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. P

    Hello,I am doing a project on Riemann, but. .

    Hello, I am doing a project on Riemann but I don't understand some of his contributions. For instance, I do not understand what Riemann zeta-function measures or what it does, what the Riemann hypothesis states, or what the difference is between Riemannian geometry and hilbert geometry. Any...
  2. 4

    Convergence and Divergence of Riemann Sum Series with Varying p

    For what values of p>0 does the series Riemann Sum [n=1 to infinity] 1/ [n(ln n) (ln(ln n))^p] converge and for what values does it diverge? How do i do this question? Would somebody please kindly show me the steps? Do i use the intergral test?
  3. W

    Why is the Cauchy Riemann relation important for complex differentiability?

    The cauchy Riemann relations can be written: \frac{\partial f}{\partial \bar{z}}=0 Is there an 'easy to see reason' why a function should not depend on the independent variable [itex]\bar{z}[/tex] to be differentiable?
  4. C

    Riemann Sum Question: Is my Answer Correct?

    Hello, just going through some Riemann sum problems before I hit integrals and I am like 99% sure that this answer from my text is wrong but I want to make sure. It's not really an important question so if you have better things to do, help the next guy :) But checking this over would be...
  5. L

    Riemann zeta function generalization

    "Riemann zeta function"...generalization.. Hello my question is if we define the "generalized" Riemann zeta function: \zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s} which is equal to the usual "Riemann zeta function" if we set h=1, x=0 ,then my question is if we can extend the definition...
  6. S

    Riemann Prime Counting Function

    Can anyone tell me if Riemann's Prime Counting function can be solved by residue integration? Here it is: J(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{ln(\zeta(s))x^s}{s}ds which has the solution: J(x)=li(x)-\sum_{\rho}li(x^\rho)-ln(2)+ \int_x^{\infty}\frac{dt}{t(t^2-1)ln(t)} I...
  7. W

    What is the definition of S in the proof for convergence of Riemann sum?

    In a book of mine, the author proceeds to the proof that a Riemann sum in a interval [a,b] must converge by proving that for S_m and S_n (m>n) where the span of the subdivisions is suffiencienly small, then |S_m - S_n)| < e(b-a) Where e can assume infinitly small values in dependence of...
  8. benorin

    An elementary problem equivalent to the Riemann hypothesis

    Let H_{n}=\sum_{k=1}^{n}\frac{1}{k} be the nth harmonic number, then the Riemann hypothesis is equivalent to proving that for each n\geq 1, \sum_{d|n}d\leq H_{n}+\mbox{exp}(H_{n})\log H_{n} where equality holds iff n=1. The paper that this came from is here: An Elementary Problem...
  9. quasar987

    Riemann sum of derivative (something like that)

    Hi, maybe someone can help. When I think about it, I'm pretty sure that the following is true: Let c be a curve parametrized by t\in [a,b], let \sigma = \{t_0,...,t_N\} be a partition of [a,b] and \delta_{\sigma}=\max_{0\leq k \leq N-1}(t_{k+1}-t_k). Also define \Delta t_k=t_{k+1}-t_k Then...
  10. S

    Evaluating Exact Area using Riemann Sum and Limit on y = 2x^2 + 4 from 0 to 1

    Use the Riemann sum and a limit to evaluate the exact area under the graph of y = 2x^2 + 4 on [0, 1] I know how to do this normally but now they ask to do it w/ a limit and I'm not sure how. (LaTex corrected by HallsofIvy.)
  11. M

    Prove Riemann Integrability of Function f on [0,1]

    Given a function f: [0,1] \to \mathbb{R}. Suppose f(x) = 0 if x is irrational and f(x) = 1/q if x = p/q, where p and q are relatively prime. Prove that f is Riemann integrable.
  12. E

    Exploring the Role of Riemann Tensor in General Relativity

    A doubt..why einstein Chose Riemann Tensor for GR?..i know its covariant derivative is zero and all that..but Why Riemann tensor?...was not other tensor avaliable or simpler than that?..i studied that and found that for Geodesic deviation ( i didn,t understand that concept..sorry) the Riemann...
  13. S

    What are the applications of the Riemann Mapping Theorem?

    so i know what it is (i think lol) ... but what are its applications?
  14. N

    Is the Calculation of Area Using Riemann Sum Correct?

