Ramanujan Expansions of Arithmetic Functions of Several Variables over Fq[T]

Tian Fang QI

Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (5) : 891-906.

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Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (5) : 891-906. DOI: 10.12386/A20210053

Ramanujan Expansions of Arithmetic Functions of Several Variables over Fq[T]

  • Tian Fang QI
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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 Fq[T], and show that the multi-variable arithmetic functions defined on Fq[T] can also be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums.

Key words

arithmetic function / Ramanujan sum / polynomial ring / finite fields / Zeta function

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Tian Fang QI. Ramanujan Expansions of Arithmetic Functions of Several Variables over Fq[T]. Acta Mathematica Sinica, Chinese Series, 2022, 65(5): 891-906 https://doi.org/10.12386/A20210053

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