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.
"Prime polynomials in short intervals and in arithmetic progressions." Duke Math. J. 164 (2) 277 - 295, 1 February 2015. https://doi.org/10.1215/00127094-2856728