Riemann Definition and 619 Threads

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?

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...

\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...

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...

I have two questions: Why hasn't the hypothesis been proved yet? Is it because we don't know why re(s) has to be 1/2 and thus can't prove it, or is it because we know why re(s) has to be 1/2 but we just don't know how to prove it. Why exactly does re(s) have to be 1/2? \zeta...

This is driving me nuts. (I originally posted this to the coursework section, but in thinking about this, I felt that it might not be the right place (this is for a term paper, not really any ongoing coursework, so there). Hope I'm not imposing ... I feel quite embarrassed on this one, since it...

I still don't understand a few things. Let's say we had a non-trivial zero counting function, Z_n(n), for the riemann zeta function. Couldn't we fairly easily prove the riemann hypothesis by evaluating \zeta (\sigma+iZ_n), solving for \sigma , then proving it for all n using induction...

Homework Statement Show that when f(z)=x^3+i(1-y)^3, it is legitimate to write: f'(z)=u_x+iv_x=3x^2 only when z=i Homework Equations Cauchy riemann equations: u_x=v_y , u_y=-v_x f'(z)=u_x+i*v_y The Attempt at a Solution u=x^3 v=(1-y)^3 u_x=3*x^2 v_y=-3*(1-y)^2...

Alright, I started doing Riemann sums and I am ripping my hair out in frustration. I just can't wrap my head around how I'm supposed to do it, and my woefully vague textbook isn't helping either. I'm wondering how I'm supposed to solve a Riemann sum question with sigma notation (no limits), and...

Can someone show me the steps to evaluating \zeta(c + xi), where 0 \leq c<1?

It is a standard fact that at any point p in a Riemannian space one can find coordinates such that \left.g_{\mu\nu}\right|_p = \eta_{\mu\nu} and \left.\partial_\lambda g_{\mu\nu}\right|_p. Consider the Taylor expansion of g_{\mu\nu} about p in these coordinates: g_{\mu\nu} = \eta_{\mu\nu}...

Hello folks, this is going to be a bit longish, but please bear with me, I'm going nuts over this. For a term paper I am working through a paper on higher dimensional spacetimes by Andrew, Bolen and Middleton. You can http://arxiv.org/abs/0708.0373" . My problem/confusion is in...

Homework Statement (My first post on this forum) Background: I am teaching myself General Relativity using Dirac's (very thin) 'General Theory of Relativity' (Princeton, 1996). Chapter 11 introduces the (Riemann) curvature tensor (page 20 in my edition). Problem: Dirac lists several...

Hi, Given a large number of test particles N, it should be possible to determine the Riemann curvature tensor by tracking their motion as they undergo geodesic deviation. Is there a minimum number N that will achieve this in any situation, or does it vary from problem to problem? How...

Hi everyone, hope this is the right place to put this :) I have just finished "Theory of Functions" Vol. 1 & 2 by Konrad Knopp. I'd like to continue with a book that picks up where the second volume it left off. (Especially would be nice is a more "modern" book) The second volume is about...

Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct. Please could you also guide me on Q1 & 4. Q1...

I have a question concerning the Riemann Hypothesis, a conjecture about the distribution of zeros of the Riemann-zeta function. the trivial zeros (s=-2, s= -4, s=-6) arent much of a concern as the NON-trivial zeros, where any real part of the non-trivial zero is = 1/2. What i am having...

Lagarias’ equivalence to the Riemann hypothesis should be discussed, i.e., if hn := n-th harmonic number := 1/1 + 1/2 + · · · + 1/n, and σn := divisor function of n := sum of positive divisors of n, then if n > 1, hn + ehn ln hn > σn. There is a $1,000,000 prize for the proof of this at...

http://arxiv.org/abs/0806.0892"

i am a high school student i want to prove the riemann hypothesis but i do not how to start:confused:

Hello I plan on applying to the university of waterloo next year and due to the fact that many of my marks are not that great (failed gr 10 math) I decided to start a site to showcase my ability in math and programing. For those of you who are interested I wrote a program to graph regions of...

f(x) = x , if x is rational = 0 , if x is irrational on the interval [0,1] i just wanted to check if my reasoning is right. take the equipartition of n equal subintervals with choices of t_r's as r/n for each subinterval. calculating the integral as limit of this sum (and...

Can anyone tell we how this: http://mathworld.wolfram.com/RiemannSeriesTheorem.html can be proved? The book that I read it in said that it was "beyond the scope of the book". It one of the coolest theorems I've read about. For example, it means that for any number (pi, phi, ...) there's...

[SOLVED] Riemann Sum with Fortran 90 My assignment: Use Reimann Sums to estimate pi to 6 decimal places (ie: you can stop when successive iterations yield a change of less than 0.000001. For the Reimann Sums solution, an iteration equals 2X the number of segments as the trial before. Print out...

Suppose you have a spacetime with an observer at rest at the origin, and the surface at t = 0 going through the origin, and passing through the surface there are geodesics along increasing time. Then as you get a small ways away from the surface, the geodesics start to deviate from each other...

[SOLVED] Riemann sum Important stuff: \sum i^2 = \frac{n(n+1)(2n+1)}{6} \sum i = \frac{n(n+1)}{2} And the solution: (Where I write "lim" I mean limit as n-->infinity. Where I write the summation sign I mean from i=1 to n.) lim \sum t^2 + 6t - 4 \Delta t \Delta t = \frac{5 -...

