Geodesic
Periodic multiplicative arithmetic functions
Prikladnaâ diskretnaâ matematika, no. 4 (2024), pp. 30-35

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The notions of a periodic multiplicative function, the main modulus of such function, and the simplest periodic multiplicative function have been introduced. The basic properties of periodic multiplicative functions are studied, and a complete description of such functions through Dirichlet characters is given. In particular, it has been proven that any periodic multiplicative function other than unitary can be uniquely represented as a product of the simplest periodic multiplicative functions, and the principal modules of such functions are powers of prime numbers, the product of which is the canonical decomposition of the principal module of the original function. Based on this representation, the study of periodic multiplicative functions is reduced to the study of the simplest periodic multiplicative functions. The obtained results lead to a complete description of periodic multiplicative functions.
Keywords: arithmetic function, periodic multiplicative function, Dirichlet character, Dirichlet $L$-function. 
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     author = {E. V. Kaygorodov},
     title = {Periodic multiplicative arithmetic functions},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {30--35},
     publisher = {mathdoc},
     number = {4},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2024_4_a2/}
}
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E. V. Kaygorodov. Periodic multiplicative arithmetic functions. Prikladnaâ diskretnaâ matematika, no. 4 (2024), pp. 30-35. http://geodesic.mathdoc.fr/item/PDM_2024_4_a2/