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Prime Number Races

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arxiv math/0408319 v1 pith:UNTCT4ZM submitted 2004-08-24 math.NT

classification math.NT
keywords formprimesbeenmadenumberothersprimeraces
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This is a survey article on prime number races. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less than x then there are of the form 4n+1. Similar observations have been made with primes of the form 3n+2 and 3n+1, with primes of the form 10n+3/10n+7 and 10n+1/10n+9, and many others besides. More generally, one can consider primes of the form qn+a, qn+b, qn+c, >... for our favorite constants q, a, b, c, ... and try to figure out which forms are "preferred" over the others. In this paper, we describe these phenomena in greater detail and explain the efforts that have been made at understanding them.

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  1. Unimodular Fake Mobius Functions

    math.NT 2025-12 conditional novelty 7.0 of 10

    For multiplicative functions generated by a fixed sequence of prime-power values, an explicit formula extracts the critical-line contributions of ζ(s)^z ζ(2s)^w and gives a criterion for persistent, apparent, or absent bias.

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