Correlation of multiplicative functions over function fields

Abstract

In this article we study the asymptotic behaviour of the correlation functions over polynomial ring F q [x]. Let M n,q and P n,q be the set of all monic polynomials and monic irreducible polynomials of degree n over F q respectively. For multiplicative functions ψ 1 and ψ 2 on F q [x], we obtain asymptotic formula for the following correlation functions for a fixed q and n → ∞where h 1 , h 2 are fixed polynomials of degree < n over F q . As a consequence, for real valued additive functions ψ1 and ψ2 on F q [x] we show that for a fixed q and n → ∞, the following distribution functions{P ∈ P n,q : ψ1 (P + h 1 ) + ψ2 (P + h 2 ) ≤ x} converges weakly towards a limit distribution.PRANENDU DARBAR AND ANIRBAN MUKHOPADHYAY than q, which we call the large degree limit, or when q is much larger than n, which we call the large finite field limit.

Correlations of multiplicative functions in function fields

“…Now the claim follows from the Turán-Kubilius inequality for h (see [3,Lemma 7] for the function field version of this 9 ).…”

“…When working over F q [t], we have the generalized Riemann hypothesis at our disposal, arising from an application of Weil's Riemann hypothesis for curves over finite fields (see [31, p. 134]). 3 Lemma 3.1 (Rhin). Let N ≥ 1.…”

“…In[3], the Turán-Kubilius inequality was stated for the linear forms G → G + B, but the same proof works for any linear forms G → W G + B.…”

Correlations of multiplicative functions in function fields

“…Since 𝐻 ⩾ 𝐾 and 𝐾 is large, we may bound this from above using (53) and the prime polynomial theorem, obtaining by induction on 𝑟 we find that 𝐻 𝑟 ⩽ 𝐶 0 (𝑟 log 𝑟 + 𝑟 1 𝜀 ) for some absolute constant 𝐶 0 > 0 whenever 𝑟 ⩾ 𝐾 ⩾ 1000 log(1∕𝜀). Therefore, 𝜀 4 ∑ 𝐾⩽𝑟⩽exp(exp(𝐾∕3)) 1 10𝐶 0 𝑟 log 𝑟 ⩽ 𝐶 𝑘,𝑏 + 𝜀.…”

“…We apply the Taylor approximation Summing this expression over 𝐺 ∈  ⩽𝑁 , then applying the Cauchy-Schwarz inequality followed by the Turán-Kubilius inequality for ℎ (see [4,Lemma 7] for the function field version of this † ), † In [4], the Turán-Kubilius inequality was stated for the linear forms 𝐺 ↦ 𝐺 + 𝐵, but the same proof works for any linear forms 𝐺 ↦ 𝑊𝐺 + 𝐵. When 𝑀 is large enough, we thus have 𝑒 𝐴 ℎ (𝑀,𝑁) = 𝑒 𝑖ℑ 𝑓 (𝑀,𝑁) + 𝑂(𝔻(𝑓, 1; 𝑀, ∞) 2 + 𝑀 −1∕2 ), which, when combined with (64) yields the claim.…”