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. K

    Riemann zeta function - one identity

    Let p_n be number of Non-Isomorphic Abelian Groups by order n. For R(s)>1 with \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s} we define Riemann zeta function. Fundamental theorem of arithmetic is biconditional with fact that \zeta(s)=\prod_{p} (1-p^{-s})^{-1} for R(s)>1. Proove that for R(s)>1 is...
  2. polygamma

    MHB Understanding Riemann Integrals of $\ln\ x$

    $\displaystyle \int_{0}^{1} \ln \ x \ dx $ is not a proper Riemann integral since $\ln \ x $ is not bounded on $[0,1]$. Yet $ \displaystyle \int_{0}^{1} \ln \ x \ dx = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \left(\frac{k}{n} \right)$. Is this because $\ln \ x$ is monotone on $(0,1]$?
  3. J

    Understanding Riemann Zeta functions for s=1/3

    Hi guys I'm trying to understand Riemann Zeta functions particularly for s=1/3 I know \zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s} and converges for Re(s)>1 Ok, but what about for s=1/3, then \zeta(1/3)=\sum_{n=1}^\infty\frac{1}{n^{1/3}}= \sum_{n=1}^\infty\frac{1}{\sqrt[3]{n}}...
  4. T

    What is the Integral Being Approximated by the Given Riemann Sum?

    Homework Statement For which integral, is the below example, a Riemann sum approximation.? The example is: 1/30( sqrt(1/30) + sqrt(2/30) + sqrt(3/30)+...+sqrt(30/30)) A. Integral 0 to 1 sqrt(x/30) B. Integral 0 to 1 sqrt(x) C. (1/30) Integral 0 to 30 sqrt(x) D. (1/30) Integral 0 to 1...
  5. O

    No. of Independent Components of Riemann Tensor in Schwartzchild Metric

    In general 4d space time, the Riemann tensor has 20 independent components. However, in a more symmetric metric, does the number of independent components reduce? Specifically, for the Schwartzchild metric, how many IC does the corresponding Riemann tensor have? (I think it is 4, but I...
  6. H

    Definite Integral of the Natural Log of a Quadratic using Riemann Sums

    Homework Statement Use the form of the definition of the integral to evaluate the following: lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6] Homework Equations x_{i} = 2 + \frac{4}{n}i Δx = \frac{4}{n} Ʃ^{n}_{i=1}i^{2} =...
  7. D

    Easy way of calculating Riemann tensor?

    Homework Statement Is there any painless way of calculating the Riemann tensor? I have the metric, and finding the Christoffel symbols isn't that hard, especially if I'm given a diagonal metric. Out of 40 components, most will be zero. But how do I know how to pick the indices of...
  8. D

    Riemann tensor, Ricci tensor of a 3 sphere

    Homework Statement I have the metric of a three sphere: g_{\mu \nu} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\sin^2\theta \end{pmatrix} Find Riemann tensor, Ricci tensor and Ricci scalar for the given metric. Homework Equations I have all the formulas I need, and I...
  9. F

    Riemann Sum Calculation for f(x)=x on [0,2] with n=8

    Let Pn denote the partition of the given interval [a,b] into n sub intervals of equal length Δxi = (b-a)/n Evaluate L(f,Pn) and U(f,Pn) for the given functions f and the given values of n. f(x)=x on [0,2], with n=8 2.My solution x0 = 0, x1 = 1/4, x2 = 1/2, x3 = 3/4, x4 = 1, x5 = 5/4...
  10. J

    Determining Riemann surface geometry of algebraic functions

    Hi, given the algebraic function: f(z,w)=a_n(z)w^n+a_{n-1}(z)w^{n-1}+\cdots+a_0(z)=0 how can I determine the geometry of it's underlying Riemann surfaces? For example, here's a contrived example: f(z,w)=(w-1)(w-2)^2(w-3)^3-z=0 That one has a single sheet manifold, a double-sheet...
  11. P

    Prove Riemann Sum: (ex-1)/x for x > 0

    Homework Statement Prove that: lim n->inf1/n*Ʃn-1k=0ekx/n = (ex-1)/x x>0 Homework Equations That was all the information provided. Any progress i have made is below. I didn't want to add any of that to this section because this is all speculation on my part so far. The Attempt at a...
  12. L

    How is the Riemann tensor proportinial to the curvature scalar?

    My professor asks, "Double check a formula that specifies how Riemann tensor is proportional to a curvature scalar." in our homework. The closet thing I can find is the relation between the ricci tensor and the curvature scalar in einstein's field equation for empty space.
  13. P

    Riemann Integrability Question

    This isn't a homework question. My adviser has me studying basic analysis and has lately pushed me towards the following question: "Let f be any continuous function. Can we prove that there exists a SEQUENCE of step functions that converges UNIFORMLY to f?" I have noticed this idea is...
  14. Y

    Is there any good reference on Riemann Surface and Riemann Theta Function?