    I was hoping someone could check my answer? Use the limit of a Riemann sum to find the area of the region bounded by the graphs of y=2x^3+1, y=0, x=0, x=2. Area=2
  15. B

    Exploring the Riemann Hypothesis and Its Implications for Prime Numbers

    let \zeta(z)=\sum_{n \in \mathbb{N}} n^{-z} ~ {{a+ib}}>1 then, \zeta(z)=0 iff z=-2n where n is a natural number. pi(x)=\int_0^\infty\frac{dx}{\xS[x+1]} gamma(x+) where S[x+1]= \sum_{n \in \mathbb{N}} n^-{x+1} I have discovered that pi(x)=\int_a^b\frac{dx}/logx = 1/log b+ 2/log b...
  16. P

    Understanding Zeros of the Riemann Zeta Function

    They claim that the trivial 0s (zeta(z)=0) occur when z is a negative odd integer (with no imaginary component). But it seems obviously wrong. Take z=-2 zeta(-2)=1+1/(2^-2)+1/(3^-2)... =1+4+9... Obviously this series will not equal 0. Where have I gone wrong? Have I misunderstood...
  17. P

    Calculating the Riemann Sum of (cos1)^x

    hi, is it possible to find the riemann sum of (cos1)^x? it looks divergent to me can someome please help me... even if it is convergent, i don't know how to find the sum of a trigonometric function
  18. T

    Proving Riemann Integrability of Bounded Function w/ Countable Discontinuities

    Is it possible to show by induction that f:[a,b]->R, a bounded function, is Riemann integrable if f has a countable number of discontinuities? I'm told this is usually done with Lebesgue integrals, but I don't see why an inductive proof of this using Riemann/Darboux integrals can't work.
  19. T

    Proving that a function is Riemann integrable

    "Let (r_n) be any list of rational numbers in [0,1]. Let (a_n) be a sequence such that 1>a_1>a_2>...>a_n>... converging to 0. Let f be defined such that f=0 if x is irrational =a_n if x is equal to r_x Prove that f is Riemann integrable." We are doing integrals from Darboux' approach, so no...
  20. T

    Prove that this function is Riemann integrable

    Let f:[a,b]->R be bounded. Further, let it be continuous on [a,b] except at points a1, a2, ...,an,... such that a1>a2>a3>...>an>...> a where an converges to a. Prove that f is Riemann integrable on [a,b]. It suffices to prove that f is integrable on [a,a1) (I've worked out that part). And...
  21. X

    Proof w/ natural log and Riemann Sum

    Problem states: (A) Use mathematical induction to prove that for x\geq0 and any positive integer n. e^x\geq1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!} (B) Use part (A) to show that e>2.7. (C) Use part (A) to show that \lim_{x\rightarrow\infty} \frac{e^x}{x^k} =...
  22. X

    Area - Riemann Sum/Integration problem

    This problem is in the section of my book called "The Natural Logarithmic Function" so I am guessing that I would have to use a natural log somewhere...but anyways...the problem says: For what values of m do the line y=mx and the curve y=\frac{x}{(x^2+1)} enclose a region? Find the area of the...
  23. B

    Unifying Curvatures with Riemann Tensor

    I always wonder how the definitions of curvatures of curves and surfaces be unified by the Riemann Tensor symbols. For surfaces, I know R_{1,2,1,2} corresponds to the Gaussian curvature of a surface. How come R_{1,1,1,1}=0 and not corresponds to the curvature of a curve in \RE^2 or in \Re^3...
  24. W

    Anybody tried solving Riemann Hypothesis?

    Just wondering if any of the posters seriously tried solving Riemann Hypothesis. And if yes, then what kinds of problems did you run into?
  25. dextercioby

    Find # of Independent Components of Riemann Curvature in D Dimensions

    How do i find the number of independent components of the Riemann curvature tensor in D space-time dimensions. One is given that the Riemann tensor is an (2,2) irreducible rep of GL(4, \mathbb{R}) and obeys Bianchi I R_{[\mu\nu|\rho]\lambda}=0 Been trying this problem for 3 days and...
  26. -Job-

    What is the Riemann Hypothesis and why is it so difficult to solve?

