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. E

    Can someone help me understand and evaluate the Riemann zeta function?

    I still don't understand a few things. Let's say we had a non-trivial zero counting function, Z_n(n), for the riemann zeta function. Couldn't we fairly easily prove the riemann hypothesis by evaluating \zeta (\sigma+iZ_n), solving for \sigma , then proving it for all n using induction...
  2. B

    Cauchy-Riemann Equations and Complex Derivatives: A Homework Problem

    Homework Statement Show that when f(z)=x^3+i(1-y)^3, it is legitimate to write: f'(z)=u_x+iv_x=3x^2 only when z=i Homework Equations Cauchy riemann equations: u_x=v_y , u_y=-v_x f'(z)=u_x+i*v_y The Attempt at a Solution u=x^3 v=(1-y)^3 u_x=3*x^2 v_y=-3*(1-y)^2...
  3. G

    Trouble with Riemann sums

    Alright, I started doing Riemann sums and I am ripping my hair out in frustration. I just can't wrap my head around how I'm supposed to do it, and my woefully vague textbook isn't helping either. I'm wondering how I'm supposed to solve a Riemann sum question with sigma notation (no limits), and...
  4. E

    Evaluating the Riemann Zeta Function: Step-by-Step Guide for \zeta(c + xi)

    Can someone show me the steps to evaluating \zeta(c + xi), where 0 \leq c<1?
  5. J

    Riemann curvature tensor as second derivative of the metric

    It is a standard fact that at any point p in a Riemannian space one can find coordinates such that \left.g_{\mu\nu}\right|_p = \eta_{\mu\nu} and \left.\partial_\lambda g_{\mu\nu}\right|_p. Consider the Taylor expansion of g_{\mu\nu} about p in these coordinates: g_{\mu\nu} = \eta_{\mu\nu}...
  6. W

    Robertson-Walker metric in higher dimensions (and problematic Riemann tensor)

    Hello folks, this is going to be a bit longish, but please bear with me, I'm going nuts over this. For a term paper I am working through a paper on higher dimensional spacetimes by Andrew, Bolen and Middleton. You can http://arxiv.org/abs/0708.0373" . My problem/confusion is in...
  7. Q

    Unraveling the Mysteries of the Riemann Curvature Tensor

    Homework Statement (My first post on this forum) Background: I am teaching myself General Relativity using Dirac's (very thin) 'General Theory of Relativity' (Princeton, 1996). Chapter 11 introduces the (Riemann) curvature tensor (page 20 in my edition). Problem: Dirac lists several...
  8. J

    Determination of Riemann curvature tensor from tidal forces

    Hi, Given a large number of test particles N, it should be possible to determine the Riemann curvature tensor by tracking their motion as they undergo geodesic deviation. Is there a minimum number N that will achieve this in any situation, or does it vary from problem to problem? How...
  9. B

    Book recs please - complex analysis, riemann surfaces, multi-valued functions

    Hi everyone, hope this is the right place to put this :) I have just finished "Theory of Functions" Vol. 1 & 2 by Konrad Knopp. I'd like to continue with a book that picks up where the second volume it left off. (Especially would be nice is a more "modern" book) The second volume is about...
  10. M

    Cauchy Riemann & Taylor Expansion.

    Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct. Please could you also guide me on Q1 & 4. Q1...
  11. I

    Riemann Hypothesis: Question on Critical Line

    I have a question concerning the Riemann Hypothesis, a conjecture about the distribution of zeros of the Riemann-zeta function. the trivial zeros (s=-2, s= -4, s=-6) arent much of a concern as the NON-trivial zeros, where any real part of the non-trivial zero is = 1/2. What i am having...
  12. C

    Lagarias’ equivalence to the Riemann hypothesis

    Lagarias’ equivalence to the Riemann hypothesis should be discussed, i.e., if hn := n-th harmonic number := 1/1 + 1/2 + · · · + 1/n, and σn := divisor function of n := sum of positive divisors of n, then if n > 1, hn + ehn ln hn > σn. There is a $1,000,000 prize for the proof of this at...
  13. C

    Proof of the Riemann Hypothesis

    http://arxiv.org/abs/0806.0892"
  14. H

    Prove Riemann Hypothesis: High School Student Guide

    i am a high school student i want to prove the riemann hypothesis but i do not how to start:confused:
  15. D

    Program for graphing Riemann zeta function

    Hello I plan on applying to the university of waterloo next year and due to the fact that many of my marks are not that great (failed gr 10 math) I decided to start a site to showcase my ability in math and programing. For those of you who are interested I wrote a program to graph regions of...
  16. S

    Riemann Integrability of f(x) = x on [0,1]

    f(x) = x , if x is rational = 0 , if x is irrational on the interval [0,1] i just wanted to check if my reasoning is right. take the equipartition of n equal subintervals with choices of t_r's as r/n for each subinterval. calculating the integral as limit of this sum (and...
  17. daniel_i_l

    What is the proof for the Riemann Series Theorem?

