Comments on: Fourier uniformity of bounded multiplicative functions in short intervals on average
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoFri, 18 Jan 2019 20:18:02 +0000
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By: Terence Tao
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-511434
Fri, 18 Jan 2019 20:18:02 +0000http://terrytao.wordpress.com/?p=10843#comment-511434At AIM we pretty much convinced ourselves that the argument extends to polynomial phases, and we have a sketch of an argument that it also works for nilsequences. Also it looks hopeful that the condition can be relaxed a bit, probably to and further than that if we assume RH. The nilsequence extension also seems to have some implications for sign patterns of the Liouville function and for various polynomial correlations of Liouville. Eventually we’ll try to write up these things properly, but at present everything is just in sketchy note form.
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By: Uwe Stroinski
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-511433
Fri, 18 Jan 2019 18:59:29 +0000http://terrytao.wordpress.com/?p=10843#comment-511433Today I was on a workshop where Tanja Eisner was scheduled to give a short talk about Sarnaks conjecture and (I guess) this last AIM workshop. Unfortunately she could not make it. Can you say a sentence or two about what you think what the situation is and whether there are there any promising approaches?
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By: Anonymous
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508672
Thu, 13 Dec 2018 00:30:47 +0000http://terrytao.wordpress.com/?p=10843#comment-508672The math exchange argument is an old flawed one (that seems to reappear once in a while either there or on mathoverflow) that misuses the complex logarithm and the residue theorem to make an integral vanish in thin air
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By: Anonymous
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508667
Wed, 12 Dec 2018 19:27:56 +0000http://terrytao.wordpress.com/?p=10843#comment-508667Another proposed one-page proof of RH was recently given by Atiyah.
One should not be “too excited” by such proposed proofs.
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By: Excited
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508662
Wed, 12 Dec 2018 11:24:07 +0000http://terrytao.wordpress.com/?p=10843#comment-508662It seems the Riemann Hypothesis has been proven ! https://math.stackexchange.com/q/3034495/507152
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By: Terence Tao
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508528
Sat, 08 Dec 2018 19:43:09 +0000http://terrytao.wordpress.com/?p=10843#comment-508528We haven’t checked the details of this yet, but I am optimistic that most of the argument should be generalisable to polynomial phases, and thence to nilsequences. This is certainly something we plan to look at next week during the AIM workshop.
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By: Nick Cook
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508525
Sat, 08 Dec 2018 19:25:22 +0000http://terrytao.wordpress.com/?p=10843#comment-508525 I think some references to display (8) should be to display (6).

[Corrected, thanks – T.]

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By: Will Sawin
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508519
Sat, 08 Dec 2018 16:14:41 +0000http://terrytao.wordpress.com/?p=10843#comment-508519What goes wrong if you try to apply the same argument to polynomial phases? One might naively hope that everything works if you just raise p to a power at every step, but this seems unlikely.
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By: Terence Tao
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508474
Fri, 07 Dec 2018 20:57:44 +0000http://terrytao.wordpress.com/?p=10843#comment-508474The typical diameter of this “graph” should be logarithmic in size, though in our actual argument we don’t actually work with such long paths and proceed by using a “mixing lemma” (analogous to the “expander mixing lemma” in the theory of expander graphs) instead to show that any two large sets are highly connected to each other by this graph, which is all we really need anyway.
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By: Allan van Hulst
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508458
Fri, 07 Dec 2018 13:46:19 +0000http://terrytao.wordpress.com/?p=10843#comment-508458Typo: “that ths quantity”. By the way, do you think it would be possible to derive a practical upper bound for the length of these “fairly short paths” in the graph-based perspective applied in the latter part of the proof?
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