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Proposition: Explicit Formula for the Euler Function

If $n=\prod_{k=1}^rp_1^{e_1}\cdots p_r^{e_1}$ is the factorization of the natural number $n\ge 1,$ then the Euler function $\phi(n)$ has the formula $$\phi(n)=n\prod_{k=1}^r\left(1-\frac{1}{p_k}\right)=\prod_{k=1}^r p_k^{e_k-1}({p_k}-1).$$

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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927