What is Riemann: Definition and 615 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

    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...
  2. 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...
  3. L

    Understanding the Riemann Tensor and its Properties in Differential Geometry

    i need to show that R_{abc}{}^{e} g_{ed} + R_{abd}{}^{e} g_{ce}=(\nabla_a \nabla_b - \nabla_b \nabla_a) g_{cd} = 0 ok well i know that R_{abc}{}^{d} \omega_d=(\nabla_a \nabla_b - \nabla_b \nabla_a) \omega_c so i reckon that R_{abc}{}^{e} g_{ed} = (\nabla_a \nabla_b - \nabla_b \nabla_a)...
  4. S

    Lebesgue integral over the Riemann integral

    You always see in books that one advantage of the Lebesgue integral over the Riemann integral is that a sequence of continuous functions f_n does not have to converge unifomly to a function f to have: integral of the limit of the sequence = the limit of the integrals of functions in the...
  5. N

    Navigating Research on Riemann Hypothesis

    Let me start off by saying I have not yet had a formal course in Number Thoery and have only read briefly on the subject...hence the question: How close (in terms that would be understood by someone in my position) is the math community to proving the Riemann Hypothesis? I'm assuming there...
  6. L

    How many independent components does the Riemann curvature tensor have?

    (i) show that R_{abcd}+R_{cdab} (ii) In n dimensions the Riemann tensor has n^4 components. However, on account of the symmetries R_{abc}^d=-R_{bac}^d R_{[abc]}^d=0 R_{abcd}+-R_{abdc} not all of these components are independent. Show that the number of independent components is...
  7. P

    Equation with Riemann curvature tensor

    Can anyone prove the following formula: R_{abf}^{\phantom{abf}e} \Gamma_{cd}^f = R_{abc}^{\phantom{abc}f} \Gamma_{fd}^e + R_{abd}^{\phantom{abd}f} \Gamma_{cf}^e I found it in "General Relativity" by Wald (in slightly different notation).
  8. 2

    Can someone explain zeros and zeta function for Riemann Hypothesis? (Yr13)

    Hi, I'm Yr 13 and just wanted to do some further reading/exploring. So i understand that the zeta function is something to do with summing up like this: 1/ (1^s) + 1/(2^s) etc etc Now, I just want to know what are non-trivial zeros and trivial zeros? I just want to be able to understand this...
  9. N

    Query on Cauchy Riemann Condition question

    Dear Friends and Colleagues! I have this practise question:- Show that z(sin(z))(cos(z)) statisfies the Cauchy-Riemann Conditions for analyticity for all values of z. Does 1/[z(sin(z))(cos(z))] statisify simiar conditions? Calculate the derivative of 1/[z(sin(z))(cos(z))] at z=0, +...
  10. D

    Legendre and Riemann: A Conjecture Comparison

    I recall reading somewhere that Legendre's conjecture implies the Riemann Hypothesis. But the Wiki article suggests that Legendre imposes lighter bounds on the density of primes than does RH, so I would think the other way around, if anything. Thanks for any enlightenment.
  11. F

    Riemann integral of arcsinh (have the answer, want an explanation)

    Homework Statement Given the following sum, turn it into an integral: \lim_{n \to \infty}\Sigma^n_{k=1}\dfrac{1}{n\sqrt{1+(k/n)^2}} Homework Equations The answer says =\int^2_1\dfrac{1}{\sqrt{1+x^2}} The Attempt at a Solution I understand how to get the equation, but why...
  12. I

    Accurate Proof and varification for Riemann Hypothesis

    Accurate Proof verification of Riemann’s Hypothesis Riemann Hypothesis states that \int \frac{1}{ln (x)} has a root at \frac{1}{2} when s=2 The time series expansion of Log function is, [tex] \ln(x) = \frac {[x-1}{[x-2}+ \frac{1){3} \frac{x-3}{x-4} + \frac{1}{5}\frac{x-5}{x-6}+……...
  13. H

