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. D

    Inverse of the Riemann Zeta Function

    Homework Statement I wish to prove that for s>1 $$ \sum\limits_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}. $$ The Attempt at a Solution (1) I first showed that $$ \prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}. $$ It was a given theorem in the text that $$...
  2. chisigma

    MHB A curiosity about the Riemann Zeta Function....

    Recently some interesting material about the Riemann Zeta Function appeared on MHB and I also contributed in the post... http://mathhelpboards.com/challenge-questions-puzzles-28/simplifying-quotient-7235.html#post33008 ... where has been obtained the expression... $\displaystyle \zeta (s) =...
  3. V

    Using parallel propagator to derive Riemann tensor in Sean Carroll's

    Hello all, In Carroll's there is a brief mention of how to get an idea about the curvature tensor using two infinitesimal vectors. Exercise 7 in Chapter 3 asks to compute the components of Riemann tensor by using the series expression for the parallel propagator. Can anyone please provide a...
  4. tom.stoer

    Riemann surfaces over algebraic surfaces

    Suppose we have one polynom ##P(r_1, r_2, \ldots, r_n) = 0## in n complex variables. This defines a n-1 dimensional complex algebraic surface. Suppose that for each variable we have ##r_i = e^{ip_i}## with complex p. In the case n=1 of one variable r this results in the complex logarithm...
  5. J

    Bounded derivative Riemann integrable

    Assume that a function f:[a,b]\to\mathbb{R} is differentiable in all points of its domain, and that the derivative f':[a,b]\to\mathbb{R} is bounded. Is the derivative necessarily Riemann integrable? This what I know: Fact 1: Assume that a function is differentiable at all points of its domain...
  6. polygamma

    MHB Another integral representation of the Riemann zeta function

    Here is another integral representation of $\zeta(s)$ that is valid for all complex values of $s$. It's similar to the first one, but a bit harder to derive.$ \displaystyle \zeta(s) = 2 \int_{0}^{\infty} \frac{\sin (s \arctan t)}{(1+t^{2})^{s/2} (e^{2 \pi t} - 1)} \ dt + \frac{1}{2} +...
  7. S

    Riemann sum where n = 3 for both left and right endpoint estimates

    Homework Statement Compute the integral that is highlighted in MyWork.jpg using Riemann sums using left and right endpoints. Homework Equations ##x_i* = a + i Δx## ##*x_i = a + i Δx - Δx## ##Σ_{i=1}^{n} i = n(n+1)/2## ##Σ_{i=1}^{n} i^2 = n(n+1)(2n+1)/6## The Attempt at a Solution My...
  8. polygamma

    MHB An integral representation of the Riemann zeta function

    Show that $\displaystyle \zeta(s) = \frac{2^{s-1}}{1-2^{1-s}} \int_{0}^{\infty} \frac{\cos (s \arctan t)}{(1+t^{2})^{s/2} \cosh \left( \frac{\pi t}{2} \right)} \ dt $The cool thing about this representation is that it is valid for all complex values of $s$ excluding $s=1$. This integral is...
  9. Mandelbroth

    Understanding Riemann Surfaces and Their Applications in Complex Analysis

    I'll preface this by saying that this just isn't getting through to me. I know the material, but my brain feels like it doesn't, for lack of a better word, "fit." Looking back at Riemann surfaces in complex analysis after familiarizing myself with some differential geometry really makes them...
  10. Philosophaie

    Formulating x^n Coordinate System for Non-Rectangular/Spherical Riemann Manifold

    I want to be able to formulate x^{n} coordinate system. x^{n} =(x^{1}, x^{2}, x^{3}, x^{4}) How do you do this when the Riemann Manifold is not rectangular or spherical? Also how do you differentiate with respect to "s" in that case. \frac{dx^n}{ds}
  11. P

    Why Do These Riemann Tensor Terms Cancel Each Other Out?

    I was working on the derivation of the riemann tensor and got this (1) ##\Gamma^{\lambda}_{\ \alpha\mu} \partial_\beta A_\lambda## and this (2) ##\Gamma^{\lambda}_{\ \beta\mu} \partial_\alpha A_\lambda## How do I see that they cancel (1 - 2)? ##\Gamma^{\lambda}_{\ \alpha\mu}...
  12. MarkFL

    MHB Maya's question at Yahoo Answers regarding a Riemann sum and definite integral

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  13. E

    What is the Riemann Curvature Tensor for Flat and Minkowski Space?

