Duke Mathematical Journal
- Duke Math. J.
- Volume 164, Number 2 (2015), 277-295.
Prime polynomials in short intervals and in arithmetic progressions
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 is about . The second says that the number of primes in the arithmetic progression , for , is about , where is the Euler totient function.
More precisely, for short intervals we prove: Let be a fixed integer. Then
holds uniformly for all prime powers , degree monic polynomials and , where is either , or if , or if further and . Here , and denotes the number of prime polynomials in . We show that this estimation fails in the neglected cases.
For arithmetic progressions we prove: let be a fixed integer. Then
holds uniformly for all relatively prime polynomials satisfying , where is either or if and is a constant. Here is the number of degree prime polynomials and is the number of such polynomials in the arithmetic progression .
We also generalize these results to arbitrary factorization types.
Duke Math. J., Volume 164, Number 2 (2015), 277-295.
First available in Project Euclid: 30 January 2015
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Primary: 11T06: Polynomials
Bank, Efrat; Bary-Soroker, Lior; Rosenzweig, Lior. Prime polynomials in short intervals and in arithmetic progressions. Duke Math. J. 164 (2015), no. 2, 277--295. doi:10.1215/00127094-2856728. https://projecteuclid.org/euclid.dmj/1422627049