Riemann Definition and 618 Threads

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. A

    I Infinite Series - Divergence of 1/n question

    I understand mathematically several ways to test whether an infinite series converges or diverges. However, I came across one particular equation that is stumping me, ## \sum_{n=1}^{\infty} 1/n ##. I understand how to mathematically apply series tests to show it diverges. But intuitively, I...
  2. hongseok

    B Can Riemann integrable be defined using the epsilon delta non method?

    Is it correct to define Riemann integrability as follows: 'For any ϵ>0, there exists a δ>0 such that if the maximum interval length of the partition is less than δ, then the difference between the upper and lower Riemann sums is less than or equal to ϵ'? I wanted to define Riemann integrability...
  3. V

    B Hopf fibration, Bloch sphere and quantum rotations - interactive site

    There is a very interesting topic in mathematics called the Hopf fibration. In 1931, Heinz Hopf published his work on the construction he discovered in topology, which in history was called "Hopf fibration". The essence of this design was based on the geometric designs of William Kingdon...
  4. ergospherical

    I Null energy condition constrains the metric

    Another GR question... in the thick of revision season. I would appreciate a sketch of how to approach the problem. You basically are given a metric, involving a positive function ##A(z)##, $$g = A(z)^2(-dt^2 + dx^2 + dy^2) + dz^2$$The game is to figure out somehow that the null-energy...
  5. L

    I Riemann integrability and uniform convergence

    Was reading the Reimann integrals chapter of Understanding Analysis by Stephen Abbott and got stuck on exercise 7.2.5. In the solutions they went from having |f-f_n|<epsilon/3(a-b) to having |M_k-N_k|<epsilon/3(a-b), but I’m confused how did they do this. We know that fn uniformly converges to...
  6. AndreasC

    I What is Riemann's approach to classifying 2d surfaces?

    I was reading Bernhardt Riemann's old foundational text on abelian functions, and I found a part that really confused me. What he is trying to do is set up an invariant to classify 2d surfaces as simply connected, multiply connected, etc via some kind of "connectivity number". From the text, I...
  7. D

    Will the Riemann hypothesis be solved by 2100?

    What do u think?
  8. PeaceMartian

    I What are the Zeta Function and the Riemann Hypothesis?

    What is the zeta function and the Riemann hypothesis.
  9. MrFlanders

    A GW Binary Merger: Riemann Tensor in Source & TT-Gauge

    In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...
  10. crememars

    Finding a definite integral from the Riemann sum

    Hi! I am having trouble finalizing this problem. The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n. Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n]. I can guess that the function is...
  11. crememars

    Identifying variables from Riemann sum limits

    Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...
  12. S

    I Geometry of series terms of the Riemann Zeta Function

    This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i## The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...
  13. S

    I Reconciling 2 expressions for Riemann curvature tensor

    I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in the notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ An index-based expression is also given in (eq...
  14. H

    I Original definition of Riemann Integral and Darboux Sums

    Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows: $$ S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$ A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a...
  15. Eclair_de_XII

    I Relating integration of forms to Riemann integration

    Partition each closed interval ##[a_i,b_i]## in the Cartesian product, ##A##. Denote the partition for the i-th closed interval as ##\{x_i^1,\ldots,x_i^{k_i}\}##. The Cartesian product of the partitions forms a partition of ##A## (think: a lattice of points that coincide with the points of each...
  16. G

    I Calculate Contraction 2nd & 4th Indices Riemann Tensor

    How to calculate the contraction of second and fourth indices of Riemann tensor?I can only deal with other indices.Thank you！
  17. MevsEinstein

    B Does this help solve the Riemann Hypothesis?

    Hello PF! If ##\Re (s)## is the real part of ##s## and ##\Im (s)## is the imaginary part, then t is very easy to prove that $$\zeta (s) = \zeta ( \Re (s) ) \zeta ( \Im (s)i) - \displaystyle\sum_{n=1}^\infty \frac{1}{n^{\Re (s)}} [\displaystyle\sum_{k \in S, \mathbb{Z} \S = n}...
  18. WMDhamnekar

    MHB How to prove this corollary in Line Integral using Riemann integral

    . Let C be a smooth curve with arc length L, and suppose that f(x, y) = P(x, y)i +Q(x, y)j is a vector field such that $|| f|(x,y) || \leq M $ for all (x,y) on C. Show that $\left\vert\displaystyle\int_C f \cdot dr \right\vert \leq ML $ Hint: Recall that $\left\vert\displaystyle\int_a^b g(x)...
  19. Graham87

    Numerical Analysis - Richardson Extrapolation on Riemann Sum

    I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?). For example sin(x) for [0,1], n with trapezoidal rule = 0.420735... With my own...
  20. F

    Insights The History and Importance of the Riemann Hypothesis

    Continue reading...
  21. physicsuniverse02

    Does anyone know which are Ricci and Riemann Tensors of FRW metric?

