On elementary estimates of arithmetic sums for polynomial rings over finite fields

Abstract

In this paper, a simple and elementary method is given for deriving estimates of sums of arithmetic functions in Fq[t]. The method is the function field analogue of a result first proved by Stefan A. Burr in 1973 in the number field case. A novelty of this paper is that we are able to extend Burr's result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions dr(f ) with an error term that improves the one given by Burr.

Correlation of multiplicative functions over Fq[x]$\mathbb {F}_q[x]$: A pretentious approach

“…Let 𝑓 ∶ ℕ → 𝕌 be multiplicative. Then 𝑀 𝑓 (𝑥) = 𝑜 (1) unless there exists 𝑡 ∈ ℝ such that 𝔻(𝑓, 𝑛 𝑖𝑡 , ∞) < ∞ in which case, as 𝑥 → ∞, we have…”

“…If g 𝑗 = 𝜆, Liouville's function and 𝐹 𝑗 (𝑥) = 𝑥 + ℎ 𝑗 , 𝑗 = 1, 2, … , 𝑘 for distinct natural numbers ℎ 𝑗 's, then a famous conjecture of Chowla asserts that 𝑀 𝑘 (𝑥) = 𝑜 (1) as 𝑥 → ∞. Chowla's conjecture remains open for any ℎ 1 , … , ℎ 𝑘 with 𝑘 ⩾ 2.…”

“…Kátai [14] was the first to study the asymptotic behavior of the sum (1) for some specific g 𝑗 's and 𝐹 𝑗 (𝑥)'s, but did not provide error terms. Stepanauskas [29] obtained asymptotic formulas for sum (1) with explicit error terms when 𝐹 𝑗 (𝑥)'s are linear polynomials and g 𝑗 are "close to 1" (which is much stronger condition than "pretend to be 1"). In [6], the first author studied the asymptotic behavior of the sum (1) with explicit error term when 𝐹 𝑗 's are polynomials of degree ⩾ 2 and g 𝑗 's are close to 1.…”

Correlation of multiplicative functions over function fields

“…. ψ k (P + h k ) (2) as the parameter q n = |M n,q | is large (and n > deg(h i ) for all i to avoid technical difficulties). This parameter can be large, in particular, either when n is much larger In the large degree limit i.e.…”

“…In this paper, we will investigate the asymptotic behaviour of the above sums (1) and (2) for k = 2 i.e. the following 2-point correlation functions in large degree limit S 2 (n, q) := f ∈Mn,q…”