Correlation of multiplicative functions over function fields

In this article we study the asymptotic behaviour of the correlation functions over polynomial ring $\mathbb{F}_q[x]$. Let $\mathcal{M}_{n, q}$ and $\mathcal{P}_{n, q}$ be the set of all monic polynomials and monic irreducible polynomials of degree $n$ over $\mathbb{F}_q$ respectively. For multiplicative functions $\psi_1$ and $\psi_2$ on $\mathbb{F}_q[x]$, we obtain asymptotic formula for the following correlation functions for a fixed $q$ and $n\to \infty$ \begin{align*} &S_{2}(n, q):=\displaystyle\sum_{f\in \mathcal{M}_{n, q}}\psi_1(f+h_1) \psi_2(f+h_2), \\ &R_2(n, q):=\displaystyle\sum_{P\in \mathcal{P}_{n, q}}\psi_1(P+h_1)\psi_2(P+h_2), \end{align*} where $h_1, h_2$ are fixed polynomials of degree $<n$ over $\mathbb{F}_q$. As a consequence, for real valued additive functions $\tilde{\psi_1}$ and $\tilde{\psi_2}$ on $\mathbb{F}_q[x]$ we show that for a fixed $q$ and $n\to \infty$, the following distribution functions \begin{align*} &\frac{1}{|\mathcal{M}_{n, q}|}\Big|\{f\in \mathcal{M}_{n, q} : \tilde{\psi_1}(f+h_1)+\tilde{\psi_2}(f+h_2)\leq x\}\Big|,\\ & \frac{1}{|\mathcal{P}_{n, q}|}\Big|\{P\in \mathcal{P}_{n, q} : \tilde{\psi_1}(P+h_1)+\tilde{\psi_2}(P+h_2)\leq x\}\Big| \end{align*} converges weakly towards a limit distribution.Comment: 24 pages; Comments are welcom

A Note about Iterated Arithmetic Functions

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $\alpha$, $f(p^{\alpha})<p^{\alpha}$ and $f(p)\vert f(p^{\alpha})$. Suppose also that any prime that divides $f(p^{\alpha})$ also divides $pf(p)$. Define $f(0)=0$, and let $H(n)=\displaystyle{\lim_{m\rightarrow\infty}f^m(n)}$, where $f^m$ denotes the $m^{th}$ iterate of $f$. We prove that the function $H$ is completely multiplicative.Comment: 5 pages, 0 figure

On the existence of primitive completely normal bases of finite fields

Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. We prove that there exists a primitive element of $\mathbb{F}_{q^n}$ that produces a completely normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, provided that $n=p^{\ell}m$ with $(m,p)=1$ and $q>m$

The arithmetic derivative and Leibniz-additive functions

An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$f(mn)=f(m)h_f(n)+f(n)h_f(m)$$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative $D$; namely, $D$ is Leibniz-additive with $h_D(n)=n$. In this paper, we study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function $f$ is totally determined by the values of $f$ and $h_f$ at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions

Character Sums, Gaussian Hypergeometric Series, and a Family of Hyperelliptic Curves

We study the character sums $$\phi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } \psi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left((x^{m}+a)(x^{n}+b)\right)$$ where $\phi$ is the quadratic character defined over $\mathbb{F}_q$. These sums are expressed in terms of Gaussian hypergeometric series over $\mathbb{F}_q$. Then we use these expressions to exhibit the number of $\mathbb{F}_q$-rational points on families of hyperelliptic curves and their Jacobian varieties