254A, Notes 1: Elementary multiplicative number theory

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In analytic number theory, an arithmetic function is simply a function $latex {f: {bf N} rightarrow {bf C}}&fg=000000$ from the natural numbers $latex {{bf N} = {1,2,3,dots}}&fg=000000$ to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than $latex {{bf R}}&fg=000000$ or $latex {{bf C}}&fg=000000$, as in this previous blog post, but we will restrict attention here to the classical situation of real or complex arithmetic functions.) Experience has shown that a particularly tractable and relevant class of arithmetic functions for analytic number theory are the multiplicative functions, which are arithmetic functions $latex {f: {bf N} rightarrow {bf C}}&fg=000000$ with the additional property that

$latex displaystyle f(nm) = f(n) f(m) (1)&fg=000000$

whenever $latex {n,m in{bf N}}&fg=000000$ are coprime. (One also considers arithmetic functions that are not genuinely multiplicative, such as the logarithm function $latex {L(n) := log n}&fg=000000$ or the

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