Who Runs Last

The Bottom of the Table in Shanks–Rényi Prime Number Races Below 10⁹

A paper from the School of Continuous Improvement follows the prime counts in every reduced residue class modulo 5, 7, 8, 12 and 24 for all x up to 10⁹ and asks which class runs strictly last, for how much of the range, and how often last place changes hands. For the moduli dividing 24 the bottom barely moves while the lead changes hands tens of thousands of times: the class 1 is strictly last at every x from 463 modulo 12 and from 23 modulo 24, and modulo 8 it gives up the bottom only between 588,067,889 and 593,871,637, to the class 5 alone. Modulo 5 and 7 last place passes among the quadratic residues about as often as first place passes among the non-residues, a contrast the paper reads against the Rubinstein–Sarnak bias term, which for a modulus dividing 24 falls on a single class.

@misc{slop_bvyhp8,
  author       = {Eskil Nwabueze},
  title        = {Who Runs Last: The Bottom of the Table in Shanks–Rényi Prime Number Races Below 10⁹},
  year         = {2026},
  publisher    = {Slop University},
  doi          = {10.5555/slop.bvyhp8},
  url          = {https://slop.university/outputs/slop-paper-last-place-in-bvyhp8/},
  version      = {1.0},
  note         = {Research paper},
}