    Russian Dolls Matryoshka Approach to Riemann Hypothesis

    Well we know what matryoshka dolls are? Those nested dolls one inside another. I am a mere laymen and amateur that's why I am using descriptive terms instead of math rigor. So what should the approach be: If RH nest Hilbert-Polya conjecture, then what things nest HP conjecture? And ad...
  14. F

    Analysis Riemann Integral problem

    Homework Statement Suppose α(x) increases on [a,b] a≤ x_0 ≤b, α is continuous at x_0, f(x_0) =1 , at all other x in [a,b] f(x)=0. denote ('x knot' as x_0) Prove that f is Riemann Integrable and that ∫fdα=0. Homework Equations Can anyone check my proof or suggest a good method...
  15. Loren Booda

    Riemann proof effect on factoring composites

    To what degree would proving the Riemann hypothesis facilitate the factoring of large composites? In other words, how much would a complete (as opposed to "hit-or-miss") knowledge of primes help to reduce the operations needed to factor large composites?
  16. L

    The Riemann Hypothesis for High School Students

    Hi All, I would like to present what I believe to be a simple way to convey the essence of the Riemann Hypothesis to High School students. I hope you like it, and reply with suggestions for further improvements. Note for teachers: the rationale behind the graphs lays with the geometric...
  17. G

    How to Use Riemann Sums and Integrals to Estimate and Evaluate Functions

    Homework Statement (x, f(x)) (2,1) (3,4) (5,-2) (8,3) (13,6) A) Estimate f '(4). Show work. B) Evaluate the Intergral from 2 to 13 of (3 - 5f '(x))dx. show work C) Use left riemann sum with subintervals indicated bye the data in the table to apporoximate the intergral from 2 to 13 of...
  18. benorin

    Works of Riemann related to Astronomy that I can understand?

    Looking for a connection to astronomy for a history of mathematics report on 19th century mathematicians. I don't do astronomy. Would like to know of any mathematical developments of Riemann that are used in astronomy that I can understand (which expressly excludes any thing with the word tensor...
  19. P

    Riemann Integral Identification from Sum

    Hi There Everyone I am studying undergraduate calculus in first year. My question regards the rules for identifying a limit sum as a Riemann sum and therefore a definite integral. The book we are using says that when choosing \inline \large c_{i} for some f(x) , if \inline \large x_{i -...
  20. F

    What Does 'To Lowest Order' Mean in the Derivation of Riemann Curvature Tensor?

    http://www.mth.uct.ac.za/omei/gr/chap6/frame6.html" is a derivation of the components of the riemann curvature tensor. the problem is that i can't understand the transition between eq97 and eq89 . what does "To lowest order " mean ?
  21. I

    Riemann Integral - little proof help

    Homework Statement Suppose f is integrable for all x in[a,b] and f(x)>C ( C is some constant), Must show that 1/f is also integrable. Homework Equations f is integrable implies Upf-Lpf<\epsilon for some partition in [a,b] The Attempt at a Solution Therefore, I must come up...
  22. M

    Monotonicity of the riemann integral

    Hi everyone, For integrable f,g:\left[a,b\right]\rightarrow\mathbb{R} with f(x)\leq g(x) for all x\in\left[a,b\right], it's a basic property of the riemann integral that \[\int_a^b f(x)\,dx \leq \int_a^b g(x)\,dx\] My question is whether the strict version of this inequality holds...
  23. D

    Real Analysis: Riemann Measurable

    Homework Statement Assume S contained in R2 is bounded. Prove that if S is Riemann measurable, then so are its interior and closure 2. The attempt at a solution Proof: If S is Riemann measurable, its boundary is a zero set. Since the boundary of each open U in the int(S) is part of...
  24. M

    (Riemann) Integrability under composition of functions

    Homework Statement I've been looking at how integrable functions behave under composition, and I know that if f and g are integrable, f(g(x)) is not necessarily integrable, but it -is- necessarily integrable if f is continuous, regardless of whether g is. So I was wondering, what about if g is...
  25. D