    Can anyone tell we how this: http://mathworld.wolfram.com/RiemannSeriesTheorem.html can be proved? The book that I read it in said that it was "beyond the scope of the book". It one of the coolest theorems I've read about. For example, it means that for any number (pi, phi, ...) there's...
  18. T

    Fortran What is the first approximation for Riemann Sum using Fortran 90?

    [SOLVED] Riemann Sum with Fortran 90 My assignment: Use Reimann Sums to estimate pi to 6 decimal places (ie: you can stop when successive iterations yield a change of less than 0.000001. For the Reimann Sums solution, an iteration equals 2X the number of segments as the trial before. Print out...
  19. L

    Lie derivative and Riemann tensor

    Suppose you have a spacetime with an observer at rest at the origin, and the surface at t = 0 going through the origin, and passing through the surface there are geodesics along increasing time. Then as you get a small ways away from the surface, the geodesics start to deviate from each other...
  20. P

    Evaluating a Riemann Sum for $\int^{-2}_{5} t^2 + 6t - 4 dt$

    [SOLVED] Riemann sum Important stuff: \sum i^2 = \frac{n(n+1)(2n+1)}{6} \sum i = \frac{n(n+1)}{2} And the solution: (Where I write "lim" I mean limit as n-->infinity. Where I write the summation sign I mean from i=1 to n.) lim \sum t^2 + 6t - 4 \Delta t \Delta t = \frac{5 -...
  21. M

    Summation - Riemann Intergral -

    [SOLVED] Summation - Riemann Intergral - URGENT Homework Statement Im working on the upper and lower riemann sums of f(x) = exp(-x) where Pn donates the partition of [0,1] into n subintervals of equal length (so that Pn = {0,1/n,2/n,...,1}) Homework Equations The Attempt at...
  22. R

    Proving Riemann Integrability of a Function to Zero

    Homework Statement Prove that the function specified below is Riemann integrable and that its integral is equal to zero. Homework Equations f(x)=1 for x=1/n (n is a natural number) and 0 elsewhere on the interval [0,1]. The Attempt at a Solution I have divided the partition into...
  23. Shaun Culver

    Advice on complex analysis, Riemann surface & complex mappings.

    Could anybody please give advice for the study of complex analysis, Riemann surfaces & complex mappings. These subjects form the content of chapters 7 & 8 of Roger Penrose's "The Road to Reality". Any advice will do: maybe suggestions on books to supplement the learning, or books to further my...
  24. S

    Proving a Function is Riemann Integrable

    Homework Statement Let f, g : [a, b] \rightarrow R be integrable on [a, b]. Then, prove that h(x) = max{f(x), g(x)} for x \in [a, b] is integrable. 1 Homework Equations Definition of integrability: for each epsilon greater than zero there exists a partition P so that...
  25. C

    Contraction of riemann tensor

    why does the einstein field tensor have the riemann tensor contracted? I am confused as to what purpose it serves. I have seen an explanation that it gets rid of extra information about spacetime or something like that. and also is the Ricci scalar added to einstein tensor so that the covariant...
  26. B

    Lie groups as riemann manifolds

    What Lie groups are also Riemann manifolds? thanks
  27. O

    Restrictions on Riemann components

    My crude understanding of GR in outline is that spacetime curvature is described by the way the components of the Riemann tensor vary from point to point in spacetime, that such variation is controlled by Einstein's field equations, and that the source of curvature is the energy-momentum tensor...
  28. quasar987

    Something a little odd - Lebesgue vs Riemann integral

    In the Riemann theory for a function f defined on all of R, we define its improper integral over R as the sum of two limits: \int_{-\infty}^{+\infty}f(x)dx = \lim_{a\rightarrow -\infty}\int_{a}^0f(x)dx+\lim_{b\rightarrow +\infty}\int_{0}^b f(x)dx and in general, this is not equal to...
  29. S

    Find the Riemann sum for this integral

    Find the Riemann sum for this integral using the right-hand sums for n=4 Find the Riemann sum for this same integral, using the left-hand sums for n=4 Sorry the integral is attatched. I don't know how to get it onto here.
  30. M

    Exact Definition of Riemann Integral

    Hi I recently stumbled upon this: I know that the Riemann Integral is defined for every piecewise continouus curve. But now suppose you´re asked the following: you are given f(x,y)=\frac{xy^3}{(x^2+y^2)^2} with additional Definition f(0,0)=0. ( It´s a textbook problem :) ) Now surely...
  31. B

    Please HELP Don't Understand Simple Concept on Riemann Sums

    Please HELP...Don't Understand Simple Concept on Riemann Sums Can someone please explain this to me... The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. That is, ||Triangle|| approaches 0 implies that n approaches infinity. I thought...
  32. Gib Z

    Using Riemann Upper sums to solve limits

    I often see people use the Riemann definition of the integral to solve a certain limit-series computation, but they usually just skip a step that I can follow one way but not the other. Given the integral, I can see the limit-series that comes from it, but when trying to find the integral from...
  33. P

    How to Describe the Riemann Surface Associated with a Complex Function?

    Hi guys, My gf is doing honours and is having some trouble with one question on her assignment for complex analysis. She is really stuck and I've only done this topic at an undergraduate level so I have no idea. Neither of us have done any subjects in Topology so we don't know what to do...
  34. P

    String Theory & Riemann Hypothesis: Is There a Connection?

