Riemann Definition and 619 Threads

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.

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

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?

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

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...

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...

According to Wikipedia, the Zeta Riemann Function is defined as follows: \begin{equation} \zeta(z) = \sum_{k=1}^{\infty}\frac{1}{k^{z}}, \forall z \in \mathbb{C}, Re[Z] > 1. \end{equation} Well, the trivial zeros are the negative even numbers. Is that a consequence of the following...

I'm trying to evaluate the derivative of the Riemann zeta function at the origin, \zeta'(0), starting from its integral representation \zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\frac{1}{e^t-1}. I don't want to use a symbolic algebra system like Mathematica or Maple. I am able to...

Let's say I want to calculate the Ricci tensor, R_{bd}, in terms of the contractions of the Riemann tensor, {R^a}_{bcd}. There are two definitions of the Riemann tensor I have, one where the a is lowered and one where it is not, as above. To change between the two all that I have ever seen...

Homework Statement Calculate the limit with Riemann. Homework Equations \displaystyle\lim_{n \to{+}\infty}{\displaystyle\frac{pi}{4}\cdot{} \displaystyle\sum_{k=0}^n{tan^2(\displaystyle\frac{k\cdot{} pi}{4n})\cdot{}\displaystyle\frac{1}{n}}} The Attempt at a Solution I don't know how to...

Homework Statement Identify an=the summation from k=1 to n of (2n)/(4k2+1) as a Riemann sum of an appropriate function on an appropriate interval and find the limit as n approaches infinity of an. Homework Equations There is no interval givien so I assume its from 0 to 1. The...

Hi all, I'm trying to follow through some of my notes of a GR course. The notes are working towards a specific expression and the following line appears: R^{\alpha \beta}_{\gamma \delta ; \mu} + R^{\alpha \beta}_{\delta \mu ; \gamma} + R^{\alpha \beta}_{\mu \gamma ; \delta}=0 Which by...

Homework Statement I am trying to show that for a vector field Va which satisfies V_{a;b}+V_{b;a} that V_{a;b;c}=V_eR^e_{cba} using just the below identities. Homework Equations V_{a;b;c}-V_{a;c;b}=V_eR^e_{abc}(0) R^e_{abc}+R^e_{bca}+R^e_{cab}=0 (*) V_{a;b}+V_{b;a}=0 (**) The Attempt at a...

Regarding "Riemann integration defination" Hi, I did not understand the following: We have : Partition is always a "finite set". A function f is said to Riemann integrable if f is bounded and Limit ||P|| -> 0 L(f,P) = Limit||P|| -> 0 U(f,P) where L(f,P) and U(f,P) are...

Homework Statement If a function f is of bounded variation on [a,b], show it is Riemann integrable Homework Equations Have proven f to be bounded S(P) is the suprenum of the set of Riemann integrals of a partition (Let's say J) s(P) is the infinum of J S(P) - s(P) < e implies f...

Homework Statement What does it mean by this: The cauchy riemann equations are never satisfied when x and y are different from zero and when x=y=0 . Looking at the example of f(z)= l z l = \sqrt{x^{2}+y^{2}} Homework Equations The Attempt at a Solution

we know that \Gamma (s)= \int_{0}^{\infty}dxe^{-x}x^{s-1} however every factor of the Riemann Zeta can be obtained also from a Mellin transform \int_{0}^{\infty}dxf(x)x^{s-1} =(1-p^{-s})^{-1} where f(x) is the distribution \sum_{n=0}^{\infty}x \delta (x-p^{-n}) is there any...

Homework Statement Let f be a holomorphic function in the unit disc D1 whose real part is constant. Prove that the imaginary part is also constant. Homework Equations Cauchy Riemann equations The Attempt at a Solution Hi guys, I'm working through the basics again. I think here we...

Can some please draw a comparison between Riemann Integration and normal definite integration in terms of accuracy.

The Riemannian curvature tensor has the following symmetries: (a) Rijkl=-Rjikl (b) Rijkl=-Rijlk (c) Rijkl=Rklij (d) Rijkl+Rjkil+Rkijl=0 This is surely trivial, but I do not see how to prove that Rijkl=-Rjilk. :( Thanks.

How would the world benefit from the Riemann hypothesis being solved? Mathematicians have been trying to solve this for over 100 years, but have been unable to due to it's mind-boggling complexity and difficulty. What would the world benefit if this theorem was to be solved?

Would the field of the number theory collapse or flourish if the Riemann Hypothesis is proved as true?

"not riemann integrable" => "not lebesgue integrable"?? Hi, In general if a function is not Riemann integrable does this mean the function is also not Lebesgue integrable? Why or why not? I know that if the the function is Riemann integrable then its Lebesgue integrable, but I can't find...

When we talk about "Hilbert space" in (undergraduate) QM, we are typically talking about the space of square-integrable functions so that we can make sense out of \int_{-\infty}^{\infty} |\psi(\vec r,t)|^2 d^3x. But are we talking about Riemann-integrable functions or Lebesgue-integrable...

Homework Statement Proove that: R_{abcd} = R_{cdab} Homework EquationsThe Attempt at a Solution I'm not sure whether to expand the following equations any further (using the definitions for the christoffel symbols) and hope that I can re-label repeated indexes at a later stage or if there is...

Homework Statement If f,g are Riemann integrable on [a,b], then for c,d real numbers, (let I denote the integral from a to b) I (cf + dg) = c I (f) + d I (g) Homework Equations The Attempt at a Solution I have the proofs for c I(f) = I (cf) and I (f+g) = I (f)...

