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==Notation==
In this article, <math display="inline">\sum_p f(p)</math> and <math display="inline">\prod_p f(p)</math> mean that the sum or product is over all [[prime number]]s:
<math display="block">\sum_p f(p) = f(2) + f(3) + f(5) + \cdots</math>
and
<math display="block">\prod_p f(p)= f(2)f(3)f(5)\cdots.</math>
Similarly, <math display="inline">\sum_{p^k} f(p^k)</math> and <math display="inline">\prod_{p^k} f(p^k)</math> mean that the sum or product is over all [[prime power]]s with strictly positive exponent (so {{math|1=''k'' = 0}} is not included):
<math display="block">\sum_{p^k} f(p^k) = \sum_p\sum_{k > 0} f(p^k) = f(2) + f(3) + f(4) +f(5) +f(7)+f(8)+f(9)+\cdots.</math>
The notations <math display="inline">\sum_{d\mid n} f(d)</math> and <math display="inline">\prod_{d\mid n} f(d)</math> mean that the sum or product is over all positive divisors of ''n'', including 1 and ''n''. For example, if {{math|1=''n'' = 12}}, then
<math display="block">\prod_{d\mid 12} f(d) = f(1)f(2) f(3) f(4) f(6) f(12). </math>
The notations can be combined: <math display="inline">\sum_{p\mid n} f(p)</math> and <math display="inline">\prod_{p\mid n} f(p)</math> mean that the sum or product is over all prime divisors of ''n''. For example, if ''n'' = 18, then
<math display="block">\sum_{p\mid 18} f(p) = f(2) + f(3), </math>
and similarly <math display="inline">\sum_{p^k\mid n} f(p^k)</math> and <math display="inline">\prod_{p^k\mid n} f(p^k)</math> mean that the sum or product is over all prime powers dividing ''n''. For example, if ''n'' = 24, then
<math display="block">\prod_{p^k\mid 24} f(p^k) = f(2) f(3) f(4) f(8). </math>
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