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PRIME POLYNOMIALS IN SHORT INTERVALS AND IN ARITHMETIC PROGRESSIONS
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-4602-8362
2015 (English)In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 164, no 2, p. 277-295Article in journal (Refereed) Published
##### Abstract [en]

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 < x in the arithmetic progression p equivalent to a (mod d), for d < x(1-delta), is about pi(x)/phi(d), where phi is the Euler totient function. More precisely, for short intervals we prove: Let k be a fixed integer Then pi(q)(I(f,is an element of)) similar to #I(f,is an element of)/k, q -> 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) <= is an element of, where is an element of(0) is either 1/k, or 2/k if p vertical bar k(k-1), or 3/k if further p = 2 and deg f ' <= 1. Here I (f, is an element of) = {g is an element of F-q[t] vertical bar deg( f - g) <= is an element of deg f }, and pi(q)(I(f, is an element of)) denotes the number of prime polynomials in I (f, 6). We show that this estimation fails in the neglected cases. For arithmetic progressions we prove: let k be a fixed integer. Then pi(q)(k; D,f) similar to pi(q)(k)/phi(D), 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.

##### Place, publisher, year, edition, pages
2015. Vol. 164, no 2, p. 277-295
Mathematics
##### Identifiers
ISI: 000349513200003Scopus ID: 2-s2.0-84922496038OAI: oai:DiVA.org:kth-161612DiVA, id: diva2:797919
##### Note

QC 20150325

Available from: 2015-03-25 Created: 2015-03-13 Last updated: 2017-12-04Bibliographically approved

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Rosenzweig, Lior

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