TY - JOUR

T1 - UNIFORM ESTIMATES FOR ALMOST PRIMES OVER FINITE FIELDS

AU - Elboim, Dor

AU - Gorodetsky, Ofir

N1 - Publisher Copyright:
© 2022 American Mathematical Society

PY - 2022

Y1 - 2022

N2 - We establish a new asymptotic formula for the number of polynomials of degree n with k prime factors over a finite field Fq. The error term tends to 0 uniformly in n and in q. Previously, asymptotic formulas were known either for fixed q, through the works of Warlimont [Arch. Math. (Basel) 60 (1993), pp. 58-72] and Hwang [Random Structures Algorithms 13 (1998), pp. 17-47], or for small k, through the work of Arratia, Barbour and Tavaré [Math. Proc. Cambridge Philos. Soc. 114 (1993), pp. 347-368]. As an application, we estimate the total variation distance between the number of cycles in a random permutation on n elements and the number of prime factors of a random polynomial of degree n over Fq. The distance tends to 0 at rate 1/(q√log n). Previously this was only understood when either q is fixed and n tends to ∞, or n is fixed and q tends to ∞, by results of Arratia, Barbour and Tavaré.

AB - We establish a new asymptotic formula for the number of polynomials of degree n with k prime factors over a finite field Fq. The error term tends to 0 uniformly in n and in q. Previously, asymptotic formulas were known either for fixed q, through the works of Warlimont [Arch. Math. (Basel) 60 (1993), pp. 58-72] and Hwang [Random Structures Algorithms 13 (1998), pp. 17-47], or for small k, through the work of Arratia, Barbour and Tavaré [Math. Proc. Cambridge Philos. Soc. 114 (1993), pp. 347-368]. As an application, we estimate the total variation distance between the number of cycles in a random permutation on n elements and the number of prime factors of a random polynomial of degree n over Fq. The distance tends to 0 at rate 1/(q√log n). Previously this was only understood when either q is fixed and n tends to ∞, or n is fixed and q tends to ∞, by results of Arratia, Barbour and Tavaré.

UR - http://www.scopus.com/inward/record.url?scp=85130874846&partnerID=8YFLogxK

U2 - 10.1090/proc/15870

DO - 10.1090/proc/15870

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AN - SCOPUS:85130874846

SN - 0002-9939

VL - 150

SP - 2807

EP - 2822

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 7

ER -