# 254A, Notes 1: Elementary multiplicative number theory

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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