## Abstract

In this paper we establish function field versions of two classical conjectures on rime numbers. The first says that the number of primes in intervals (x, x + ^{ε}) is about x^{ε}/log x. The second says that the number f primes p < x in the arithmetic progression p = a (mod d), for d <x^{1-δ}, is about π(x)/ϕ(d), where ϕ is the Euler totient function. holds uniformly for all prime powers q, degree k monic polynomials f ε F_{q}t and ε_{0}(f, q)≤ ε, where ε_{0} is either 1/k, or 2/k if p k(k - 1), or 3/k if further p = 2 and deg f' ≤ 1. Here I(f,ε)= {g ε F_{q} t | deg(f - g) ≤ ε deg f}, and π_{q}(I(f, ε)) denotes the number of prime polynomials in I(f,ε). We show that this estimation ails in the neglected cases. holds uniformly for all relatively prime polynomials D, f ε F_{q}t satisfying ||D||≤ q^{k(1-δ0)}, where δ_{0} is either 3/k or 4/k if p = 2 and (f/D)' is a constant. Here π_{q}(k) is the number of degree k prime polynomials and π_{q}(k;D, f) is the number of such polynomials in the arithmetic progression P= f (mod D). We also generalize these results to arbitrary factorization types.

Original language | English |
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Pages (from-to) | 277-295 |

Number of pages | 19 |

Journal | Duke Mathematical Journal |

Volume | 164 |

Issue number | 2 |

DOIs | |

State | Published - 2015 |

## All Science Journal Classification (ASJC) codes

- General Mathematics