    Evaluating Riemann Sum f(x,y) - 4x^2+y

    The problem says: evaluate 4x^2+y by breaking into four congruent subrectangles and evaluating at the midpoints, 1=<x<=5 0=<y<=2 When i setup the rectangles these are my coordinates: (1,1/2),(1,3/2),(3,1/2),(3,3/2) and delta A = 2 My answer comes out to be 168...
  26. D

    Proving Riemann Integral: Non-Negative f(x)=0 $\rightarrow \int^{b}_{a}f=0$

    Homework Statement Suppose f(x):[a,b]\rightarrow\Re is bounded, non-negative and f(x)=0. Prove that \int^{b}_{a}f=0. Homework Equations The Attempt at a Solution I am trying to use the idea that lower sums are zero, and show that the upper sums go to zero as the norm of the...
  27. D

    Riemann geometry and hydrostatic

    I read in Elie CARTAN book : "la Géométrie des espace de Riemann" that when R = cte, you can compare space-time to hydrostatic description of a liquid. Is it true ?
  28. G

    Riemann Geometry: Where is the Flaw in My Thinking?

    One of the axioms of Riemann's geometry holds that there are no parallel lines and that any two lines meet. Since Riemann's geometry fits for that of a sphere, any two great circles of the sphere should intersect. However, if we were to take 2 longitudinal lines, then it is possible that these...
  29. L

    Integrals (Riemann-Darboux, Riemann, Lebesgue,etc)

    Homework Statement My book presents the Riemann-Darboux integral. It has a small supplemental section on the Riemann integral. Then a later section on the Riemann-Stieljes integral. Then a later chapter on the Lebesgue integral. A supplementary text that I have has a section on...
  30. M

    Calculating the Area of a Strip Using a Riemann Sum

    Homework Statement Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown in the figure below where the upper line is defined by 6x + y = 12 and the other line is defined by y=x^2-4. The figure, which I can't get on here, is just the...
  31. Y

    Riemann sum question, with picture

    Homework Statement http://img4.imageshack.us/img4/898/integerqj5.jpg Homework Equations The Attempt at a Solution It does appear to be a Riemann sum, I figured the 1/n is probably the width of the intervals and the sum in brackets is related to the sums of the heights of the rectangles. But my...
  32. A

    Is the characteristic function of the irrationals Riemann integrable on [a,b]?

    The characteristic function of the RATIONALS is a well-known example of a bounded function that is not Riemann integrable. But is the characteristic function of the IRRATIONALS (that is, the function that is 1 at every irrational number and 0 at every rational number) Riemann integrable on an...
  33. T

    Grander than the Riemann Hypothesis

    ...But it may not exist yet. Has any mathematician thought about producing a formula or function which spits out all the prime numbers? i.e 1->2, 2->3, 3->3, 4->5, 5->7, 6->11 etc. Any attempts been made? What the closest that people have thought?
  34. L

    Q on Riemann Domains: Is P Injective on U?

    If P is the projection map from a Riemann domain M \rightarrow C^n, and U is a connected subset of M with P(U)=B, where B is a ball in C^n, then is P injective on U, so it's a homeomorphism on U? P is locally a homeomorphism by definition. It would be related to B being simply connected...
  35. science_rules

    Area approximation and (riemann?) sums

    Homework Statement I am a first-year physics student learning calculus. my question is about the approximation of the area of a region bounded by y = 0. Homework Equations Use rectangles (four of them) to approximate the area of the region bounded by y = 5/x (already did this one), and y =...
  36. J

    Mathematica Help with Right and Left Riemann Sums

    Homework Statement I am given a left riemann sum program module in Mathematica and need to convert it into the right riemann sum. The program takes values for x and f/x and the partition and graphs on a certain interval provided. leftRiemannGraph[f_, a_, b_, n_] := Module[{expr}, expr[1]...
  37. S

    Convergence of Riemann Sums for Limit of Series

    Homework Statement I need to find the following: \lim_{n\rightarrow\infty}\left(\frac{1^2+2^2+3^2+...+n^2}{n^3} \right) Homework Equations The Attempt at a Solution I know I could do the sum of the series to find the result but I would like to use Riemann sums. I think I have to start by...
  38. W

