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 Tao
Wed, 12 Dec 2018 11:24:07 +0000
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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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By: foobar
https://terrytao.wordpress.com/2018/12/05/fourier-uniformity-of-bounded-multiplicative-functions-in-short-intervals-on-average/#comment-508432
Fri, 07 Dec 2018 09:06:25 +0000http://terrytao.wordpress.com/?p=10843#comment-508432 A typo nit: Joni Teräväinen, not Joni Teravainen. (It appears to be correct on the other blog post.)

[Corrected, thanks – T.]

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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-508361
Wed, 05 Dec 2018 23:47:44 +0000http://terrytao.wordpress.com/?p=10843#comment-508361Yes; we believe in fact that our arguments should extend to cover the range for some absolute constant , and on the Riemann hypothesis one can hope to get as far as for any that goes to infinity. However we have not checked these claims in detail (particularly the latter). Next week Maksym and I will be at the AIM workshop on the Sarnak and Chowla conjectures, together with several other experts in this area, and we may be able to sort out exactly how far these techniques can be pushed then.
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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-508360
Wed, 05 Dec 2018 23:28:36 +0000http://terrytao.wordpress.com/?p=10843#comment-508360Is it possible in theorem 1 to make for some explicit sufficiently slowly decreasing function (as ) ?
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