Arithmetic function: Difference between revisions
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imported>Morinator link domain |
imported>Anita5192 →Dirichlet convolutions: copy edit for consistency: d → \delta, in two places. |
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=== ''J''<sub>''k''</sub>(''n'') – Jordan totient function === | === ''J''<sub>''k''</sub>(''n'') – Jordan totient function === | ||
'''[[Jordan totient function|''J''<sub>''k''</sub>(''n'')]]''', the Jordan totient function, is the number of ''k''-tuples of positive integers all less than or equal to ''n'' that form a coprime (''k'' + 1)-tuple together with ''n''. It is a generalization of Euler's totient, {{math|1=''φ''(''n'') = ''J''<sub>1</sub>(''n'')}}. | '''[[Jordan's totient function|''J''<sub>''k''</sub>(''n'')]]''', the Jordan totient function, is the number of ''k''-tuples of positive integers all less than or equal to ''n'' that form a coprime (''k'' + 1)-tuple together with ''n''. It is a generalization of Euler's totient, {{math|1=''φ''(''n'') = ''J''<sub>1</sub>(''n'')}}. | ||
<math display="block">J_k(n) = n^k \prod_{p\mid n} \left(1-\frac{1}{p^k}\right) | <math display="block">J_k(n) = n^k \prod_{p\mid n} \left(1-\frac{1}{p^k}\right) | ||
= n^k \left(\frac{p^k_1 - 1}{p^k_1}\right)\left(\frac{p^k_2 - 1}{p^k_2}\right) \cdots \left(\frac{p^k_{\omega(n)} - 1}{p^k_{\omega(n)}}\right) | = n^k \left(\frac{p^k_1 - 1}{p^k_1}\right)\left(\frac{p^k_2 - 1}{p^k_2}\right) \cdots \left(\frac{p^k_{\omega(n)} - 1}{p^k_{\omega(n)}}\right) | ||
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=== Logarithmic derivative === | === Logarithmic derivative === | ||
<math>\operatorname{ld}(n)=\frac{D(n)}{n} = \sum_{\stackrel{p\mid n}{p\text{ prime}}} \frac {v_p(n)} {p}</math>, where <math>D(n)</math> is the arithmetic derivative. | <math>\operatorname{ld}(n)=\frac{D(n)}{n} = \sum_{\stackrel{p\mid n}{p\text{ prime}}} \frac {v_p(n)} {p}</math>, where <math>D(n)</math> is the [[arithmetic derivative]]. | ||
== Neither multiplicative nor additive == | == Neither multiplicative nor additive == | ||
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=n\sum_{\delta\mid n}\frac{\mu(\delta)}{\delta}. | =n\sum_{\delta\mid n}\frac{\mu(\delta)}{\delta}. | ||
</math> Möbius inversion | </math> Möbius inversion | ||
: <math>\sum_{ | : <math>\sum_{\delta \mid n } J_k(\delta) = n^k.</math> <ref>see references at [[Jordan's totient function]]</ref> | ||
:: <math> | :: <math> | ||
J_k(n) | J_k(n) | ||