bio  website  math.ucla.edu/~tao 

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Professor of Mathematics at UCLA
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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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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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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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Zeta function double product
Any particular reason why this product is being considered? 
Dec 11 
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A sumofdeterminants 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 
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CauchySchwarz proof of Sidorenko for 3edge path (BlakleyRoy inequality)
Lovasz may be referring to this article of Carbery: ams.org/mathscinetgetitem?mr=2073287 , reproving an inequality similar to the above of Katz and myself, ams.org/mathscinetgetitem?mr=1739220 , proven via the pigeonhole principle and the tensor power trick. 
Dec 5 
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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 
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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/mathscinetgetitem?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 BlondelPortier obstruction applies here. Nevertheless this is reason for pessimism regarding a polynomial time algorithm here. 
Nov 28 
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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 SkolemMahlerLech 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 nonpolynomial time) to describe it in general. 
Nov 26 
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Small residue classes with small reciprocal
The paper linked to by Matt Young appears to have moved to nieuwarchief.nl/serie5/pdf/naw52000014380.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 
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Regularity of random Fourier series
Khintchine's inequality (plus an epsilon of Sobolev embedding) gives $C^{kn/2\varepsilon}$ (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 
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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? 