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]

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I am a professor of mathematics at ETH Zürich since 2008.

One thought on “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=’http://math.etsu.edu/LaTeXMathML/LaTeXMathML.js’ type=’text/javascript’/>
    <link href=’http://math.etsu.edu/LaTeXMathML/LaTeXMathML.standardarticle.css’ 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 http://math.etsu.edu/LaTeXMathML/ (i could not do this because it is disallowed by blogspot.com).
    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!

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