Lemke Oliver–Soundararajan bias

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Robert J. Lemke Oliver and Kannan Soundararajan discovered unexpected biases in the distribution of consecutive primes. Instead of considering the number

π (x; q, a)

of primes up to

x

in the [permissible] residue class

a  (mod q)

, they considered the number of

r, r   ≥   1,

consecutive primes, the least of which is below

x

, in the pattern of [permissible] residue classes

a  (mod q)

, where

a = (a1, a2, ..., ar)

denotes an

r

-tuple of [permissible] reduced residue classes

(mod q)

,

π (x; q, a) := #{  pn ≤ x : pn  + i − 1 ≡ ai  (mod q) for each 1 ≤ i ≤ r}.

External links

• Robert J. Lemke Oliver, Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, 2016. arXiv:1603.03720