[SOLVED] Summation - Riemann Intergral - URGENT Homework Statement Im working on the upper and lower riemann sums of f(x) = exp(-x) where Pn donates the partition of [0,1] into n subintervals of equal length (so that Pn = {0,1/n,2/n,...,1}) Homework Equations The Attempt at...

Homework Statement Prove that the function specified below is Riemann integrable and that its integral is equal to zero. Homework Equations f(x)=1 for x=1/n (n is a natural number) and 0 elsewhere on the interval [0,1]. The Attempt at a Solution I have divided the partition into...

Could anybody please give advice for the study of complex analysis, Riemann surfaces & complex mappings. These subjects form the content of chapters 7 & 8 of Roger Penrose's "The Road to Reality". Any advice will do: maybe suggestions on books to supplement the learning, or books to further my...

Homework Statement Let f, g : [a, b] \rightarrow R be integrable on [a, b]. Then, prove that h(x) = max{f(x), g(x)} for x \in [a, b] is integrable. 1 Homework Equations Definition of integrability: for each epsilon greater than zero there exists a partition P so that...

why does the einstein field tensor have the riemann tensor contracted? I am confused as to what purpose it serves. I have seen an explanation that it gets rid of extra information about spacetime or something like that. and also is the Ricci scalar added to einstein tensor so that the covariant...

What Lie groups are also Riemann manifolds? thanks

My crude understanding of GR in outline is that spacetime curvature is described by the way the components of the Riemann tensor vary from point to point in spacetime, that such variation is controlled by Einstein's field equations, and that the source of curvature is the energy-momentum tensor...

In the Riemann theory for a function f defined on all of R, we define its improper integral over R as the sum of two limits: \int_{-\infty}^{+\infty}f(x)dx = \lim_{a\rightarrow -\infty}\int_{a}^0f(x)dx+\lim_{b\rightarrow +\infty}\int_{0}^b f(x)dx and in general, this is not equal to...

Find the Riemann sum for this integral using the right-hand sums for n=4 Find the Riemann sum for this same integral, using the left-hand sums for n=4 Sorry the integral is attatched. I don't know how to get it onto here.

Hi I recently stumbled upon this: I know that the Riemann Integral is defined for every piecewise continouus curve. But now suppose you´re asked the following: you are given f(x,y)=\frac{xy^3}{(x^2+y^2)^2} with additional Definition f(0,0)=0. ( It´s a textbook problem :) ) Now surely...

Please HELP...Don't Understand Simple Concept on Riemann Sums Can someone please explain this to me... The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. That is, ||Triangle|| approaches 0 implies that n approaches infinity. I thought...

I often see people use the Riemann definition of the integral to solve a certain limit-series computation, but they usually just skip a step that I can follow one way but not the other. Given the integral, I can see the limit-series that comes from it, but when trying to find the integral from...

Hi guys, My gf is doing honours and is having some trouble with one question on her assignment for complex analysis. She is really stuck and I've only done this topic at an undergraduate level so I have no idea. Neither of us have done any subjects in Topology so we don't know what to do...

Could the maths of string theory or versions of it lead insight into the Riemann hypothesis as, for a start both are about mathematics in the complex plane. Anyone working on this connection at the moment?

In the course I'm taking, we are already done with Lebesgue integration on R, and while we have proven that for continuous fonctions, the Riemann integral and the Lebesgue integral give the same output, we have not investigated further the correspondance btw the two. So I have some questions...

What is the difference between the two? Does the Newton integral arise from the fundalmental theorem of calculus and the Riemann integral is the Newton integral but more rigorously defined?

Can somebody explain me about the trivial zeros? Why \zeta(-2) = \zeta(-4) = \zeta(-6) = 0 = \zeta(k) So \zeta(k) \sum_{n=1}^{ \infty} \frac{1}{n^k} = 0 ?

Homework Statement Let f:[a,b] -> R R being the set of real numbers If f^3 is Reimann-integrable, does that imply that f is? Homework Equations If f is Riemann-Integrable, then it has upper/lower step functions, such that the difference between the upper and lower sums is less...

Are there notes on the net or books that give a gentle introduction on Riemann surfaces ( say undergraduate math or math for physicists type level)? Always read of the importance and beauty of Riemann surfaces but can't find surveys or intros for outsiders. Same for elliptic curves...

Hello I'm having some difficulty in finding sums which relate to Riemann integrals. The first one seems pretty simple.. a finite calculation of what would otherwise be the harmonic series i.e. 1/k from k=n to k=(2n-1). I can't see an easy way of finding a formula in terms of n, however...

http://www.arsmathematica.net/archives/2007/05/22/latest-on-latest-paper-on-arxiv/

Can someone elaborate on the relationship of the Riemann zeros and primes? How are the zeros harmonic to the primes? The quotes below mention the 'sum of its complex zeros' and 'other sums over prime numbers'. Can someone clarify this? From Answers.com: "The zeros of the Riemann...

Can someone please explain to what exactly the Riemann Hypothesis is? My friend said it is something to do with imaginary numbers and how they behave in a certain interval- just wondering.

Homework Statement The following sum \sqrt{9 - \left(\frac{3}{n}\right)^2} \cdot \frac{3}{n} + \sqrt{9 - \left(\frac{6}{n}\right)^2} \cdot \frac{3}{n} + \ldots + \sqrt{9 - \left(\frac{3 n}{n}\right)^2} \cdot \frac{3}{n} is a right Riemann sum for the definite integral. Solve as n->infinity...