The function field Sathe--Selberg formula in arithmetic progressions and `short intervals'

Abstract

We use a function field analogue of a method of Selberg to derive an asymptotic formula for the number of (square-free) monic polynomials in Fq[X] of degree n with precisely k irreducible factors, in the limit as n tends to infinity. We then adapt this method to count such polynomials in arithmetic progressions and short intervals, and by making use of Weil's 'Riemann hypothesis' for curves over Fq, obtain better ranges for these formulae than are currently known for their analogues in the number field setting. Finally, we briefly discuss the regime in which q tends to infinity.

Uniform estimates for almost primes over finite fields

“…uniformly for 1 ≤ k ≤ (q − δ) log n. This is an analogue of (1.4); see also Car [Car82] and Afshar and Porritt [AP19]. Our Theorem 1.3 implies (1.6).…”

Sums of random multiplicative functions over function fields with few irreducible factors

“…We shall also need an asymptotic estimate for |P k (n)| that is valid when k = o(log n). This so-called Sathe-Selberg formula for function fields was established by Afshar and Porritt [1] in a strong form.…”

“…The strategy follows that of Harper [4] with appropriate modifications and simplifications as mentioned earlier in the introduction. First, notice P 1 (n) is the set of irreducible monic polynomials of degree n, so S (1) (n) is a sum of |P 1 (n)| independent random variables uniform on {±1}. Thus, the classical central limit theorem implies that S (1) (n) converges in distribution to N (0, 1) as n → ∞.…”

“…log n by assumption, this implies that k − k r ≤ k − 1 ≤1 3 log(n/2) ≤ 1 3 log(n − n r ). Thus, by the inductive hypothesis, the above is≪ r q 2n k−r+1 kr=1 ⌊n/r⌋ nr=1 |P kr (n r )| 2 − n r ) + c) 2k−2kr−2r+2 (n − n r ) 2 (k − k r − r + 1)!…”

On the size of primitive sets in function fields

“…We obtain results analogous to the results of Erdős, Besicovitch and Martin and Pomerance in this setting. In Corollary 2.4 we show that the lower density of a primitive set in the function field will also always be 0 by considering a sum analogous to (1). We then give a construction (Theorem 3.1) of a set with upper density arbitrarily close to q−1 q .…”