Comments on: 254A, Notes 4: Some sieve theory
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoTue, 18 Dec 2018 23:56:27 +0000
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By: Terence Tao
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-508999
Tue, 18 Dec 2018 23:56:27 +0000http://terrytao.wordpress.com/?p=7997#comment-508999No, one can only hope to get here (one can only make good use of the portion of the sieve of Eratosthenes up to primes of size about ). Note for instance that the number of primes in does not go to zero in the limit (indeed, we believe it equal to 2 infinitely often!)
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By: Anonymous
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-508973
Tue, 18 Dec 2018 14:31:00 +0000http://terrytao.wordpress.com/?p=7997#comment-508973Exercise 26 (Approximations to Legendre’s conjecture):

I guess it should be O(y/logx) instead of O(y/logy), shouldn’t it?

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By: Terence Tao
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-472684
Wed, 28 Sep 2016 16:02:13 +0000http://terrytao.wordpress.com/?p=7997#comment-472684There has been a lot of work on counting smooth numbers in short intervals, see e.g. this paper of Croot http://people.math.gatech.edu/~ecroot/smooth6.pdf for some results and references.
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By: Anonymous
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-472664
Wed, 28 Sep 2016 08:44:31 +0000http://terrytao.wordpress.com/?p=7997#comment-472664I have a question which is somewhat related to the Brun-Titchmarsh inequality. It would be great if you can give me a reference. Does the following statement hold: Almost all natural numbers in the interval has the property that all prime factors of are at most , where grows slowly to infinity with . Please ignore (delete) my question if you think it is inappropriate for this post. Thanks!
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By: Anonymous
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-472579
Sun, 25 Sep 2016 19:54:53 +0000http://terrytao.wordpress.com/?p=7997#comment-472579”which by the asymptotic $\sum_p \frac{1}{p^s} = \log(s-1) + O(1)$ for $s > 1$” should be ”…. $-\log(s-1)$…” (missing minus sign).

[Corrected, thanks – T.]

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By: Anonymous
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-472578
Sun, 25 Sep 2016 19:44:53 +0000http://terrytao.wordpress.com/?p=7997#comment-472578In the definition of $G(t_1,t_2)$ (after (44)) and $E_p(t_1,t_2)$: $1+p^{\frac{1+it_2}{\log R}}$ should be $p^{1+\frac{1+it_2}{\log R}}$.

[Corrected, thanks – T.]

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By: manhtuankhtn
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-472574
Sun, 25 Sep 2016 16:19:55 +0000http://terrytao.wordpress.com/?p=7997#comment-472574In the definition of $G(t_1,t_2)$ after equation (44): should the term $dt_1dt_2$ removed?

[Corrected, thanks – T.]

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By: Notes on the Bombieri asymptotic sieve | What's new
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-470708
Sun, 17 Jul 2016 16:54:14 +0000http://terrytao.wordpress.com/?p=7997#comment-470708[…] where denotes the derivative of . Note the loss of that had previously been pointed out. In the arguments that follows I will be a little brief with the details, as they are standard (see e.g. this previous post). […]
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By: A cheap version of Halasz’s inequality | What's new
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-462305
Tue, 24 Nov 2015 05:24:48 +0000http://terrytao.wordpress.com/?p=7997#comment-462305[…] This follows for instance from the fundamental lemma of sieve theory (see e.g. Corollary 19 of this blog post). (One can also use the Selberg sieve or the large […]
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By: The logarithmically averaged Chowla and Elliott conjectures for two-point correlations; the Erdos discrepancy problem | What's new
https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/#comment-459427
Sat, 19 Sep 2015 00:46:23 +0000http://terrytao.wordpress.com/?p=7997#comment-459427[…] such as (2) or (3) are known to be subject to the “parity problem” (discussed numerous times previously on this blog), which roughly speaking means that they cannot be proven solely using […]
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