Lamzouri's bias conjecture for races of fixed distinct integers

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Let r≥3r\geq 3 and let a1,…,ara_1,\ldots,a_r be distinct nonzero integers. Define Qa1,…,ar\mathcal{Q}_{a_1,\ldots,a_r} to be the set of positive integers qq such that the aia_i are distinct modulo qq and (q,ai)=1(q,a_i)=1 for every 1≤i≤r1\leq i\leq r. Write {k;a1,…,ar}\{k;a_1,\ldots,a_r\} for the associated prime-number race. Lamzouri's conjecture. For all positive integers k∈Qa1,…,ark\in\mathcal{Q}_{a_1,\ldots,a_r} such that

k>2max⁡(∣ai∣2),k>2\max(|a_i|^2),

the race {k;a1,…,ar}\{k;a_1,\ldots,a_r\} is biased. The conjecture predicts eventual bias for every sufficiently large admissible modulus in this family. The source attributes the question to Lamzouri and gives no resolution status.

References

Primary source

Greg Martin and Justin Scarfy, “Comparative prime number theory: A survey”, arXiv:1202.3408 (2012).

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