Euler's totient function

:For other meanings, see Euler function (disambiguation).

Inequalities

Some inequalities involving the φ function are:

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:

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arphi (n) > rac {n} {e^gamma; log log n + rac {3} {log log n}}

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for n > 2, where γ is Euler's constant,

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:

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arphi(n) ge sqrt{rac {n} {2} }

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for n > 0,

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and

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:

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arphi(n) ge sqrt{n}

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for n > 6.

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For prime n, clearly φ(n) = n-1. For composite n we have

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:

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arphi(n) le n-sqrt{n}

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(for composite n).

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A pair of inequalities combining the φ function and the σ divisor function are:

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:

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rac {6 n^2} {pi^2} < arphi(n) sigma(n) < n^2

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for n > 1.

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