Comments on: A combinatorial subset sum problem associated with bounded prime gaps
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoThu, 06 Jul 2017 08:07:41 -0800
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By: Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges | What's new
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-483423
Thu, 06 Jul 2017 08:07:41 +0000http://terrytao.wordpress.com/?p=6783#comment-483423[…] step, which is again standard, is the use of the Heath-Brown identity (as discussed for instance in this previous blog post) to split up into a number of components that have a Dirichlet convolution structure. Because the […]
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By: A conjectural local Fourier-uniformity of the Liouville function | What's new
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-462896
Wed, 02 Dec 2015 21:51:11 +0000http://terrytao.wordpress.com/?p=6783#comment-462896[…] (e.g. Vaughan’s identity or Heath-Brown’s identity), as discussed for instance in this previous post (which is focused more on the von Mangoldt function, but analogous identities exist for the […]
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By: Derived multiplicative functions | What's new
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-419785
Wed, 24 Sep 2014 22:29:12 +0000http://terrytao.wordpress.com/?p=6783#comment-419785[…] to , where and are arbitrary parameters and denotes the -fold convolution of , and discussed in this previous blog post; this is the top order component […]
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By: Terence Tao
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-270809
Tue, 11 Feb 2014 03:28:14 +0000http://terrytao.wordpress.com/?p=6783#comment-270809Use hypotheses (ii) and (iii) of Definition 11, rather than hypothesis (i). (iii) gives the case when q is prime, and (ii) then gives the general squarefree case as a consequence.

More generally, bounds on multiplicative functions should usually be proven by first considering the prime case (or the prime power case, if one is not restricted to squarefree numbers).

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By: Stijn Hanson
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-270773
Mon, 10 Feb 2014 23:56:45 +0000http://terrytao.wordpress.com/?p=6783#comment-270773ok, I roughly see how those bounds help (presumably by using for all but how exactly do you get ? I’m looking at the bounds in the definition and the closest asymptotic bound I can easily see is . Sorry, I’m probably being really thick.
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By: Terence Tao
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-270587
Sun, 09 Feb 2014 23:41:36 +0000http://terrytao.wordpress.com/?p=6783#comment-270587One can bound by and by .
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By: Stijn Hanson
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-270580
Sun, 09 Feb 2014 23:03:29 +0000http://terrytao.wordpress.com/?p=6783#comment-270580Sorry, could you expand a little on how the “crude bounds” give the last displayed equation of Proposition 14?
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By: Terence Tao
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-248275
Thu, 17 Oct 2013 20:03:11 +0000http://terrytao.wordpress.com/?p=6783#comment-248275Hmm, that is a numbering issue; one can use for instance Lemma 8 from the previous post. (These estimates can also be found in standard multiplicative number theory texts, e.g. Corollary 2.15 of Montgomery-Vaughan. A good rule of thumb regarding these sorts of estimates is that the number of divisors of a large number is typically of size ; this is closely related to the Erdos-Kac theorem, which is a precise version of the assertion that the number of prime factors of a number is typically of size .)
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By: Stijn Hanson
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-248273
Thu, 17 Oct 2013 19:46:34 +0000http://terrytao.wordpress.com/?p=6783#comment-248273In Lemma 8 it proves equation (10) using a simple bound and then Proposition 5 from “The elementary Selberg sieve and bounded prime gaps” but that’s a proof of MPZ from EH and doesn’t appear to contain any sums of the required form (requiring either squarefree-ness or some coprimality condition on the summands). Is there some numbering/linking mix-up here or am I just not reading it properly?
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By: An improved Type I estimate | What's new
https://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/#comment-239913
Sun, 28 Jul 2013 02:55:09 +0000http://terrytao.wordpress.com/?p=6783#comment-239913[…] Again, these conditions are implied by (8). The claim then follows from the Heath-Brown identity and dyadic decomposition as in this previous post. […]
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