    Homework Statement Show that all components of Riemann curvarue tensor are equal to zero for flat and Minkowski space. Homework Equations The Attempt at a Solution ## (ds)^2=(dx^1)^2+(dx^2)^2+...+(dx^n)^2 \\ R_{MNB}^A=\partial _{N}\Gamma ^{A}_{MB}-\partial _{M}\Gamma ^{A}_{NB}+\Gamma...
  14. 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...
  15. Fernando Revilla

    MHB Solving an Integral with a Right Endpoint Riemann Sum

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  16. Z

    Riemann curvature scalar, Ricci Scalar.What does they measure ?

    hello Can you perhaps explain what does the Riemann curvature scalar R measure? or is just an abstract entity ? What does the Ricci tensor measure ? I just want to grasp this and understand what they do. cheers, typo: What DO they measure in the title.
  17. S

    What is the Form of Riemann Tensor in 3D?

    hi Riemann tensor has a definition that independent of coordinate and dimension of manifold where you work with it. see for example Geometry,Topology and physics By Nakahara Ch.7 In that book you can see a relation for Riemann tensor and that is usual relation according to Christoffel...
  18. J

    Mapping of algebraic function to Riemann surface?

    When we map the algebraic function, w(z), to a Riemann surface we essentially create a new "Riemann" coordinate system over a surface that is called the "algebraic function's Riemann surface". This mapping allows one to create single-valued functions, f(z,w), of the coordinate points over...
  19. Z

    Riemann Curvature Scalar Differs in Landau & MTW

    hello For the same Friedmann metric, Landau (Classical theory of fields) finds a value for the Riemann curvature scalar which is given in section 107 : R = 6/a3( a + d2(a)/dt2) whereas in MTW , in box 14.5 , equation 6 , its value is : R = 6(a-1 d2(a)/dt2 + a-2 (1 + (d(a)/dt)2 ) ) The...
  20. alyafey22

    Mathematica Plotting Riemann Surfaces with Wolfram Mathematica 8.0

    Does anybody know how to plot the Riemann surface of complex functions on wolfram Mathematica 8.0 ?
  21. M

    Uniqueness of value of Riemann Integral(proof)

    The proof is in the document. I highlighted the main points that I am questioning in the document. I am questioning the fact that A = B... (The following is in the document) |A-B|=ε where they define the value of ε to be a positive arbitrary real number (ε>0). And for A = B that means ε must...
  22. M

    Second derivative of a metric and the Riemann curvature tensor

    I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...
  23. S

    Riemann Integrability of Composition

    Homework Statement Let ψ(x) = x sin 1/x for 0 < x ≤ 1 and ψ(0) = 0. (a) If f : [-1,1] → ℝ is Riemann integrable, prove that f \circ ψ is Riemann integrable. (b) What happens for ψ*(x) = √x sin 1/x? Homework Equations I've proven that if ψ : [c,d] → [a,b] is continuous and for every set...
  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. I

    Riemann integral is zero for certain sets

    Homework Statement The question is: Let ##\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}##. Prove that if ##E\subset\pi## is a closed Jordan domain, and ##f:E\rightarrow\mathbb{R}## is Riemann integrable, then ##\int_{E}f(x)dV=0##. Homework Equations n/a...
  26. I

    How to prove if Riemann integrable then J-integrable

    Homework Statement Let ##E\subset\mathbb{R}^n## be a closed Jordan domain and ##f:E\rightarrow\mathbb{R}## a bounded function. We adopt the convention that ##f## is extended to ##\mathbb{R}^n\setminus E## by ##0##. Let ##\jmath## be a finite set of Jordan domains in ##\mathbb{R}^n## that...
  27. I

    MHB Riemann integrable then J-integrable

    Let $E\subset\mathbb{R}^n$ be a closed Jordan domain and $f:E\rightarrow\mathbb{R}$ a bounded function. We adopt the convention that $f$ is extended to $\mathbb{R}^n\setminus E$ by $0$. Let $\jmath$ be a finite set of Jordan domains in $\mathbb{R}^n$ that cover $E$. Define $M_J=sup\left \{...
  28. I

