Riemann Definition and 618 Threads

If the zeta function intersects the critical line when the real part is 1/2, then it will intersect some other line when some other real part is used. Isn't the Riemann Hypothesis just based on a particular convention for the critical line?

I ran into some issues when trying to calculate the lower Riemann sum of f\left(x\right)={x}^{3}, x\in[0,1] I am asked to use the standard partition {P}_{n} of [0,1] with n equal subintervals and evaluate L(f,{P}_{n}) and U(f,{P}_{n}) What I did: L(f,{P}_{n}) =...

"Physicists are attempting to map the distribution of the prime numbers to the energy levels of a particular quantum system." https://www.quantamagazine.org/20170404-quantum-physicists-attack-the-riemann-hypothesis/

What do you think of the following paper about the Riemann Zeta Function? http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.130201

I've thought of a new way (at least I never read it anywhere) of counting the independent components of the Riemann tensor, but I am not sure whether my arguments are valid, so I would like to ask whether my argument is sound or total bonkers. The Riemann tensor gives the deviation of a vector A...

Hello! I read some stuff about the Riemann hypothesis and the formulation seems pretty clear. I also read that many proof of it (well basically all of them) are wrong. I was just wondering in which way are they wrong? (I haven't find a page with the wrong proofs, together with explanations of...

Just a couple questions. Problem 2: Just would like to know if this is the correct approach for this problem. Problem 3: I am just wondering if I can use Problem 2 to prove the first part of Problem 3? Because to me, they seem very similar. Problem 4: Would I use the MVT for integrals...

I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?

Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get...

Homework Statement This is a combination of two questions, one being the continuation of the other 3) Calculate the DFT of the sequence of measurements \begin{equation*} \{ g \}_{k=0}^{5} = \{ 1,0,4,-1,0,0 \} \end{equation*} 4a) Draw the DFT calculated in question 3 on the complex plane. 4b)...

Hi. I try to solve the integral $$\int_{0}^{1} x^{x} dx$$ Through sums of riemann But I came to the conclusion that the result is 0 that is wrong $$\int_{0}^{1} x^{x} dx = \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}$$ $$= \lim_{n\rightarrow...

1. Homework Statement I want to show that the non-trival zeros of the Riemann Zeta function all lie in the critical strip ## 0 < Re(s) < 1## and further to this that they are symmetric about the line ##Re(s)= 1/2 ## where ## \zeta(s) = \sum\limits^{\infty}_{n=1}n^{-s}## With the functional...

The problem I want to calculate $$\sum^n_{k=1} \frac{4}{1+ \left(\frac{k}{n} \right)^2} \cdot \frac{1}{n}$$ when ##n \rightarrow \infty## The attempt ## \sum^n_{k=1} \underbrace{f(\epsilon)}_{height} \underbrace{(x_k-x_{k-1})}_{width} \rightarrow \int^b_a f(x) \ dx ##, when ##n \rightarrow...

How to proof that εμνρσ Rμνρσ =0 ? Thanks.

Homework Statement Show that ## \int\limits^{\infty}_0 \sum\limits^{\infty}_{n=1}e^{-\pi n^{2} t} t^{s-1} dt = \sum\limits^{\infty}_{n=1}(\pi n^{2} )^{-s} \int\limits^{\infty}_{0} e^{-t} t^{s-1} dt ## Homework Equations ##\Phi (t)=\theta(it/2)=1+2\sum\limits^{\infty}_{n=1}e^{-\pi n^2 t}. ##...

Found an article online detailing a proof of the Riemann Hypothesis: << link deleted by mentor - unacceptable source >>

Hello~ For usual Riemann curvature tensors defined: ##R^i_{qkl},## I read in the book of differential geometry that in 3-dimensional space, Ricci curvature tensors, ##R_{ql}=R^i_{qil}## can determine Riemann curvature tensors by the following relation...

Homework Statement Determine ##\int_{0}^{2}\sqrt{x}dx## using left riemann sums Homework Equations ##\int_{a}^{b}f(x)dx = \lim_{n\rightarrow \infty}\sum_{i=0}^{n-1}(\frac{b-a}{n})f(x_i)## The Attempt at a Solution [/B] ##\frac{b-a}{n}=\frac{2-0}{n}=\frac{2}{n}## ##\int_{0}^{2}\sqrt{x}dx =...

Homework Statement ## g(s) = \sum\limits^{\infty}_{n=1} 1/n^{-s}, ## Show that ##g(s)## converges uniformly for ## Re(s>1) ## Homework Equations Okay, so I think the right thing to look at is the Weistrass M test. This tells me that if I can find a ##M_{n}##, a real number, such that for...

In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that...

The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as: $$ R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n} $$ My question is that it seems that...

Cauchy integral theorem states that the contour integration of a complex harmonic function along a closed simply connected path=0. What if this simply connected path is drawn over a Riemann surface of function like ##f(z)=\sqrt z##. Will that be possible in the first place? and will the...

What 20 index combinations yield Riemann tensor components (that are not identically zero) from which the rest of the tensor components can be determined?

I do not get the conceptual difference between Riemann and Ricci tensors. It's obvious for me that Riemann have more information that Ricci, but what information? The Riemann tensor contains all the informations about your space. Riemann tensor appears when you compare the change of the sabe...