    I just need to compare my results of the Ricci and Riemann Tensors of FRW metric, but only considering the spatial coordinates.
  22. abhinavabhatt

    A Anti-self Dual Part (2,2) Riemann Curvature Tensor

    i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.
  23. F

    Insights The Extended Riemann Hypothesis and Ramanujan’s Sum

    Continue reading...
  24. BiGyElLoWhAt

    I A couple questions about the Riemann Tensor, definition and convention

    According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...
  25. Math Amateur

    MHB How can we prove the inequality for the supremum and infimum of f*g and f*g?

    I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ... I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof...
  26. Math Amateur

    MHB Understanding Riemann Integrable Functions: Interpreting D&K Pages 427-428

    I am not sure of the overall purpose of the concepts developed below regarding Riemann integrable functions with compact support ... nor am I sure of the details ... so I am sketching out the meaning as I understand it in 2 dimensions and depicting the relevant entities in diagrams ... I am...
  27. ergospherical

    I Finding Riemann Components: Packages & Solutions

    I'm working on a problem involving some hypothetical spacetimes (i.e. no tables/data-sheets available) and need to calculate a bunch of ##R_{\mu \nu \rho \sigma}## and ##R_{\mu \nu}## values, as well as ##R##. The metrics contain some arbitrary functions ##f(x^i)## of the spatial co-ordinates...
  28. Eclair_de_XII

    B Riemann integrability of functions with countably infinitely many dis-

    We show that there is a partition s.t. the upper sum and the lower sum of ##f## w.r.t. this partition converge onto one another. Let ##\epsilon>0##. Define a sequence of functions ##g_n:[a,b]\setminus(\{a_n\}_{n\in\mathbb{N}}\cup\{y_0\})## s.t. ##g_n(x)=|f(x)-f(a_n)|##. Suppose there is a...
  29. R

    Cauchy Riemann complex function real and imaginary parts

    Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...
  30. M

    I Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates

    Hello everyone, in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...
  31. Narasoma

    I Riemann Curvature: Understanding Parallel Transport on 1D Rings

    Everyone who is currently studying GR must be familiar with this picture. We find Riemann curvature by paraller transport a "test vector" around and see whether the vector changes its direction. My question. How does it work with one dimensional Ring? A geomteric ring is intuitively curved but...
  32. snypehype46

    Riemann curvature coefficients using Cartan structure equation

    To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation: $$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$ and using the tetrad formalism to compute the coefficients of the...
  33. C

    A Induced Metric for Riemann Hypersurface in Euclidean Signature

    We know in Lorentzian signature spacetime, in the case of timelike or spacelike hypersurfaces ##\Sigma## with \begin{align} n^\alpha n_\alpha=\epsilon=\pm1 \end{align} where ##\epsilon=1## for timelike and ##-1## for spacelike. We can define a tensor ## h_{\alpha\beta}## on ##\Sigma## by...
  34. S

    Riemann Sum to find the time to fill a container

    (a) I imagine there are several rectangles to represent the area under graph of p vs t then I try to make equation for the total area. Since the question asks about time when the container holds 22 fewer liters than it does at time t = 9, I think the total area of rectangles starting from t = b...
  35. evinda

    MHB How Do Riemann Integrals Handle Function Splits and Summations?

    Hello! (Wave) I am looking at the Riemann integral and I have two questions. Theorem: Let $f: [a,b] \to \mathbb{R}$ bounded and $c \in (a,b)$. Then $f$ is integrable in $[a,b]$ iff it is integrable in $[a,c]$ and in $[c,b]$. In this case we have $\int_a^b f=\int_a^c f + \int_c^b f$. At the...
  36. stevendaryl

    Insights Computing the Riemann Zeta Function Using Fourier Series

    Continue reading...
  37. BWV

    I Lebesgue vs Riemann on the rationals

    So Riemann integrals on ℚ can be <> 0 but Lebesgue integrals on ℚ all have measure zero?
  38. Killtech

    I Is There a Distinction Between Riemann Metrics in Physics?