    MHB Riemann integral is zero for certain sets

    The question is:Let $\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}$. Prove that if $E\subset\pi$ is a closed Jordan domain, and $f:E\rightarrow\mathbb{R}$ is Riemann integrable, then $\int_{E}f(x)dV=0$.(How to relate the condition it's Riemann integrable to the value is $0$...
  29. L

    I'm trying to write a program that plots the riemann zeta function

    I saw a picture of what it might look like when I was researching it, but I'm confused about something. The picture's caption said that the complex coordinates were darkened as their value got larger, leading to a helpful graph, but I do not understand what scale they used. For my program, I...
  30. H

    Finding limit of sum using Riemann Sums

    Find the limit limn→∞∑i=1 i/n^2+i^2 by expressing it as a definite integral of an appropriate function via Riemann sums ...?
  31. A

    Riemann Integrabe Step Functions

    Suppose that g:[a,b]\rightarrow[c,d] is Riemann integrable on [a,b] and f:[c,d]\rightarrow ℝ is Riemann integrable on [c,d]. Prove that f\circ g is Riemann integrable on [a,b] if either f or g is a step function. The proof for g being a step function seems easy enough, but the other way seems...
  32. S

    Disjoint intervals for Riemann Integral

    So the beginning of Rudin's Real and Complex Analysis states that the Riemann integral on an interval [a,b] can be approximated by sums of the form \Sigma\stackrel{i=1}{n}f(ti)m(Ei) where the Ei are disjoint intervals whose union is the whole interval. At least when I learned it, the Riemann...
  33. S

    How are Riemann surfaces graphed

    I am creating a python application to graph riemann surfaces, from the wikipedia article it says the functions are graphed as x real part of the complex domain, y, imaginary part of the complex domain, z, the real part of f(z), but how does one represent the imaginary part with the color, would...
  34. D

    Difference between Riemann-Stieltjes and Riemann Integral

    Hi all, Homework Statement Is the difference between riemann stieltjes integral and riemann integral that in riemann integral, the intervals are of equal length and in riemann stieltjes, the partitions are defined by the integrator function? If not so what exactly is it that integrator...
  35. micromass

    Geometry Algebraic Curves and Riemann Surfaces by Miranda

    Author: Rick Miranda Title: Algebraic Curves and Riemann Surfaces Amazon Link: https://www.amazon.com/dp/0821802682/?tag=pfamazon01-20 Prerequisities: Complex Analysis, Differential Geometry, Abstract Algebra Level: Grad Table of Contents: Preface Riemann Surfaces: Basic Definitions...
  36. S

    Riemann Zeta Function Zeros

    I was looking at the Wolfram Alpha page on the Riemann Zeta Function Zeros which can be found here, http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html At the top of the pag there are three graphs each with what looks to be a hole through the graph. Now I know the graph is an Argand...
  37. tom.stoer

    Hehl on gauge aspects of spacetime - beyond Riemann

    http://arxiv.org/abs/1204.3672 Gauge Theory of Gravity and Spacetime Friedrich W. Hehl (U Cologne and U of Missouri, Columbia) (Submitted on 17 Apr 2012) The advent of general relativity settled it once and for all that a theory of spacetime is inextricably linked to the theory of gravity...
  38. stripes

    Riemann integral of characteristic function.

    Homework Statement The characteristic function of a set E is given by χe = 1 if x is in E, and χe = 0 if x is not in E. Let N be a natural number, and {an, bn} from n=1 to N, be any real numbers. Use the definition of the integral (Riemann) to show that \int \sum b_{n} X_{ \left\{ a_{n}...
  39. L

    Express e^x from 1 to 8 as a Riemann Sum. Please, check my work?