    Hi, Currently, I need to read some reference about Integrable System, but I am stuck in Riemann Surface, genus, divisors, and Riemann Theta Functions. This makes me anxious. Is there introduction or pedagogical reference on this topic? I think I can spend some time read it during winter...
  15. S

    Real Analysis Riemann Integration

    Suppose we have: f(x)= 1 if 0\leq x \leq 1 AND 2 if 1\leq x \leq 2 Using the definition, show that f is Riemann integrable on [0, 2] and find its value? I have a general idea of how to complete this question using partitions and the L(f,P) U(f,P) definition, but am not quite receiving the...
  16. T

    Comparison of Riemann integral to accumulation function

    Let f:[0,1]→ℝ be an increasing function. Show that for all x in (0,1], \frac{1}{x}\int_{0}^{x}f (t) \,dt \le \int_{0}^{1}f (t) \,dt So by working backwards I got to trying to show that (1-x)\int_{0}^{1}f (t) \,dt \le \int_{x}^{1}f (t) \,dt . While I know both sides are equal at x=1, the...
  17. T

    Does the Improper Riemann Integral Converge or Diverge for p<1?

    I know that the improper integral \int_2^\infty \left(\frac{1}{x\log^2x}\right)^p \, dx converges for p=1, but does it diverge for p>1? How do you show this?
  18. Y

    What is the Genus of Riemann Surface?

    I learned something about genus in Topology. The concept Genus in Topology is intuitive and lucid. Now I am confronted with the Genus in Riemann Surface. I do not know what is Genus on Riemann Surface. Is it relevant to "singularity"? Anyone can help me make it a bit clear? Thanks.
  19. S

    Definite integral using Riemann sums?

    I'm reviewing my Calc 1 material for better understanding. So, I was reading about the area under a curve and approximating it using Riemann sums. Now, I understand the method, but I was a little confused by finding xi*. I know there is a formula for it xi*=a+Δx(i). What does the "i" stand for...
  20. T

    Analytic Functions Cauchy & Riemann Equations

    Hi, this is fairly fundamental and basic, but I cannot seem to make sense of it I know z = x + iy and hence a function of this variable would be in the form h = f(z). BUT I do not understand why f(z) = u(x,y) + iv(x,y) why so? in z = x + iy, x is the real part and iy is the imaginary...
  21. F

    A problem involving Riemann Integrals

    I've been having some trouble with a maths problem and I hoped someone might be able to help. We don't seem to have been taught most of what we need to do this, I understand Riemann integrals but what we've been taught and what they're asking for is just different. I could do with a...
  22. L

    Riemann zeta functionpole question?

    The simple pole at on is due to that its value of course is not closed due to it is an infinite value. My question is: is this value of infinity, positive or negative. or both??
  23. O

    Riemann Hypothesis and Goldbach Conjecture Proof?

    Hey guys, I saw these just showed up on arXiv, published by some unknown who claims to have invented his own number system and is not affiliated with any academic institutions. What do you make of this? http://arxiv.org/abs/1110.3465 http://arxiv.org/abs/1110.2952
  24. P

    Prove 2-dimensional Riemann manifold is conformally flat

    Homework Statement Establish the theorem that any 2-dimensional Riemann manifold is conformally flat in the case of a metric of signature 0. Hint: Use null curves as coordinate curves, that is, change to new coordinate curves \lambda = \lambda(x0, x1), \nu = \nu(x0, x1) satisfying...
  25. Z

    Riemann Hypothesis equivalence

    let be the function \sum_{\rho} (\rho )^{-1} =Z and let be the sum S= \sum_{\gamma}\frac{1}{1/4+ \gamma ^{2}} here 'gamma' runs over the imaginary part of the Riemann Zeros then is the Riemann Hypothesis equivalent to the assertion that S=2Z ??
  26. M

    Trivial zeros in the Riemann Zeta function

    Hello, I have read in many articles that the trivial zeros of the Riemann zeta function are only the negative even integers (-2, -4, -6, -8, -10, ...). The reason why these are the only ones is that when substituting them in the functional equation, the function is 0 because...
  27. M

    Comp Sci Need fortran help Trapazoid riemann sum

    Alright, I cannot seem to get this subroutine to return the correct sums for the trapezoidal rule... Where do I need to fix? SUBROUTINE atrap(i) USE space_data IMPLICIT NONE INTEGER :: i, j REAL :: f_b1, f_b2, f_x1, f_x2, trap_area REAL :: delta_x trap_area = 0 f_b1 = lower f_b2...
  28. A

    Question related to Riemann sums, sups, and infs of bounded functions

    Can someone give me an example of a bounded function f defined on a closed interval [a,b] such that f does not attain its sup (or inf) on this interval? Obviously, f cannot be continuous, but for whatever reason (stupidity? lack of imagination?) I can't think of an example of a discontinuous...
  29. R

    Programming details on the computation of the Riemann zeta function using Aribas

    (1) Let s be a complex number like s = a + b i, then we define \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} Our aim: to compute ζ(\frac{1}{2}+14.1347 i) with the help of the programming language Aribas (2) Web Links Aribas...
  30. 1

