Completely multiplicative function: Difference between revisions

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==Definition==

A '''completely multiplicative function''' (or '''totally multiplicative function''') is an [[arithmetic function]] (that is, a function whose [[Domain (mathematics)|___domain]] is the [[natural number]]s), such that ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'') ''f''(''b'') holds ''for all'' positive integers ''a'' and ''b''.<ref>{{cite book|last=Apostol|first=Tom|title=Introduction to Analytic Number Theory|year=1976|publisher=Springer|isbn=0-387-90163-9|pages=30}}</ref>

Without the requirement that ''f''(1) = 1, one could still have ''f''(1) = 0, but then ''f''(''a'') = 0 for all positive integers ''a'', so this is not a very strong restriction.