Is it true that a function is Riemann integrable on a bounded interval only if it's equal to a continuous function almost everywhere? I'd imagine this is the case, given the Riemann-Lebesgue lemma, which says that a function is RI iff its set of discontinuities has measure zero. (So the...
  40. M

    Showing the Difference Between an Ito Integral & Riemann Integrals.

    Hi Everyone, A problem I have here. I am trying to solve a problem involving Ito Integrals and Riemann interals. Homework Statement Prove \int^{T}_{0} tdW(t) = TW(T) -\int^{T}_{0} W(t)dt Homework Equations I want to solve this question WITHOUT using Ito's Lemma directly.The Attempt at a...
  41. F

    Riemann Sum and partition

    Hello everyone, I have passed my integral calculus class and it's been a little while so I don't really remember everything. Can anyone help me out with this? Homework Statement Let f(x) = sqrt(x), x E [0,1] and P=\left \{ 0,\left ( \frac{1}{n} \right )^{2}, \left ( \frac{2}{n} \right...
  42. W

    Calc 1 Riemann Sums w/ velocity and distance

    Homework Statement This is somewhat a repost... except I have figured out some of it and I have cleaned up the question. Your task is to estimate how far an object traveled during the time interval 0<= t >= 8 , but you only have the following data about the velocity of the object...
  43. W

    Riemann sums with velocity and distance.

    Homework Statement I really need help starting this problem as I am not sure what to do. Your task is to estimate how far an object traveled during the time interval 0t8 , but you only have the following data about the velocity of the object. time (sec) 0 1 2 3 4 5 6 7 8...
  44. P

    Integration - Riemann Sums

    Homework Statement The following table shows the power produced by a 600kW wind turbine at the given wind speed and the number of hours the wind blows at that speed. a) Plot the power characteristic as a function of wind speed. b) Plot the wind duration curve as a function of wind speed...
  45. T

    Fortran Find Area Under Curve w/ Fortran Riemann Sums

    Hello, for my FORTRAN class it wants me to use the method of Riemann's sums to find the area under the curve for the function f(x) = -(x-3)**2 +9, and stop when successive iterations yield a change of less than 0.0000001. I know I am going to have to used double precision. I am just confused...
  46. M

    Riemann Zeta function zeros

    Hi: ____________________________________________________________________ Added Nov.3, 2009 (For anyone who can't read the formula below (probably everyone) and who might have an interest in the subject: - the derivation of two simple equations that locate all the zeros of the zeta...
  47. B

    Riemann Sum: Solve for Area Under Curve 0 to 18

    Riemann sum help! Homework Statement Use Riemann sum with ci= i3/n3 f(x)= \sqrt[3]{x} +12 from x=0 to x=18 n= 6 subintervals Approximate the sum using Riemann's Sum Homework Equations \Sigma f(ci) \Delta xi is the equation for riemanns sum i think The Attempt at a Solution i...
  48. C

    (revised+re-post)Upper and Lower sums & Riemann sums

    http://img156.imageshack.us/i/17818455.jpg/ http://img215.imageshack.us/i/53355598.jpg/ http://img509.imageshack.us/i/11493310.jpg/ If you look at the above, I have underlined the problem that I am having. So, my first question is, where are these inequalities coming from? If you do have...
  49. C

    Upper and Lower sums & Riemann sums

    Homework Statement Homework Equations The Attempt at a Solution I have attachments that can answer the above template, and please look at the attachments if you are trying to help me. I have two questions regarding upper and lower sums & Riemann sums. So, the attachments 1 & 2 are...
  50. L

    Two Dimensional Riemann Tensor

    show that in two dimensions, the Riemann tensor takes the form R_{abcd}=R g_{a[c}g_{d]b}. i've expanded the RHS to get R g_{a[c}g_{d]b}=\frac{R}{2!} [g_{ac} g_{db} - g_{ad} g_{cb}]=\frac{1}{2} R_e{}^e [g_{ac} g_{db} - g_{ad} g_{cb}] but i can't seem to simplify it down. this is problem...
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