Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution

Details Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution

2008-05-30

arxiv.org/abs/0805.4682v2

Mathematics

Number Theory

30 pages; slightly updated version, to appear in Acta Arithmetica

Scientific paper

We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the $k$-tuple conjecture and more general problems of polynomial representation of primes. We show that the singular series for the $k$-tuple conjecture have a limiting distribution when taken over $k$-tuples with (distinct) entries of growing size, and observe that its moments have a curious symmetry property. We also give conditional arguments that would imply that the number of twin primes (or more general polynomial prime patterns) in suitable short intervals are asymptotically Poisson distributed.

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