In this paper, we investigate the multiplicative structure of the ring of complex valued arithmetic functions defined on the ring of Gaussian integers, and show that this ring is a unique factorization domain.

Gaussian integers, arithmetic functions, factorization.

Received: July 15, 2024; Accepted: September 25, 2024; Published: March 7, 2025

How to cite this article: Jaki Chowdhury, Factorization in the ring of arithmetic functions on Gaussian integers, JP Journal of Algebra, Number Theory and Applications 64(2) (2025), 117-132. https://doi.org/10.17654/0972555525007

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] E. Alkan, A. Zaharescu and M. Zaki, Arithmetical functions in several variables, Int. J. Number Theory 1(3) (2005), 383-399.
[2] E. Alkan, A. Zaharescu and M. Zaki, Unitary convolution for arithmetical functions in several variables, Hiroshima Math. J. 36(1) (2006), 113-124.
[3] E. Alkan, A. Zaharescu and M. Zaki, Multidimentional averages and Dirichlet convolution, Manuscripta Math. 123 (2007), 251-267.
[4] D. Bhattacharjee, On some rings of arithmetical functions, Georgian Math. J. 6(4) (1999), 299-306.
[5] T. B. Carroll and A. A. Gioia, On a subgroup of the group of multiplicative arithmetic functions, J. Austral. Math. Soc. Ser. A 20(3) (1975), 348-358.
[6] E. D. Cashwell and C. J. Everett, The ring of number-theoretic functions, Pacific J. Math. 9 (1959), 975-985.
[7] P. Haukkanen, Comparison of regular convolutions, Portugal. Math. 46(1) (1989), 59-69.
[8] T. MacHenry, A subgroup of the group of units in the ring of arithmetic functions, Rocky Mountain J. Math. 29(3) (1999), 1055-1065.
[9] R. Migliorato, Prime elements in the ring of arithmetical functions, Atti Soc. Peloritana Sci. Fis. Mat. Natur. 25(3-4) (1979), 255-283 (in Italian).
[10] W. Narkiewicz, On a class of arithmetical convolutions, Colloq. Math. 10 (1963), 81-94.
[11] W. Narkiewicz, Some unsolved problems, Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), pp. 159-164, Bull. Soc. Math. France, Mem. No. 25, Soc. Math. France, Paris, 1971.
[12] F. Pellegrino, Operatori lineari dell’anello delle funzioni aritmetiche, Rend. Mat. e Appl. (5) 25 (1966), 308-332 (in Italian).
[13] H. Scheid, Über ordnungstheoretische Funktionen, J. Reine Angew. Math. 238 (1969), 1-13.
[14] H. Scheid, Funktionen über lokal endlichen Halbordnungen, I, Monatsh. Math. 74 (1970), 336-347.
[15] H. Scheid, Funktionen über lokal endlichen Halbordnungen, II, Monatsh. Math. 75 (1971), 44-56.
[16] A. Schinzel, A property of the unitary convolution, Colloq. Math. 78(1) (1998), 93-96.
[17] E. D. Schwab, Möbius categories as reduced standard division categories of combinatorial inverse monoids, Semigroup Forum 69(1) (2004), 30-40.
[18] E. D. Schwab, The Möbius category of some combinatorial inverse semigroups, Semigroup Forum 69(1) (2004), 41-50.
[19] E. D. Schwab and G. Silberberg, A note on some discrete valuation rings of arithmetical functions, Arch. Math. (Brno) 36 (2000), 103-109.
[20] E. D. Schwab and G. Silberberg, The valuated ring of the Arithmetical functions as a power series ring, Arch. Math. (Brno) 37 (2001), 77-80.
[21] V. Sitaramaiah, Arithmetical sums in regular convolutions, J. Reine Angew. Math. 303/304 (1978), 265-283.
[22] J. Snellman, Truncations of the ring of arithmetical functions with unitary convolution, Int. J. Math. Game Theory Algebra 13(6) (2003), 485-519.
[23] T. Steinberger, A. Zaharescu and M. Zaki, Arithmetical functions in several variables, Int. J. Number Theory 9(8) (2013), 1923-1932.
[24] B. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212-216.
[25] K. L. Yokom, Totally multiplicative functions in regular convolution rings, Canad. Math. Bull. 16 (1973), 119-128.
[26] A. Zaharescu and M. Zaki, Derivations and generating degrees in the ring of arithmetical functions, Proc. Indian Acad. Sci. Math. Sci. 117(2) (2007), 167-175.
[27] A. Zaharescu and M. Zaki, Factorization in certain rings of arithmetical functions, Kumamoto J. Math. 21 (2008), 29-39.
[28] A. Zaharescu and M. Zaki, An algebraic independence result for Euler products of finite degree, Proc. Amer. Math. Soc. 137(4) (2009), 1275-1283.
[29] A. Zaharescu and M. Zaki, Valuations on the ring of arithmetical functions, Acta Math. Univ. Comenian. (N.S.) 78(1) (2009), 115-119.
[30] A. Zaharescu and M. Zaki, An ABC analog for arithmetical functions, J. Ramanujan Math. Soc. 25(4) (2010), 345-354.