    In terms of diff geo it seems like an obvious fact, that a manifold can be equipped with quite a variety of different Riemann metrics. But when it comes to physics (relativity theory in particular) it seems there is a very specific metric singled out. Now i do not entirely understand the...
  39. Leo Liu

    Midpoint Riemann sum approximation

    Can someone please explain why the formula for midpoint approximation looks like the equation above instead of something like $$M_n=(f(\frac{x_0+x_1}2)+f(\frac{x_1+x_2}2)+\cdots+f(\frac{x_{n-1}+x_n}2))\frac{b-a}n$$? Thanks in advance!
  40. SchroedingersLion

    A Compute Lebesgue integral as (improper) Riemann integral

    Hello everyone, in a solution to my measure theory assignment, I have seen the equation $$ \int_{\mathbb{R}}^{} \frac {1}{|x|}\, d\lambda(x)=\infty $$ with ##\lambda## as the 1⁻dim Lebesgue measure. I was wondering how that integral was evaluated as we had never proven any theorem that states...
  41. E

    I Riemann Curvature Tensor on 2D Sphere: Surprising Results

    I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped. I was surprised at this, because it implies that the curvature...
  42. S

    I Category Theory and the Riemann Hypothesis

    YouTube has been suggesting videos about category theory of late, and I have spent some time skimming through them, without really understanding where it's all going. A question came to mind, namely: It seems reasonably conceivable that group theory could perhaps supply a vital key to the...
  43. R

    I A thought about the Riemann hypothesis

    This is the Riemann Zeta function ##Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}##. It equals 0 only at the negative integers on the real axis and numbers of form ##1/2+x i##. The series can be expanded to this: $$\sum_{n=1}^{\infty} \frac{1}{n^s} = \sum_{n=1}^{\infty} \frac{1}{n^{1/2 + xi}} =...
  44. G

    MHB Interval of the Riemann integral value

    Hello everyone, I have to find an interval of this Riemann integral. Does anybody know the easiest way how to do it? I think we need to do something with denominator, enlarge it somehow. My another guess is the integral is always larger than 0 (A=0) because the whole function is still larger...
  45. jk22

    I Riemann Tensor, Stoke's Theorem & Winding Number

    I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve. If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?
  46. B

    Is the Function Analytic? Testing the Cauchy Riemann Equations

    I tested the first function with the Cauchy Riemann equations and it seemed to fail that test, so I don't believe that function is analytic. However, I'm really not sure how to show that it is or is not analytic using the definition of the complex derivative.
  47. J

    A Riemann Tensor Formula in Terms of Metric & Derivatives

    Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula. Thanks.
  48. JD_PM

    I Computing Riemann Tensor: 18 Predicted Non-Trivial Terms

    I want to compute the Riemann Tensor of the following metric $$ds^2 = dr^2+(r^2+b^2)d \theta^2 +(r^2+b^2)\sin^2 \theta d \phi^2 -dt^2$$ Before going through it I'd like to try to predict how many non-trivial components we'd expect to get, based on the Riemann tensor basic rule: It is...
  49. V

    Python Calculating Riemann Sums on Python w/ Numpy

    import numpy as np def num_int(f,a,b,n): dx=(b-a)/n x=np.arange(a,b,step=dx) y=f(x) return y.sum()*dx def rational_func(x): return 1/(1+x**2) print(num_int(rational_func,2,5,10)) Here is my code for the left endpoint, I know this code works because I compared it to an...
  50. K

    I Riemann integrability with a discontinuity

    So, I know that a function is integrable on an interval [a,b] if ##U(f,P_n)-L(f,P_n)<\epsilon ## So I find ##U(f,P_n## and ##L(f,P_n## ##L(f,P_n)=5(3-\frac{1}{n}-0)+5(3+\frac{1}{n}-(3-\frac{1}{n}))+7(4-(3+\frac{1}{n}))=22-\frac{2}{n} ##...
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