Prime polynomial values of linear functions in short intervals

Abstract

In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. We prove an asymptotic formula for the number of simultaneous prime polynomial values of n linear functions, in the limit of a large finite field.

Sums of two squares in short intervals in polynomial rings over finite fields

“…The proof of Theorem 1.1 follows the pattern of similar proofs in the literature, like in [ABR15, BB15,BBR15,Ent14]. The main ingredient is an explicit Chebotarev theorem, which we recall now.…”

Prime polynomial values of quadratic functions in short intervals

“…Hence, the Galois group G = S n 1 × ... × S nr . Rest of the proof follows from [Theorem 3.1, [5]]. Thus…”

“…Bank, Bary-Soroker and Rosenzweig [6] obtained the result on counting prime polynomials in the short interval I(A, h) for the primitive linear function f (t) + g(t)x. In [5] the function field analogue of Hardy -Littlewood prime tuple conjecture on these primitive linear functions is resolved in short interval case.…”

The Chebotarev density theorem for function fields -- incomplete intervals

“…The distribution of primes, and more generally the distribution of factorization types, in "short intervals" in the setting of function fields over finite fields has received considerable attention [5,6,3,2,13]. For example, in [3], prime equidistribution for the family {f (x) + bx + a} a,b∈Fp was shown for f ∈ F p [x] any monic degree d polynomial (for p large.)…”

“…types of f (x) + a, for f Morse, is consistent (up to an error of size O d (p −1/2 )) with the distribution of cycle types of permutations in S d , the symmetric group on d letters, with respect to the Haar measure. (E.g., for σ = (12) ∈ S 3 , write out all trivial cycles, i.e., σ = (12)(3); the cycle type of σ is then (1,2) if we order according to cycle lengths. )…”