Animated representation of the main sum in the Riemann-Siegel formula

Another of my (very simple) computer programs that has been lying around for years:
Parametric plot representing the Dirichlet polynomial which occurs in the Riemann-Siegel formula (or the approximate functional equation) for the Riemann zeta function on the critical line, animated by varying the t-value. In the range displayed the picture is approximated well by putting together a chain of Cornu spirals (plots of the incomplete Fresnel integral): Approximations of this kind were introduced by van der Corput nearly 100 years ago by applying Poisson summation and estimating the resulting integrals with Laplace's method of stationary phase. Further up the line, another of van der Corput's methods becomes effective: a modification of the Weyl shift using a trick with Cauchy's inequality, which recovers the "subconvexity" bound, "mu(1/2) is no more than 1/6"
(you can see me trying to make a proof via Gallagher's methods look pretty 6 min 5 sec into my video "Working out more stuff on the guitar").
For those who don't know, all this is related to the MOST IMPORTANT UNSOLVED PROBLEM IN MATHS (the Riemann Hypothesis). A solution to that would have radical consequences in number theory, and could go down in history as did things like Pythagoras' Theorem. There are connections with mind-blowing ideas in physics
(see "Scrolling graph of Riemann-Siegel Z(t) with sound effects").
A weaker statement than the Riemann Hypothesis is the Lindelof Hypothesis (which basically says that the thing in the picture never spreads itself out too far). The picture represents an "exponential sum". Intimately related sums are estimated to approach the Gauss Circle Problem and the Dirichlet Divisor Problem (for example). The guy with the strongest results so far on these sums was once my PhD supervisor - but so far I have been (let's say) too absent minded to see that particular course of study through to conclusion! See the description to the video-response "Perturbed cubic Gauss sums" for a discussion of those ideas.
For all the immense effort that has been put into bounding such sums, I'm sorry to have to report that the exponents we've got in the best bounds are still embarassingly close to those (such as the classical subconvexity bound mentioned above) which were obtained near the beginning of the 20th Century!
The soundtrack consists of my twin brother (nojameson.net) playing the guitar solo for "Bohemian Rhapsody". He shows an interest in the maths too sometimes, and eventually I went through one and a half (!) proofs of the Prime Number Theorem with him. He's become quite a chess player, and has created some pretty magic squares generated by knight's tours. We've done a bunch of juggling, music and programming together as well.

Category:

Education

Tags:

• Riemann
• zeta
• function
• exponential
• sum
• Cornu spiral
• Fresnel integral
• Siegel
• Poisson summation
• van der Corput
• geometric vision
• graphics
• animation
• Lindelof
• hypothesis
• Tim Jameson
• tpjameson
• Nick Jameson
• nojameson
• Bohemian Rhapsody
• guitar solo
• approximate
• functional equation
• ortega24024
• Lancaster
• Cardiff
• university

[ortega24024]

Don't say "Sorry, '<' is not a valid character for video description".

Say "Sorry, you're going to have to type that all in again because when we have a problem with your description we'll delete what you offered us" (see description to "Scrolling graph of Riemann-Siegel Z(t) with sound effects" - I always end up saying that) !

[ortega24024]

If you found your way here from the "Where are the zeros of zeta of s?" song then when I said I'd put a load of comments all over this one what I meant was that I've put a load of comments on my video "Scrolling graph of Riemann-Siegel Z(t) with sound effects" !

[TRAJEKTORULM]

High ortega,

great video, great music, very strong-

Thomas

[MargusMartsepp]

nice