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Professor of Mathematics at UCLA

19h
awarded  Enlightened
2d
awarded  Great Answer
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comment Zeta function double product
Ah, I see. Doesn't look very promising, as there does not appear to be an Euler product factorisation. I think you're better off working with the second von Mangoldt function $\Lambda_2(n) = \Lambda(n) \log n + \Lambda*\Lambda(n)$ to count semiprimes (note that $\sum_n \frac{\Lambda_2(n)}{n^s} = \frac{\zeta''(s)}{\zeta(s)}$).
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awarded  Revival
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revised What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?
added 494 characters in body
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answered What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?
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comment Zeta function double product
Any particular reason why this product is being considered?
Dec
11
comment A sum-of-determinants identity
You are computing the determinant of a matrix $[X_1,\dots,X_n]$ updated by a rank one matrix $X_n (-1,\dots,-1)$, and the matrix determinant lemma followed by Cramer's rule gives the claim. en.wikipedia.org/wiki/Matrix_determinant_lemma en.wikipedia.org/wiki/Cramer%27s_rule
Dec
9
awarded  Nice Answer
Dec
9
comment Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)
Lovasz may be referring to this article of Carbery: ams.org/mathscinet-getitem?mr=2073287 , reproving an inequality similar to the above of Katz and myself, ams.org/mathscinet-getitem?mr=1739220 , proven via the pigeonhole principle and the tensor power trick.
Dec
5
comment How many distinct eigenvalues does a random graph have?
We now have an arXiv preprint: arxiv.org/abs/1412.1438
Nov
29
awarded  Announcer
Nov
28
comment How to test if the power of some algebraic number is the rational combination of two specific algebraic numbers?
This paper of Blondel and Portier shows that determining whether a general integer linear recurrence contains a zero is NP hard: ams.org/mathscinet-getitem?mr=1917474 . The problem here can be reformulated in terms of such decision problems, but it has a special structure, so it is unclear whether the Blondel-Portier obstruction applies here. Nevertheless this is reason for pessimism regarding a polynomial time algorithm here.
Nov
28
comment How to test if the power of some algebraic number is the rational combination of two specific algebraic numbers?
This appears to be a special case of the effective Skolem-Mahler-Lech problem, which remains unsolved in general: terrytao.wordpress.com/2007/05/25/… . It may be though that this special case has some additional structure that avoids the problems of the general case. Certainly the SML theorem tells us that the set of n here is eventually periodic, but there is no known effective algorithm (polynomial time or non-polynomial time) to describe it in general.
Nov
26
comment Small residue classes with small reciprocal
The paper linked to by Matt Young appears to have moved to nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-380.pdf (and is entitled "Arithmetic applications of Kloosterman sums", in case it moves again).
Nov
6
awarded  Great Answer
Nov
3
awarded  Nice Answer
Nov
3
comment Regularity of random Fourier series
Khintchine's inequality (plus an epsilon of Sobolev embedding) gives $C^{k-n/2-\varepsilon}$ (where we abbreviate $C^{k,\alpha}$ as $C^{k+\alpha}$) and this is basically optimal. I think this sort of analysis goes back to Paley and Zygmund: journals.cambridge.org/action/…
Nov
2
comment Simultaneous vanishing of convolutions of Mertens function with itself
I can see that Landau's theorem could be used to deduce $M_k$ changes sign infinitely often, but how could it be used to force it to be zero infinitely often?