从1976年到2017年，Wintner，Delange，Ushiroya和Tóth逐步证明了定义在整数环上的多元算术函数都可以通过Ramanujan和加以展开.这类似于经典分析中周期函数的Fourier展开.本文主要研究了有限域上一元多项式环上Ramanujan和的性质，并证明了定义在上的多元算术函数也可以通过多项式Ramanujan和以及酉多项式Ramanujan和加以展开.
Abstract
Combing Wintner, Delange, Ushiroya and Tóth's works from 1976 to 2017, we have that the multi-variable arithmetic functions defined on integer ring can be expanded through the Ramanujan sums. This is an analogue of the Fourier expansion for periodic functions in the classical analysis. In this paper we further investigate the properties of Ramanujan sums in the polynomial ring , and show that the multi-variable arithmetic functions defined on can also be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums.
关键词
算术函数 /
Ramanujan和 /
多项式环 /
有限域 /
Zeta函数
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Key words
arithmetic function /
Ramanujan sum /
polynomial ring /
finite fields /
Zeta function
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参考文献
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[9] Tóth L., Ramanujan expansions of arithmetic functions of several variables, Ramanujan J., 2018, 47(3):589-603.
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[12] Zheng Z. Y., On the polynomial Ramanujan sums over finite fields, Ramanujan J., 2018, 46(3):863-898.
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脚注
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