Homework Statement f: [0,1] -> Reals, f(x) = 3-x2 P={0,1/2,1} Find lower and upper Riemann sums, and approximate the definate integral using them and find the corresponding approximation error. Homework Equations The Attempt at a Solution So I tried finding the upper Riemann...

This could be the way to proof. remember, this is not a proof. today I found a clue to solution to Riemann hypothesis: Let it be Riemann zeta function :ζ(s) The proof that all the non trivial zeroes lie on the critical strip when s = 1/2 + it let us suppose there are other zeroes...

After reading about the Riemann Zeta Function on Wolfram Alpha (http://mathworld.wolfram.com/RiemannZetaFunction.html), it's still unclear to me how the Euler product formula is essentially equal to the limit of a p-series. Someone please enlighten me

Homework Statement I'm trying to prove that Sp|f| - sp|f| \leq Spf - spf Where P is a partition of [a,b] and f is function that is riemann integrable. Homework Equations The Attempt at a Solution So I get to a point where M = supf(x) and m = inff(x) then |M|(b-a) - |m|(b-a)...

Homework Statement Show that the function f: [0,1] -> R defined by: f(x) = 1, if x=1/k for some k f(x) = 0, else is Riemann integrable on [0,1] Homework Equations The Attempt at a Solution I attempted the problem using Cauchy's criterion but found that this function is...

Homework Statement Let A={1/n, n =natural number} f: [0,1] -> Reals f(x) = {1, x in E, 0 otherwise Prove f is riemann integrable on [0,1] Homework Equations The Attempt at a Solution Not quite sure, but I think supf = 1 and inf f= 0 no matter what partition you take, then...

1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...

Homework Statement Let f(z) be analytic on the set H. Let the modulus of f(z) be constant. Does f need be constant also? Explain. Homework Equations Cauchy riemann equations Hint: Prove If f and f* are both analytic on D, then f is constant. The Attempt at a Solution I think f need be...

I was wondering if one of the consequences of the Elliott-Halberstam conjecture would imply the Riemann Hypothesis (RH) or the Generalized Riemann Hypothesis (GRH)? Or at least if there is a connection between the Elliott-Halberstam conjecture and RH or GRH? I ask because the...

Homework Statement How to use veilbein to calculate Riemann tensor, Ricci tensor and Ricci scalar? (please give me the details) de^a+\omega_{~b}^a\wedge e^b=0, R_{~b}^a=d\omega_{~b}^a+\omega_{~c}^a\wedge\omega_{~b}^c. The metric is...

Hello everyone. I was hoping someone could clarify this "heuristic" argument I found online. First, what is the analytic function they speak of and is its derivative difficult to compute? Second, does this look like a legit argument? : If you take the derivative w.r.t s of both sides of sum...

Maybe someone will find something interesting in this paper. They have a reference to some 1995 work by Alain Connes. I didn't have time to look into this very much. Maybe it's amusing and maybe not: http://arxiv.org/pdf/1012.4665v1

Lets say we have a function of a complex variable z , f(z). I read that for the function to be differentiable at a point z0 , the CR equations are a necessary condition but not a sufficient condition. Can someone give me an example where the CR equations hold but the function is not...

For those not familiar with the term Fermi estimate/problem/question see here: http://www.vendian.org/envelope/dir0/fermi_questions.html http://en.wikipedia.org/wiki/Fermi_problem My question: Between the time that Riemann posed his famous question (in 1859) and now, how many hours have...

Riemann Integrable <--> Continuous almost everywhere? I ran across a statement somewhere in the forums saying that a function is Riemann-integrable iff it is continuous almost everywhere, i.e. if its set of discontinuities has measure 0. Is that right? What about the case of a function...

Homework Statement Rn=\sum(i*e^(-2i/n))/n^2, i=1 Identify this Riemann sum corresponding to a certain definite integral. Homework Equations The Attempt at a Solution I got till 1/n^2 [1/e^(2/n)+2/e^(4/n)+3/e^(6/n)...n/e^2] and that's it. To my understanding I should be...

Homework Statement Suppose that f:[a, b] → ℝ is a function that is zero for all x ∈ [a, b] except for the values x_1,x_2,…,x_k. Find ∫[a b](f(x)dx) and prove your result. Homework Equations Definition of a Riemann integrable function...

hmm...

A metric consistent with interval: \mathrm{d}s^2=-\mathrm{d}\tau^2+\frac{4\tau^2}{(1-\rho^2)^2}\left(\mathrm{d}\rho^2+\rho^2\mathrm{d}\theta^2+\rho^2\sin(\theta)^2\mathrm{d}\varphi^2\right) get zero for riemann's tensor, therefor must be isomorphic with minkowski tensor. But I don't find thus...

Part 1. Homework Statement The problem literally states... " The Integral. limit of n-> infinity of n*[1/(n+1)^2 + 1/(n+2)^2 + 1/(n+3)^2 + 1/(2n)^2] = 1/2 " According to the teacher, the answer is 1/2. I don't know why or how to get there. Part 2. The attempt at a solution...

Hello everyone - I'm a third year student at Cambridge university, and I've recently started taking a course on Riemann surfaces along with a number of other pure courses this year. The problem is, the lecturer of the course is of a rather sub-par standard - whilst I don't doubt he's probably...

Last summer I took Semi Riemann Geometry lesson. Almost all the definitions in Semi Riemann geometry are with the same Minkowski geometry. I don't understand what is the different between Minkowski Space an Semi Riemann Space.

Homework Statement The problem given is: Show that the function f(x) = 1/sqrt(x) is integrable on the compact interval [0,1]. Homework Equations We are only allowed to use theorems, definitions, and properties that have been covered in class or are in the book. The ones I...

Can someone tell me what a riemann invariant is in general terms please? i can't find it anywhere! thanks, lav