# 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. ### I What is Riemann's approach to classifying 2d surfaces?

I was reading Bernhardt Riemann's old foundational text on abelian functions, and I found a part that really confused me. What he is trying to do is set up an invariant to classify 2d surfaces as simply connected, multiply connected, etc via some kind of "connectivity number". From the text, I...
2. ### Will the Riemann hypothesis be solved by 2100?

What do u think?
3. ### I What are the Zeta Function and the Riemann Hypothesis?

What is the zeta function and the Riemann hypothesis.
4. ### A GW Binary Merger: Riemann Tensor in Source & TT-Gauge

In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...
5. ### Finding a definite integral from the Riemann sum

Hi! I am having trouble finalizing this problem. The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n. Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n]. I can guess that the function is...
6. ### Identifying variables from Riemann sum limits

Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...
7. ### I Geometry of series terms of the Riemann Zeta Function

This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i## The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...
8. ### I Reconciling 2 expressions for Riemann curvature tensor

I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in the notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ An index-based expression is also given in (eq...
9. ### I Original definition of Riemann Integral and Darboux Sums

Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows: $$S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$ A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a...
10. ### I Relating integration of forms to Riemann integration

Partition each closed interval ##[a_i,b_i]## in the Cartesian product, ##A##. Denote the partition for the i-th closed interval as ##\{x_i^1,\ldots,x_i^{k_i}\}##. The Cartesian product of the partitions forms a partition of ##A## (think: a lattice of points that coincide with the points of each...
11. ### I Calculate Contraction 2nd & 4th Indices Riemann Tensor

How to calculate the contraction of second and fourth indices of Riemann tensor?I can only deal with other indices.Thank you！

39. ### MHB Interval of the Riemann integral value

Hello everyone, I have to find an interval of this Riemann integral. Does anybody know the easiest way how to do it? I think we need to do something with denominator, enlarge it somehow. My another guess is the integral is always larger than 0 (A=0) because the whole function is still larger...
40. ### I Riemann Tensor, Stoke's Theorem & Winding Number

I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve. If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?
41. ### Is the Function Analytic? Testing the Cauchy Riemann Equations

I tested the first function with the Cauchy Riemann equations and it seemed to fail that test, so I don't believe that function is analytic. However, I'm really not sure how to show that it is or is not analytic using the definition of the complex derivative.
42. ### A Riemann Tensor Formula in Terms of Metric & Derivatives

Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula. Thanks.
43. ### I Computing Riemann Tensor: 18 Predicted Non-Trivial Terms

I want to compute the Riemann Tensor of the following metric $$ds^2 = dr^2+(r^2+b^2)d \theta^2 +(r^2+b^2)\sin^2 \theta d \phi^2 -dt^2$$ Before going through it I'd like to try to predict how many non-trivial components we'd expect to get, based on the Riemann tensor basic rule: It is...
44. ### Calculating Riemann Sums on Python w/ Numpy

import numpy as np def num_int(f,a,b,n): dx=(b-a)/n x=np.arange(a,b,step=dx) y=f(x) return y.sum()*dx def rational_func(x): return 1/(1+x**2) print(num_int(rational_func,2,5,10)) Here is my code for the left endpoint, I know this code works because I compared it to an...
45. ### I Riemann integrability with a discontinuity

So, I know that a function is integrable on an interval [a,b] if ##U(f,P_n)-L(f,P_n)<\epsilon ## So I find ##U(f,P_n## and ##L(f,P_n## ##L(f,P_n)=5(3-\frac{1}{n}-0)+5(3+\frac{1}{n}-(3-\frac{1}{n}))+7(4-(3+\frac{1}{n}))=22-\frac{2}{n} ##...
46. ### A Parallel transport of a 1-form aound a closed loop

Good day all. Since the gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Then If we form the Gradient vector field...
47. ### I Riemann Integration ... Existence Result .... Browder, Theorem 5.12 ....

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...
48. ### MHB Riemann Integration ... Existence Result .... Browder, Theorem 5.12 ....

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...
49. ### I The Riemann and Darboux Integrals .... Browder, Theorem 5.10 .... ....

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...
50. ### MHB The Riemann and Darboux Integrals .... Browder, Theorem 5.10 .... ....

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...