Mathematica Bohemica, first online, pp. 1-18


Semimultiplicative generalized arithmetical functions

Pentti Haukkanen

Received July 29, 2024.   Published online January 21, 2025.

Abstract:  By a generalized arithmetical function we mean a function from the set of positive integers to a ring with identity, and we say that a generalized arithmetical function $f$ is semimultiplicative if $f(n) = c_f f_M(n/a_f)$, where $c_f$ is a unit in the ring, $a_f$ is a positive integer and $f_M$ is a multiplicative generalized arithmetical function. We study basic properties of these functions, connections to Selberg multiplicative functions and to the Dirichlet convolution. Particular attention is paid to the commutativity and noncommutativity of the function values.
Keywords:  generalized arithmetical function; semimultiplicative function; Selberg multiplicative function; Dirichlet convolution
Classification MSC:  11A25

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References:
[1] E. Alkan, A. Zaharescu, M. Zaki: Arithmetical functions in several variables. Int. J. Number Theory 1 (2005), 383-399. DOI 10.1142/S179304210500025X | MR 2175098 | Zbl 1088.11005
[2] E. Alkan, A. Zaharescu, M. Zaki: Unitary convolution for arithmetical functions in several variables. Hiroshima Math. J. 36 (2006), 113-124. DOI 10.32917/hmj/1147883399 | MR 2213646 | Zbl 1101.13035
[3] T. M. Apostol: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, New York (1976). DOI 10.1007/978-1-4757-5579-4 | MR 0434929 | Zbl 0335.10001
[4] F. Bouzeffour, W. Jedidi, M. Garayev: Extended arithmetic functions. Ramanujan J. 51 (2020), 593-609. DOI 10.1007/s11139-018-0122-8 | MR 4076173 | Zbl 1444.11011
[5] P. Bundschuh, L. C. Hsu, P. J.-S. Shiue: Generalized Möbius inversion - theoretical and computational aspects. Fibonacci Q. 44 (2006), 109-116. MR 2243776 | Zbl 1189.11003
[6] M. R. Chawdhury: On the Möbius inversion formula. Punjab Univ. J. Math. 3 (1970), 29-34. MR 0271007
[7] H. Delange: On the integral-valued additive functions. J. Number Theory 1 (1969), 419-430. DOI 10.1016/0022-314X(69)90004-3 | MR 0248105 | Zbl 0209.34902
[8] J. Elliott: Ring structures on groups of arithmetic functions. J. Number Theory 128 (2008), 709-730. DOI 10.1016/j.jnt.2007.07.011 | MR 2400035 | Zbl 1204.11010
[9] M. Ferrero: On generalized convolution rings of arithmetic functions. Tsukuba J. Math. 4 (1980), 161-176. DOI 10.21099/tkbjm/1496159171 | MR 0623434 | Zbl 0468.10003
[10] P. Haukkanen: Classical arithmetical identities involving a generalization of Ramanujan's sum. Ann. Acad. Sci. Fenn., Ser. A I, Diss. 68 (1988), 1-69. MR 0964709 | Zbl 0651.10005
[11] P. Haukkanen: Extensions of the class of multiplicative functions. East-West J. Math. 14 (2012), 101-113. MR 3076471 | Zbl 1315.11006
[12] P. Haukkanen: On the Kesava Menon norm of semimultiplicative functions. Aequationes Math. 94 (2020), 71-81. DOI 10.1007/s00010-019-00660-x | MR 4060472 | Zbl 1446.11008
[13] P. Haukkanen, R. Sivaramakrishnan: Arithmetic functions in an algebraic setting. Tsukuba J. Math. 15 (1991), 227-234. DOI 10.21099/tkbjm/1496161585 | MR 1118600 | Zbl 0741.11005
[14] P. Haukkanen, L. Tóth: An analogue of Ramanujan's sum with respect to regular integers (mod r). Ramanujan J. 27 (2012), 71-88. DOI 10.1007/s11139-011-9327-9 | MR 2886490 | Zbl 1245.11012
[15] T.-X. He, L. C. Hsu, P. J. S. Shiue: On generalised Möbius inversion formulas. Bull. Aust. Math. Soc. 73 (2006), 79-88. DOI 10.1017/S0004972700038648 | MR 2206565 | Zbl 1102.11003
[16] C.-P. Lu: On the unique factorization theorem in the ring of number theoretic functions. Ill. J. Math. 9 (1965), 40-46. DOI 10.1215/ijm/1256067579 | MR 0170906 | Zbl 0128.04504
[17] J. Popken: On multiplicative arithmetic functions. Studies in Mathematical Analysis and Related Topics. Stanford University Press, Stanford (1962), 285-293. MR 0143752 | Zbl 0115.26604
[18] D. Rearick: Correlation of semi-multiplicative functions. Duke Math. J. 33 (1966), 623-627. DOI 10.1215/S0012-7094-66-03372-2 | MR 0200252 | Zbl 0154.29503
[19] D. Rearick: Semi-multiplicative functions. Duke Math. J. 33 (1966), 49-53. DOI 10.1215/S0012-7094-66-03308-4 | MR 0184897 | Zbl 0139.27002
[20] J. Sándor, B. Crstici: Handbook of Number Theory. II. Kluwer Academic, Dordrecht (2004). DOI 10.1007/1-4020-2547-5 | MR 2119686 | Zbl 1079.11001
[21] A. Selberg: Remarks on multiplicative functions. Number Theory Day. Lecture Notes in Mathematics 626. Springer, Berlin (1977), 232-241. DOI 10.1007/BFb0063067 | MR 0485750 | Zbl 0367.10041
[22] R. Sivaramakrishnan: Classical Theory of Arithmetic Functions. Pure and Applied Mathematics, 126. Marcel Dekker, New York (1989). DOI 10.1201/9781315139463 | MR 0980259 | Zbl 0657.10001

Affiliations:   Pentti  Haukkanen, Tampere University, Faculty of Information Technology and Communication Sciences, Korkeakoulunkatu 7, FI-33014 Tampere,  Finland,  e-mail:  pentti.haukkanen@tuni.fi


 
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