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.

Abstract. In this paper, we find asymptotic formula for the following sum with explicit error term:

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In this article, we study the logarithm of the central value L 1 2 , χ D in the symplectic family of Dirichlet L-functions associated with the hyperelliptic curve of genus δ over a fixed finite field F q in the limit as δ → ∞. Unconditionally, we show that the distribution of log L 1 2 , χ D is asymptotically bounded above by the Gaussian distribution of mean 1 2 log deg(D) and variance log deg (D). Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.