    Express e^x from 1 to 8 as a Riemann Sum. Please, check my work? :) 1. Express ∫1 to 8 of e^xdx as a limit of a Riemann Sum. (Please ignore the __ behind the n's. The format is not kept without it...) _____n 2. lim Ʃ f(xi)(Δx)dx x→∞ i=1 Δx= (b-a)/n = 8-1/n = 7/n xi= 1 + 7i/n ____n lim Ʃ...
  40. M

    Riemann Sum with subintervals/partition

    So I missed a class and am trying to figure out a question in my textbook but am completely lost. It goes a little something like this: Let f(x)=x3 and let P=<-2,0,1,3,4> be a partition of [-2,4]. a) Compute Riemann Sum S(f,P*) if the points <x1*,x2*,x3*,x4*>=<-1,1,2,4> are embedded in P...
  41. U

    Understanding Riemann Surfaces in Complex Analysis

    I don't mean to sound ignorant, but when reading up on complex analysis in the broad sense, I don't really see the point of introducing Riemann surfaces. It's a way of making multivalued functions single valued, but so what? I don't see the utility of such an idea, which isn't to argue there is...
  42. C

    About interesting convergence of Riemann Zeta Function

    Hi, I was playing with Riemann zeta function on mathematica. I encountered with a quite interesting result. I iterated Riemann zeta function for zero. (e.g Zeta...[Zeta[Zeta[0]]]...] It converges into a specific number which is -0.295905. Also for any negative values of Zeta function, iteration...
  43. S

    Is Riemann Zeta function related to differential equations?

    Hi. I just came back from my differential equation midterm and was surprised to see a problem with the Riemann-zeta equations on it. I think the problem went something like "Prove that \pi/6 = 1 + (1/2)^2 + (1/3)^2 + ... " The study guide did mention that "prepare for a problem or two...
  44. Z

    Calculate the elements of the Riemann tensor

    Homework Statement Compute 21 elements of the Riemann curvature tensor in for dimensions. (All other elements should be able to produce through symmetries) Homework Equations R_{abcd}=R_{cdab} R_{abcd}=-R_{abdc} R_{abcd}=-R_{bacd} The Attempt at a Solution I don't see how 21...
  45. STEMucator

    Proving Riemann Integration

    Homework Statement Suppose that f(x) is a bounded function on [a,b] If M = sup(f) and m = inf(f), prove that -M = inf(-f) and -m = sup(-f). If m>0, show that 1/f is bounded and has 1/m as its supremum and 1/M as its infimum. Let f be integrable on [a,b]. Prove that -f is integrable on [a,b]...
  46. F

    Relationship between Riemann Sum and the Integral

    Homework Statement The notation for a Riemann sum - Ʃ f(x*i)Δx - is very similar to the notation for the integral (the Ʃ becomes ∫, the f(x*i) becomes f(x) and the Δx becomes dx). \int f(x)dx = \lim_{n \to \infty}\sum_{k=0}^{n} f(x_i) Δx Is there a way to explicitly define the values on the...
  47. A

    Geometry of the Riemann, Ricci, and Weyl Tensors

    Hi, I was wondering if someone wouldn't mind breaking down the geometrical differences between the Riemann, Ricci, and Weyl tensor. My current understanding is that the Ricci tensor describes the change in volume of a n-dimensional object in curved space from flat Euclidean space and that if we...
  48. A

    How Do You Estimate a Double Integral Using Riemann Sums?

    Homework Statement If R = [0,4]x[-1,2], use a Riemann sum with m=2, n=3 to estimate the value of ∫∫(1-xy^2)dA. Take the sample points to be the lower right corners. Homework Equations NoneThe Attempt at a Solution 2*1[f(2,-1) + f(2,0) + f(2,1) + f(4,-1) + f(4,0) + f(4,1)] = some value Just...
  49. mnb96

    Poincaré disk vs Riemann sphere

    Hello, it is well-known that with stereographic projection we can obtain a 1-1 correspondence between the points of the 2d Cartesian plane (plus the point at infinity), and the points on the Riemann sphere. What is the geometrical construction that corresponds to a 1-1 mapping between the...
  50. C

    Constructing a Step Function for the Riemann Function with Restrictions on q

    Homework Statement The reimann function (if x is rational then f(x)=1/q, if x is irrational then f(X)=0)is not a step function,then, for any essilope, construct a step function k:[0,1] ~>R s.t. ||f-k||=sup{|f(t)-k(t)|:t is in[0,1]}<essilope,can someone help me to construct such step function...
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