    Mathematicians' Original Work: Riemann & Taylor Theorems

    Can anyone provide me with a website that has copies of the original works of Riemann, Taylor, famous mathematicians. I am looking for papers on proved theorems.
  39. T

    Complex Zeros in Riemann Zeta Function: Is it Possible?

    in the Riemann Zeta function, is it possible to have two complex zeros off the critical strip that both have the same imaginary part?
  40. B

    Calculating Riemann Zeta function

    Homework Statement Using method of Euler, calculate \zeta(4), the Riemann Zeta function of 4th order. Homework Equations \zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s} Finding \zeta(2): \zeta(2)=\sum_{n=1}^\infty...
  41. K

    Proof on a property of Riemann integrals

    Homework Statement Let f be continuous on [a,b] and suppose that f(x) \geq 0 for all x Є [a,b] Prove that if there exists a point c Є [a,b] such that f(c) > 0 , then \int_{a}^{b} f > 0 Homework Equations The Attempt at a Solution Using my books notation, Suppose P =...
  42. K

    Need some help with Riemann Sums.

    Need some urgent help with Riemann Sums. Homework Statement PART A: In all of this question, let I = \int ^{2}_{-2} f(x)dx where f(x) = -2x + 1 Evaluate I. PART B: Use the defintion of the definite integral to evaluate I. i.e Riemann Sum. Homework Equations The...
  43. N

    Comp Sci Calculating pi using Riemann Sum and Fortran77

    My assignment: Solve for pi using a Riemann Sum with n= 40,000,000. The function is the antiderivate of 4/(1+x^2) dx. The bounds are from 0 to 1. Solving this gives you pi. Anyone know how to do this? Preferably with fortran77?
  44. U

    Solving Riemann Sums for \int_0^{2\pi} x^{2}sin(x)\,dx | Homework Help

    Homework Statement Express the integral as a limit of Riemann sums. Do not evaluate the limit. Homework Equations \int_0^{2\pi} x^{2}sin(x)\,dx The Attempt at a Solution I really don't know where to start...any help getting me started would be highly appreciated!
  45. N

    Examples where it's Riemann integrable but no derivative exists at pts

    What is an example where it's Riemann integrable int(f(t),t,a,x) but no derivative exists at certain pts?
  46. N

    Example of Lebesgue Integral but not Riemann Integrable

    What's Example of Lebesgue Integrable function which is not Riemann Integrable?
  47. N

    Is the given function Riemann integrable?

    Here is the classic Dirichlet function: Let, for x ∈ [0, 1], f (x) =1 /q if x = p /q, p,q in Z or 0 if x is irrational. Show that f (x) is Riemann integrable and give the value of the integral. Is this actually true?
  48. S

    Real Analysis: Finding the Limit of a Riemann Sum

    Homework Statement Find the limit, as n -> infinity, of \sum_{k=1}^nk3/n4 Homework Equations Riemann sum: S(f, \pi, \sigma) = \sum_{k=1}^nf(\xi)(xk - xk-1) The Attempt at a Solution My guess is that I should try to put this sum in terms of a Riemann sum, and then taking n -> infinity will...
  49. E

    Riemann zeta function

    \zeta (s)= \frac{1}{(1-2^{1-s})} \sum_{n=0}^{\infty} \frac {1}{(2^{n+1})} \sum_{k=0}^{n}(-1)^k{n \choose k}(k+1)^{-s} Is the main problem with trying to prove the hypothesis algebraically boil down to the fact that s is an exponent to a "k" term? Would a derivation of the function that had...
  50. R

    Writing a Riemann Sum w/out Sumation signs

    Homework Statement In this problem you will calculate ∫0,4 ( [(x^2)/4] − 7) dx by using the definition ∫a,b f(x) dx = lim (n → ∞) [(n ∑ i=1) (f(xi) ∆x)] The summation inside the brackets is Rn which is the Riemann sum where the sample points are chosen to be the right-hand endpoints...
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