- #1
nomadreid
Gold Member
- 1,697
- 220
The question here is not asking for links to help understand analytic continuation or the Riemann hypothesis, but rather help in understand the bits of hand-waving in the following video’s explanations : https://www.youtube.com/watch?v=sD0NjbwqlYw (apparently narrated by the same person who does Khan Academy videos), an introduction to the Riemann hypothesis at a pre-calculus level.(But the answers do not need to be at that level.)
[1] At 9:56 he somehow reduces the 4-dimensional map of the zeta function on {s ∈ ℂ: Re(s) >1} to a two dimensional map. (It seems he is including the whole map, not just the set of values, not just from his drawings but also because in his example of a simpler 4-D map, starting at 8:53, he says that “it all happens in two dimensions” at 9:20.) So my first question: what family of curves is he referring to?
[2] Then, at 13:13, he goes back to the idea of the four-dimensional map, and says that the zeta function, being analytic (except at the pole), is then conformal except at the points where the derivative vanishes, where (at 14:19) he says that the angles get multiplied by an integer n. Then, at 14:22 he says that the points of vanishing derivative are “by far the minority”. Several questions based on this:
(a) As the number of continuous curves in C is that of the continuum, is he saying that the points of vanishing derivative is countable (not to get nitpicky with whether or not one assumes the continuum hypothesis)?
(b) is there a way to determine that integer n?
(c) At 15:46 he again goes back to representing the zeta function as a four-dimensional map, he says that to keep the continuation of an analytic function analytic “that is, that it still preserves angles everywhere” – presuming that the viewer remembers the caveat about the points of vanishing derivative. But if there is a way to determine n in (b), would it have been valid to make the caveat explicit by saying “that it still preserves angles everywhere at the points of non-zero derivative and that multiplies the angle by n otherwise.” ? Could the caveat be made simpler somehow with the concept of continuity instead?
Thanks.
[1] At 9:56 he somehow reduces the 4-dimensional map of the zeta function on {s ∈ ℂ: Re(s) >1} to a two dimensional map. (It seems he is including the whole map, not just the set of values, not just from his drawings but also because in his example of a simpler 4-D map, starting at 8:53, he says that “it all happens in two dimensions” at 9:20.) So my first question: what family of curves is he referring to?
[2] Then, at 13:13, he goes back to the idea of the four-dimensional map, and says that the zeta function, being analytic (except at the pole), is then conformal except at the points where the derivative vanishes, where (at 14:19) he says that the angles get multiplied by an integer n. Then, at 14:22 he says that the points of vanishing derivative are “by far the minority”. Several questions based on this:
(a) As the number of continuous curves in C is that of the continuum, is he saying that the points of vanishing derivative is countable (not to get nitpicky with whether or not one assumes the continuum hypothesis)?
(b) is there a way to determine that integer n?
(c) At 15:46 he again goes back to representing the zeta function as a four-dimensional map, he says that to keep the continuation of an analytic function analytic “that is, that it still preserves angles everywhere” – presuming that the viewer remembers the caveat about the points of vanishing derivative. But if there is a way to determine n in (b), would it have been valid to make the caveat explicit by saying “that it still preserves angles everywhere at the points of non-zero derivative and that multiplies the angle by n otherwise.” ? Could the caveat be made simpler somehow with the concept of continuity instead?
Thanks.
Last edited: