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What is the best known quantitative upper bound for the quantity $G(k)$?

I know that it's due to Trevor Wooley and in simplest form states that $\limsup_{k \to \infty} \frac{G(k)}{k \log k} \le 1$.

I have been digging mathsci.net and I'm not able to find it.

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At Trevor's website, I see $$G(k)\le k(\log k+\log\log k+2+o(1))$$

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    $\begingroup$ That talk is quite a few years old now; Trevor has made a lot of progress on such problems recently using efficient congruencing and I would not be surprised if better bounds were known. Certainly I recommend to the OP to look at the papers on Trevor's webpage. $\endgroup$ – Daniel Loughran Dec 9 '14 at 8:37

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