TY - JOUR
T1 - Extreme biases in prime number races with many contestants
AU - Ford, Kevin
AU - Harper, Adam J.
AU - Lamzouri, Youness
N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - We continue to investigate the race between prime numbers in many residue classes modulo q, assuming the standard conjectures GRH and LI. We show that provided n/ log q→ ∞ as q→ ∞, we can find n competitor classes modulo q so that the corresponding n-way prime number race is extremely biased. This improves on the previous range n⩾ φ(q) ϵ, and (together with an existing result of Harper and Lamzouri) establishes that the transition from all n-way races being asymptotically unbiased, to biased races existing, occurs when n= (log q) 1+o(1). The proofs involve finding biases in certain auxiliary races that are easier to analyse than a full n-way race. An important ingredient is a quantitative, moderate deviation, multi-dimensional Gaussian approximation theorem, which we prove using a Lindeberg type method.
AB - We continue to investigate the race between prime numbers in many residue classes modulo q, assuming the standard conjectures GRH and LI. We show that provided n/ log q→ ∞ as q→ ∞, we can find n competitor classes modulo q so that the corresponding n-way prime number race is extremely biased. This improves on the previous range n⩾ φ(q) ϵ, and (together with an existing result of Harper and Lamzouri) establishes that the transition from all n-way races being asymptotically unbiased, to biased races existing, occurs when n= (log q) 1+o(1). The proofs involve finding biases in certain auxiliary races that are easier to analyse than a full n-way race. An important ingredient is a quantitative, moderate deviation, multi-dimensional Gaussian approximation theorem, which we prove using a Lindeberg type method.
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U2 - 10.1007/s00208-019-01810-x
DO - 10.1007/s00208-019-01810-x
M3 - Article
AN - SCOPUS:85062705057
SN - 0025-5831
VL - 374
SP - 517
EP - 551
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 1-2
ER -