Arithmetic function: Difference between revisions

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==Notation==

<math display="inline">\sum_p f(p)</math> &nbsp; and &nbsp; <math display="inline">\prod_p f(p)</math> &nbsp; mean that the sum or product is over all [[prime number]]s:

:<math>\sum_p f(p) = f(2) + f(3) + f(5) + \cdots</math>

:<math>\prod_p f(p)= f(2)f(3)f(5)\cdots.</math>

Similarly, &nbsp; <math display="inline">\sum_{p^k} f(p^k)</math> &nbsp; and &nbsp; <math display="inline">\prod_{p^k} f(p^k)</math> &nbsp; mean that the sum or product is over all [[prime power]]s with strictly positive exponent (so ''k'' = 0 is not included):

:<math>\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>

<math display="inline">\sum_{d\mid n} f(d)</math> &nbsp; and &nbsp; <math display="inline">\prod_{d\mid n} f(d)</math> &nbsp; mean that the sum or product is over all positive divisors of ''n'', including 1 and ''n''. For example, if ''n'' = 12,

:<math>\prod_{d\mid 12} f(d) = f(1)f(2) f(3) f(4) f(6) f(12).\ </math>

The notations can be combined: &nbsp; <math display="inline">\sum_{p\mid n} f(p)</math> &nbsp; and &nbsp; <math display="inline">\prod_{p\mid n} f(p)</math> &nbsp; mean that the sum or product is over all prime divisors of ''n''. For example, if ''n'' = 18,

:<math>\sum_{p\mid 18} f(p) = f(2) + f(3),\ </math>

and similarly &nbsp; <math display="inline">\sum_{p^k\mid n} f(p^k)</math> &nbsp; and &nbsp; <math display="inline">\prod_{p^k\mid n} f(p^k)</math> &nbsp; mean that the sum or product is over all prime powers dividing ''n''. For example, if ''n'' = 24,

:<math>\prod_{p^k\mid 24} f(p^k) = f(2) f(3) f(4) f(8).\ </math>

==Ω(''n''), ''ω''(''n''), ''ν''_''p''(''n'') – prime power decomposition==