    I know this is one of the famous unsolved problems still hanging around. Could someone give me the "gist" of it, and what the implications are if it is solved one way or the other? I looked it up on Wikipedia but that didn't help me much. Has anyone any idea why it is so hard to solve (i imagine...
  27. A

    Evaluating Riemann Integrals of f(x)=x^k where k>1 is an Integer

    Please Help... Riemann Please Help! To compute the Riemann integral of f:[0,1]->R given f(x)=x^k where k>1 is an integer 1. Let m>2 and define q_m= m^(-1/m) Let P_m be the partition of [0,1] given by P_m=(0< q_m^m < q_m^(m-1)< ...< q_m <1) Explicitly evalute L(f,P_m) and U(f,P_m) 2. Show...
  28. E

    Could Riemann Hypothesis be completely false?

    Heilbronn proved that the Epstein Zeta function did not satisfy RH...but the Zeta function \zeta(s) can be put in a form of an Epstein function but a factor k..let be the functional equation for Epstein functions: \pi^{-s}\Gamma(s)Z_{Q^{-1}}(s)=|Q|^{1/2}\pi^{s-n/2}\Gamma(n/2-s)Z_{Q}(n/2-s)...
  29. benorin

    Fractional Calculus and the Riemann Zeta function

    So it is well-known that for n=2,3,... the following equation holds \zeta(n)=\int_{x_{n}=0}^{1}\int_{x_{n-1}=0}^{1}\cdot\cdot\cdot\int_{x_{1}=0}^{1}\left(\frac{1}{1-\prod_{k=1}^{n}x_{k}}\right)dx_{1}\cdot\cdot\cdot dx_{n-1}dx_{n} My question is how can this relation be extended to...
  30. E

    A proof of Riemann hypothesis (but of course the snobbish journals don,t want

    to publish it because i,m not a famous teacher,mathematician from a snob and pedant univesity of Usa of England...this is the way science improves..only by publishing works from famous mathematician..:mad: :mad: :mad: :mad: :mad: o fcourse if i were Louis de Branges or Alain Connes or other...
  31. L

    Is this Riemann Integrable?

    let f(x) = 1 when x in in [0,1) f(x) = -1/2 when x is in [1,2) f(x) = 1/3 when x is in [2, 3) and so on, in othe words its the sequence (1/n)(-1)^n, whose series obviously converges to log 2. However is f(x) Riemann integrable and equal to this series? If so, how to give an...
  32. Jameson

    How do non-trivial zeros of the Riemann Zeta Function occur?

    I've tried my best to understand the Riemann Zeta Function on my own, but I appeal to the knowledge of you guys to help me understand more. For s >1 , the Riemann Zeta Fuction is defined as: \zeta(s)=\sum_{n=1}^{\infty}n^{-s} I have no problem with this. That series obviously converges...
  33. E

    Are Riemann hypothesis and Goldbach conjecture related?

    this is a question i have i mean are RH and Goldbach conjecture related? i mean in the sense that proving RH would imply Goldbach conjecture and viceversa: RIemann hypothesis: (RH) \zeta(s)=0 then s=1/2+it Goldbach conjecture,let be n a positive integer then: 2n=p1+p2 ...
  34. E

    An strategy to prove Riemann hypothesis

    Let be the Hamitonian of a particle with mass m in the form: H=\frac{-\hbar^{2}}{2m}D^{2}\phi(x)+V(x)\phi(x) then the RH is equivalent to prove that exist a real potential V(x) of the Hamiltonian so that the values E_n H\phi=E_{n}\phi satisfy the equation \zeta(1/2+iE_{n})=0 that is...
  35. M

    Understanding Riemann Roch: A Summary of the Classical Proof by Riemann and Roch

    Summary of the classical proof by Riemann and Roch Let D = p1 + ...+pd be a divisor of distinct points on a compact connected Riemann surface X of genus g, and let L(D) be the space of meromorphic functions on X with at worst simple poles contained in the set {p1,...,pd}. For each point pj...
  36. E

    A question of roots of riemann function

    let be the quotient: Lim_{x->c}\frac{\zeta(1-x)}{\zeta(x)} where x=c is a root of riemann function... then my question is if that limit is equal to exp(ik) with k any real constant...thanks... the limit is wehn x tends to c bieng c a root of riemann constant
  37. W

    Riemann Hypothesis effect on the world

    What would be the effects on the world and to the individual or individuals if the Riemann Hypothesis was solved?
  38. P

    What is the connection between the Riemann hypothesis and prime numbers?