Has anything similar to the Riemann hypothesis ever been solved? Specifically, has anyone proven that the real part of a result of some particular function always assumes a particular value?

http://arxiv.org/abs/1202.2115 I know Arxiv isn't a real journal, but this caught my eye. Is this a meaningful physical interpretation of the Riemann hypothesis? From what I understand, the zeta function can be modeled as a wave, but attempting to solve for the real part requires infinite...

hi, I tried to take the covariant derivative of riemann tensor using christoffel symbols, but it is such a long equation that I have always been mixing up something. So, Could you share the entire solution, pdf file, or links with me? ((( I know this is the long way to derive the einstein...

I don't recall that I have seen Riemann zeta function put in the form of ##f(s)=u(s)+iv(s)##.

Homework Statement Given two spaces described by ##ds^2 = (1+u^2)du^2 + (1+4v^2)dv^2 + 2(2v-u)dudv## ##ds^2 = (1+u^2)du^2 + (1+2v^2)dv^2 + 2(2v-u)dudv## Calculate the Riemann tensor Homework Equations Given the metric and expanding it ##~~~g_{τμ} = η_{τμ} + B_{τμ,λσ}x^λx^σ + ...## We have...

Hi Community, I have the following question: I have done basic solving of limits and also of Riemann sums but never had to do them in the same question. Would I be correct in saying that I need to solve for the Riemann sum first then take the limit of the integral? Cheers Nemo

So I was reading now about the new geometries and I wanted to know if I can study the Reimann Geometry knowing that I finished high school or if I could just know about it but not about the formulas. I am so interested in the subject because it is used in astronomy.

I am not sure which is the appropriate rubric to put this under, so I am putting it in General Math. If anyone wants to move it, that is fine. Two questions, unrelated except both have to do with the Riemann zeta function (and are not about the Riemann Hypothesis). First, in...

For the function given below find a formula for the Riemann sum obtained by dividing the interval [1,5] into n equal subintervals and using the right-hand endpoint for each c subscript k. Then take a limit of thissum as n-> infinite to calculate the area under the curve over [1,5]. Below you...

I already googled this but I did not find a definite answer. Is there such a thing as a 'inverse riemann'? Specifically, where you invert the start number to be at the top of the riemann symbol and then decrement down to the end value which is on the bottom of the riemann symbol?

I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...

Can someone explain mathematically why do we say Riemann Curvature Tensor has all the information about curvature of Space Thank You

I used the expression Rabcd=-Rbacd=-Rabdc=Rcdab to reduce the number of components. I also used if a=b=0 the R=0 and if and c=d=0 then R=0. This reduced the number of components to 64. How do I get them down to 21? I know I need another equality to reduce it to 20. <<Mentor note: Fixed...

Homework Statement f(z)=z^\frac{3}{2} find the branch points, branch cuts, and Riemann sheet structure. Homework Equations none The Attempt at a Solution So, I converted this to complex exponential form r^\frac{3}{2} e^\frac{i*3*\Theta}{2} From here I mapped around a circle that was...

Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...

Hello, I am currently in my first year of college, and I already took calculus in high school. I was able to solve all the problems, but I feel like I didn't understand everything conceptually. When integrating dy/dx=x you get, ∫x dx=1/2x2. But what exactly happened to the dx, why did it...

Homework Statement Find the upper, lower and midpoint sums for $$\displaystyle\int_{-3}^{3} (12-x^{2})dx$$ $$\rho = \Big\{-3,-1,3\Big\}$$ The Attempt at a Solution For the upper: (12-(-1)^2)(-1-(-3)) + (12-(-1))(3-(-1)) =74 For the lower: (12-(-3)^2)(-1-(-3))+(12-3)(3-(-1)) =42 For midpoint...

I just thought about the critical concepts in mathematics and physics that arose in the last century: Goedel, Schroedinger, etc. My question is: Are there any physical theories that rely on the validity of the extended Riemann Hypothesis? I don't mean computer science, i.e. secure...

Hello, Since it was mentioned in my textbook, I've been trying to find Riemann's proof of the existence of definite integrals (that is, the proof of the theorem stating that all continuous functions are integrable). If anyone knows where to find it or could point me in the right direction, I...

In chapter 8 of Padmanabhan's "Gravitation: Foundations and Frontiers" titiled Black Holes, where he wants to explain that the horizon singularity of the Schwarzschild metric is only a coordinate singularity, he does this by trying to find a scalar built from Riemann tensor and show that its...

Just want to see if I actually understand what these all mean. Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...

There are two forms of Riemann functional equation. One is more symmetric and follows from the other and the duplication theorem of the Gamma function. At least, that's been claimed here...

Hello, I am working through Hughston and Tod "An introduction to General Relativity" and have gotten stuck on their exercise [7.7] which asks to prove the following non- linear wave equation for the Riemann tensor in an empty space: ∇e∇eRabcd = 2Raedf Rbecf − 2Raecf Rbedf − Rabef Rcdef I have...

How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...

Homework Statement Verify that each of the following functions is entire: f(z)=(z^2-2)e^(-x)e^(-iy) Homework Equations The Cauchy Riemann equations u(x,y) = ______ and v(x,y) = ______ u_y=-v_x u_x=v_y The Attempt at a Solution So, I've done a few of these problems and understand that to...

∫(4−7x)dx ====> Integral is from 1 to 8 Similar question but for some reason I can't get the answer after following all the steps