The Mobius inversion formula is extended to the unique factorization semigr
oup. Based on the theory of generalized function and the theory of algebrai
c numbers, we obtain the relation between high-dimensional Fourier coeffici
ents and the extended Mobius inversion formula, i.e., high-dimensional arit
hmetic Fourier transform. (C) 2001 Elsevier Science Inc. All rights reserve
d.