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===''ν''<sub>''p''</sub>(''n'') – [[P-adic order|''p''-adic valuation]] of an integer ''n''===
For a fixed prime ''p'', '''''ν''<sub>''p''</sub>(''n'')''', defined above as the exponent of the largest power of ''p'' dividing ''n'', is completely additive.
===The ''logarithmic derivative''===
<math>\operatorname{ld}(n)=\frac{D(n)}{n} = \sum_{\stackrel{p\mid n}{p\text{ prime}}} \frac {v_p(x)} {p}</math>, where <math>D(n)</math> is the arithmetic derivative.
==Neither multiplicative nor additive==
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