标题： 算术函数与对称多项式的卷积环 显示英文标题

标题： The Convolution Ring of Arithmetic Functions and Symmetric Polynomials

摘要： 受Rearick（1968）的启发，我们引入了两个新的算子，LOG和EXP。 LOG作用于广义斐波那契多项式，得到广义卢卡斯多项式。 EXP是LOG的逆运算。 具体来说，LOG将广义斐波那契多项式的卷积乘积转换为广义卢卡斯多项式的和，而EXP则将和转换为卷积乘积。 我们利用这种结构，在算术函数中建立对数和指数的理论，从而给出另一个证明，说明在卷积乘积下的乘法函数群与在加法下的加法函数群是同构的。 双曲三角函数由EXP算子构造，同样以通常的方式进行。 通常的双曲三角恒等式仍然成立。 我们在等压环中展示了新的结构和恒等式。 给定一个首一多项式，其无限伴随矩阵可以嵌入到加权等压多项式群中。 首一多项式及其伴随矩阵的导数给我们不同的矩阵和无限不同的矩阵。 不同矩阵的行列式是首一多项式的判别式，符号除外。 事实上，作用于无限伴随矩阵的LOG就是无限不同的矩阵。 我们证明了一个算术函数是局部可表示的当且仅当它是一个乘法函数。 如果一个算术函数是平凡地全局表示的，那么它既是局部又是全局可表示的。 显示英文摘要

摘要： Inspired by Rearick (1968), we introduce two new operators, LOG and EXP. The LOG operates on generalized Fibonacci polynomials giving generalized Lucas polynomials. The EXP is the inverse of LOG. In particular, LOG takes a convolution product of generalized Fibonacci polynomials to a sum of generalized Lucas polynomials and EXP takes the sum to the convolution product. We use this structure to produce a theory of logarithms and exponentials within arithmetic functions giving another proof of the fact that the group of multiplicative functions under convolution product is isomorphic to the group of additive functions under addition. The hyperbolic trigonometric functions are constructed from the EXP operator, again, in the usual way. The usual hyperbolic trigonometric identities hold. We exhibit new structure and identities in the isobaric ring. Given a monic polynomial, its infinite companion matrix can be embedded in the group of weighted isobaric polynomials. The derivative of the monic polynomial and its companion matrix give us the different matrix and the infinite different matrix. The determinant of the different matrix is the discriminate of the monic polynomial up to sign. In fact, the LOG operating on the infinite companion matrix is the infinite different matrix. We prove that an arithmetic function is locally representable if an only if it is a multiplicative function. An arithmetic function is both locally and globally representable if it is trivially globally represented.