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Averages of singular series, or: when Poisson is everywhere

Here is another post where the mediocre mathematical abilities of HTML will require inserting images with some TeX-produced text…

I have recently posted on my web page a preprint concerning some averages of “singular series” (another example of pretty bad mathematical terminology…) arising in the prime k-tuple conjecture, and its generalization the Bateman-Horn conjecture. The reason for looking at this is a result of Gallagher which is important in the original version of the proof by Goldston-Pintz-Yildirim that there are infinitely many primes p for which the gap q-p between p and the next prime q is smaller than ε times the average gap, for arbitrary small ε>0.

This result refers to the behavior, on average over h=(h_1,…,h_k), of the constant S(h) which is supposed to be the leading coefficient in the conjecture

|{n<X | n+h_i is prime for i=1,…,k}|~S(h) X(log X)-k

Gallagher showed that the average value of S(h) is equal to 1, and I’ve extended this in two ways…

[LaTeX 1]

[LaTex 2]

2 Responses to “Averages of singular series, or: when Poisson is everywhere”

  1. Dear Emmanuel, an almost-solution to the latex problem is including the following HTML

    <script src=’’ type=’text/javascript’/>
    <link href=’’ rel=’stylesheet’ type=’text/css’/>

    in the template (possibly in the head), this will automatically load a javascript file that will automatically convert $\latex$ expressions to MathML. Or maybe even better you may put the .js and .css file on your site instead that directly loading them from (i could not do this because it is disallowed by
    Of course this is not a complete solution, and the script does not support all latex expressions, but at least you won’t have to include a pgn image just for a formula.
    Best regards!


    Sunday, March 23, 2008 at 21:53 | Permalink

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