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Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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Tristan Freiberg, Department of Pure Mathematics, University of Waterloo
"Biases in the distribution of primes in intervals"
Cramér's model predicts that the primes are distributed among the positive integers as if according to a Poisson process, in the sense that, for any fixed real number \eta > 0 and integer m \ge 0, the proportion of positive integers n \le N for which the interval (n, n + \eta \log N] contains exactly m primes tends to e^{\eta}\eta^m/m! as N \to \infty. Gallagher showed that this prediction is consistent with a certain form of the HardyLittlewood prime tuples conjecture. We might therefore expect to see, when \eta is an integer (and the corresponding Poisson distribution is bimodal), more or less the same proportion of intervals containing \eta primes as \eta  1.
Contrary to this expectation, we observe in the actual numerical data that, in fact, the proportion of intervals containing \eta primes is slightly higher. This suggests that there ought to be more precise predictions, with secondary terms that account for this bias towards the mean. The purpose of this talk is to offer such predictions, and show how they would follow from certain versions of the HardyLittlewood prime tuples conjecture.
MC 5417
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Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca