Two prime races followed at every integer up to a hundred million

Primes other than $2$ fall into two classes modulo $4$, congruent to $1$ or to $3$, equally common in the long run, yet the class of is almost always ahead: Chebyshev's bias. Following the race exactly at every integer , the class of first leads at x = , and leads at only of the hundred million integers, in stretches, none after x = . Weighting each by , the class of leads on a share of the race, about a tenth of the limiting share Rubinstein and Sarnak (1994) computed under two standard hypotheses, so the race is still short of its limiting behavior this far out. Modulo , primes congruent to lead those congruent to strictly at every from to , consistent with the first reversal Bays and Hudson (1978) found near .