Journal of Integer Sequences, Vol. 20 (2017), Article 17.9.2

The Arithmetic Jacobian Matrix and Determinant

Pentti Haukkanen and Jorma K. Merikoski
Faculty of Natural Sciences
FI-33014 University of Tampere

Mika Mattila
Department of Mathematics
Tampere University of Technology
PO Box 553
FI-33101 Tampere

Timo Tossavainen
Department of Arts, Communication and Education
Lulea University of Technology
SE-97187 Lulea


Let a1,..., am be real numbers that can be expressed as a finite product of prime powers with rational exponents. Using arithmetic partial derivatives, we define the arithmetic Jacobian matrix Ja of the vector a = (a1,..., am) analogously to the Jacobian matrix Jf of a vector function f. We introduce the concept of multiplicative independence of {a1,..., am} and show that Ja plays in it a similar role as Jf does in functional independence. We also present a kind of arithmetic implicit function theorem and show that Ja applies to it somewhat analogously as Jf applies to the ordinary implicit function theorem.

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(Concerned with sequences A000040 A003415.)

Received January 27 2017; revised versions received July 11 2017; August 1 2017. Published in Journal of Integer Sequences, September 8 2017.

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