MathDB
increasing functions, multiplicative and completely multiplicative

Source: ITAMO 1998 p6

January 25, 2020
functionincreasing functionsalgebramultiplicative function

Problem Statement

We say that a function f:N→Nf : N \to N is increasing if f(n)<f(m)f(n) < f(m) whenever n<mn < m, multiplicative if f(nm)=f(n)f(m)f(nm) = f(n)f(m) whenever nn and mm are coprime, and completely multiplicative if f(nm)=f(n)f(m)f(nm) = f(n)f(m) for all n,mn,m. (a) Prove that if ff is increasing then f(n)≥nf(n) \ge n for each nn. (b) Prove that if ff is increasing and completely multiplicative and f(2)=2f(2) = 2, then f(n)=nf(n) = n for all nn. (c) Does (b) remain true if the word ”completely” is omitted?
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