函数域上双重酉除数函数的均值

The Average Value of Bi-Unitary Divisor Function in Function Fields

摘要: 设F_q为q元有限域，在函数域F_q(T)中，我们称首一多项式d为多项式f的酉因式，如果f=dδ且(d,δ)=1。若d同时为多项式f与g的酉因式，则称d是f与g的酉公因式。记(f,g)^∗∗为f与g的次数最大的首一酉公因式。我们称g是f的双重酉因式，如果f=gh且(g,h)^∗∗=1。令τ^∗∗(f)为f的双重酉因式的个数，本文研究了τ^∗∗(f)的均值，并给出了相应的渐近公式。

Abstract: Let F_q be the finite field with q elements. In the function field F_q(T), a monic divisor d of a polynomial f is called unitary, if f=dδ and (d,δ)=1. For polynomials f,g, if d is an unitary divisor of both f and g, it is called the common unitary divisor of them. Let (f,g)^∗∗ be the common unitary monic divisor of f and g, whose degree is the largest. We say a divisor g of f is bi-unitary, if f=gh and (g,h)^∗∗=1. Let τ∗∗(f) denote the number of bi-unitary divisor of f. We consider the average value of τ^∗∗(f) and give a corresponding asymptotic formula.