    Hi, When I hear about the Riemann hypothesis, it seems like the first thing I hear about it is its importance to the distribution of prime numbers. However, looking online this seems to be a very difficult thing to explain. I understand that the Riemann Hypothesis asserts that the zeroes of...
  39. S

    Riemann Integrability: Bounded Functions on [a,b]

    hello all well I was working through Riemanns Criterion : let f be a bounded function on the closed interval [a,b]. then f is riemann integrable on [a,b] if and only if , given any epsilon>0, there exist a partition P of [a,b] such that U(f,P)-L(f,P)<epsilon but there is one thing that...
  40. S

    Proving Riemann Integrability of f(x)=x^2

    hello all i just wanted to ask how would one prove that a function is riemann integrable through the definition that the lower integral has to equal the upper integral, an example on the function f(x)=x^2 would be of great help thanxs
  41. S

    Riemann Hypothesis: Explaining \zeta(s) & Diagrams

    hello all after doing a bit of research on the riemann hypothesis I came along this paragraph, in which I don't understand, especially the first sentence , how would one be able to show that? It can be shown that \zeta (s) = 0 when s is a negative even integer. The famous Riemann...
  42. I

    What progress has been made on the Riemann Hypothesis since 2004?

    http://news.uns.purdue.edu/UNS/html4ever/2004/040608.DeBranges.Riemann.html Any news since then? There are links to the papers themselves on the bottom of the page. But I can't understand much, I'm afraid.
  43. Jameson

    What is the Riemann Hypothesis and Why is it Important in Number Theory?

    I have read what MathWorld has to offer on this and I am extremely confused. Could someone please explain this as simply as possible? Or then again maybe MathWorld already did that. Also, why is this function so important? Many thanks, Jameson
  44. C

    Elements of multiple riemann integration

    Hello guys, I am following the chapter about multiple Riemann integrals in Apostol's Mathematical Analysis. Theorem 14.11 says this (I translate from spanish to english): "Let S be a Jordan measurable set. Let the function f be defined and bounded in S. Then f is Riemann integrable if and...
  45. N

    Riemann sum / difference quotient

    How does the difference quotient undo what the Riemann sum does or vice versa. In terms of the two formulas? I would assume that working a difference quotient backwards would be similar to working a Riemann sum forward, but in reality as the operations go this couldn't be further from the...
  46. T

    Proving Log 2 with Riemann Sums

    I have been working on this problem for a while. I am supposed to prove that log 2 = \lim_{n \rightarrow \infty} \frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{2^n}. The problem is that I have a hard time figuring out how I am supposed to prove that something is equal to a transcendental...
  47. S

    What is the proof of riemann integral?

    so riemann integral pretty much says that if you take closer and closer approximations, then you can find the area of whatever(not too precise, I know, but doing the rectangles and stuff). I'm looking for a proof of it, but all I can find are more general things, i.e, Riemann integeral is...
  48. G

    How to Approximate the Area of a Semi-Circle Using Riemann Sums

    Can anyone please direct me in the right way on working out the approximate area of a semi-circle with equation y = (r^2 - x^2)^0.5, by using a Riemann Sum
  49. M

    Robustness of the Riemann integral

    Could someone explain me what robustness is(in ur words), and how it works in proofs. All i kno is that basically u have two functions and u jiggle them a lot until u make them integrable if its not or destroy their integrability if they are integrable. Geometric explanation would really help...
  50. K

    Understanding Riemann Sums: Solving Lf(P) and Uf(P) Correctly

    When calculating Lf(P) and Uf(P) there can be many different answers correct? Provided that you solved it properly.
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