PRIME POLYNOMIALS IN SHORT INTERVALS AND IN ARITHMETIC PROGRESSIONS

By Efrat Bank, L. Bary‐Soroker, Lior Rosenzweig. 2013

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In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x x(is an element of)] is about x(is an element of)/logx. The second says that the number of primes p infinity holds uniformly for all prime powers q, degree k monic polynomials is an element of F-a[t] and is an element of(0)(f, q) infinity, holds uniformly for all relatively prime polynomials D, f is an element of F-q[t] satisfying parallel to D parallel to <= q(k)(1-(delta 0)) where delta(0) is either 3/k or 4/k if p = 2 and (f /D)' is a constant. Here pi(q)(k) is the number of degree k prime polynomials and pi(a)(k; D, f) is the number of such polynomials in the arithmetic progression P equivalent to f (mod D). We also generalize these results to arbitrary factorization types.

Subfields: Algebra and Number Theory, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1081718f7dee4c1eaff3503f3d496a4e5ec903e5

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Citations in the corpus

Squarefree polynomials and Mobius values in short intervals and arithmetic progressions (2015, methodology): “ill get that (3.8) holds for such ~a’s, with C = n−2. The argument in [6, Section 4] works verbatim here to prove that. An alternative argument is to use the work of Bank, Bary-Soroker and Rosenzweig [3] who prove equidistribution of cycle types of polynomials in any short interval I(A;h) for 2 ≤ h ”