    Could the maths of string theory or versions of it lead insight into the Riemann hypothesis as, for a start both are about mathematics in the complex plane. Anyone working on this connection at the moment?
  35. quasar987

    Exploring the Relationship Between Riemann and Lebesgue Integration

    In the course I'm taking, we are already done with Lebesgue integration on R, and while we have proven that for continuous fonctions, the Riemann integral and the Lebesgue integral give the same output, we have not investigated further the correspondance btw the two. So I have some questions...
  36. P

    Exploring the Differences Between Newton and Riemann Integrals

    What is the difference between the two? Does the Newton integral arise from the fundalmental theorem of calculus and the Riemann integral is the Newton integral but more rigorously defined?
  37. K

    Understanding the Trivial Zeros of the Riemann Hypothesis

    Can somebody explain me about the trivial zeros? Why \zeta(-2) = \zeta(-4) = \zeta(-6) = 0 = \zeta(k) So \zeta(k) \sum_{n=1}^{ \infty} \frac{1}{n^k} = 0 ?
  38. J

    Problem on Riemann Integrable functions

    Homework Statement Let f:[a,b] -> R R being the set of real numbers If f^3 is Reimann-integrable, does that imply that f is? Homework Equations If f is Riemann-Integrable, then it has upper/lower step functions, such that the difference between the upper and lower sums is less...
  39. R

    Riemann surface, elliptic curves

    Are there notes on the net or books that give a gentle introduction on Riemann surfaces ( say undergraduate math or math for physicists type level)? Always read of the importance and beauty of Riemann surfaces but can't find surveys or intros for outsiders. Same for elliptic curves...
  40. T

    Sums of series for Riemann integrals

    Hello I'm having some difficulty in finding sums which relate to Riemann integrals. The first one seems pretty simple.. a finite calculation of what would otherwise be the harmonic series i.e. 1/k from k=n to k=(2n-1). I can't see an easy way of finding a formula in terms of n, however...
  41. N

    Is this the end of the Riemann Hypothesis?

    http://www.arsmathematica.net/archives/2007/05/22/latest-on-latest-paper-on-arxiv/
  42. O

    Riemann Zeros and Harmonic Frequencies

    Can someone elaborate on the relationship of the Riemann zeros and primes? How are the zeros harmonic to the primes? The quotes below mention the 'sum of its complex zeros' and 'other sums over prime numbers'. Can someone clarify this? From Answers.com: "The zeros of the Riemann...
  43. W

    Explain Riemann Hypothesis: Imaginary Numbers & Interval

    Can someone please explain to what exactly the Riemann Hypothesis is? My friend said it is something to do with imaginary numbers and how they behave in a certain interval- just wondering.
  44. W

    Solving the Definite Integral Using Right Riemann Sums

    Homework Statement The following sum \sqrt{9 - \left(\frac{3}{n}\right)^2} \cdot \frac{3}{n} + \sqrt{9 - \left(\frac{6}{n}\right)^2} \cdot \frac{3}{n} + \ldots + \sqrt{9 - \left(\frac{3 n}{n}\right)^2} \cdot \frac{3}{n} is a right Riemann sum for the definite integral. Solve as n->infinity...
  45. S

    Calculating Degrees of Freedom for Riemann Tensor in D Dimensions

    How many degrees of freedom has Riemann Tensor in general D dimensions and how it can be calculated?
  46. G

    Riemann Sums infinite strips

    Use the riemann sums model to estimate the area under the curve f(x) = x^2, between x =2 and x = 10, using an infinite number of strips. Be sure to include appropriate diagriams and full explanation of the method of obtaining all numerical values, full working and justification. Does anybody...
  47. P

    Proving function is improper riemann integrable

    Homework Statement let f:[0,oo) -> R be given by f(x) = sin(x) / x for x>0 and f(0) = c. Prove that f is improper riemann integrable without computing the integral explicitly The Attempt at a Solution I've attempted to find a upperbound for f(x) such that the integral does not diverge...
  48. Gib Z

    Ramanujan Summation & Riemann Zeta Function: Negative Values

    I was wondering if anyone could tell me more about the Riemann Zeta function, esp at negative values. Especially when \sum_{n=1}^{\infty}n= \frac{-1}{12} R where R is the Ramanujan Summation Operator. Could anyone post a proof?
  49. T

    Finding Riemann Sum for f(x)=3x^2+3

    Homework Statement Find the Riemann sum associated with f(x)=3 x^2 +3 ,\quad n=3 and the partition x_0=0,\quad x_1=3,\quad x_2=4,\quad x_3=6,\qquad \mbox{ of } [0,6] (a) when x_k^{*} is the right end-point of [x_{k-1},x_k]. . (b) when x_k^{*} is the mid-point of [x_{k-1},x_k]...
  50. Apost8

    Integration or Riemann Sums: Which is More Effective for Numerical Integration?

    Is there ever a situation where it is more appropriate/advantageous to use Riemann summation as opposed to evaluating an integral, or is Riemann summation merely taught in order to help the student to understand what's going on?
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