## 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…

## One Response to “Averages of singular series, or: when Poisson is everywhere”

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!