    Riemann Sums and Integrals, feel lost without actual functions

    Homework Statement At my old university, Calculus was taught much differently than it is where I am now. My old school focused on numerical things, which this school focuses much more on pictures, abstract, etc. and it's very difficult for me. At my old school, we were given a shape...
  31. J

    Contraction of the Riemann Tensor with the Weak Field Metric

    I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found: \Gamma^0_{00}=\phi_{,0}...
  32. J

    Contraction of the Riemann Tensor with the Weak Field Metric

    I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found...
  33. Q

    Riemann Curvature: Exploring 3D Curves

    Riemann Curvature? i was watching this documentary that mentioned that riemann came up with a method to deduce whether we were on a curved surface, or on a flat surface, without leaving the surface to make the deduction. for example, for a curved 2d surface, we know it is as such as we can...
  34. O

    Riemann Sum from Indefinite Integral

    Homework Statement Consider the integral, \int _3 ^7 (\frac{3}{x} + 2) dx a) Find the Riemann Sum for this integral using right endpoints and n=4. b) Find the Riemann Sum for this integral using left endpoints and n=4. Homework Equations The sum, \sum^{n = 4} (\frac{3}{x} + 2) The graph...
  35. P

    Understanding Complex Func., Laplace Transforms & Cauchy Riemann

    I am reading a chapter on Complex Functions, Laplace Transforms & Cauchy Riemann (as part of Control theory) And I don't understand how they arrive at a particular part. [ I tried to type it out in tex, but it takes way too much time so uploaded a screenshot to flickr]...
  36. C

    Infinite riemann sums discrepancy

    Hello. I have to solve some integrals using both the standard theorem of calculus and infinite Riemann sums. \int_{1}^{7} (x^2-4x+2) dx = \lim_{n \to \infty } \sum f(x_i)\Delta x_i = \lim_{n \to \infty } \sum (x_i^2 - 4x_i + 2)6/n Evaluating the definite integral results in an answer of 30...
  37. M

    A function f that is not Riemann integrable but |f| is Riemann integrable?

    I was just going over Riemann integrability and how to prove it, and was just wondering is it possible to have a function f that is not Riemann integrable but |f| is Riemann integrable? Say on an interval [0,1] for example. (as that is what most examples I have done are on so easiest for me to...
  38. S

    Covariant derivative of riemann tensor

    what would Rabcd;e look like in terms of it's christoffels? or Rab;c
  39. D

    Riemann Zeta function of even numbers

    Given that \zeta (2n)=\frac{{\pi}^{2n}}{m} Then how do you find m with respect to n where n is a natural number. For n=1, m=6 n=2, m=90 n=3, m=945 n=4, m=9450 n=5, m=93555 n=6, m=\frac{638512875}{691} n=7, m=\frac{18243225}{2} n=8, m=\frac{325641566250}{3617} n=9...
  40. G

    A property of a riemann stieltjes integral

    Hi! While studying a text " A First Course in Real Analysis" by protter, I've been asked to prove a property of riemann stieltjes integral. The propery is as follows ; Suppose a<c<b. Assume that not both f and g are discontinuous at c. If \intfdg from a to c and \intfdg ffrom c to b exist...
  41. W

    Simple-Connectedness in Complex Plane: Def. in Terms of Riemann Sphere.

    Hello, There is a definition of simple-connectedness for a region R of the complex plane C that states that a region R is simply-connected in C if the complement of the region in the Riemann Sphere is connected. I don't know if I'm missing something; I guess we are actually consider...
  42. G

    Additivity of riemann stieltjes integral

    Homework Statement Hi! I'm studying line integral in vector calculus and I've faced a difficulty related with the additivity of line integral. I really hope to get an answer for my question through this site. (2.17) Theorem : If \intfd\phi from a to b exists and a<c<b, then \intfd\phi...
  43. J

    I really need some help with these Riemann sum problems

    Homework Statement 1. Express as a sum of riemann and write the integral to express the area of the trapezoid with vertex (0,0) , (1,3) , (3,3) , (5,0). 2. find the intersection points limited by these equations y = xsquare -3x and y = -2x +3 = 0 3. the trapezoid with vertex...
  44. QuarkCharmer

    Find Area Under Curve y=x^3 from 0 to 1: Riemann Sum Limit

    Homework Statement a.) Use definition 2 to find an expression for the area under the curve y=x^3 from 0 to 1 as a limit. b.)Evaluate the (above) limit using the sum of the cubes of the n integers. Homework Equations (\frac{n(n+1)}{2})^{2} The Attempt at a Solution For part a.) I wrote my...
  45. F

    Infinite series (i think it's riemann)

    Homework Statement \sum_{1}^{inf} k^2/(n^3+k^2) The Attempt at a Solution I think it's Riemann but i cannot find a suitable function to integrate.
  46. D

    Disproof of Riemann Hypothesis

    I think I have managed to disprove the Riemann Hypothesis. Hope I have not made a calculation error.
  47. 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?
  48. D

    What is x in the Riemann Hypothesis?

    Th Riemann Hypothesis states that every 0 lies on x=0.5. What is x here?
  49. 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...
  50. 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...
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