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Euler function

From Encyclopedia of Mathematics
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The arithmetic function $\phi$ whose value at $n$ is equal to the number of positive integers not exceeding $n$ and relatively prime to $n$. The Euler function is multiplicative, that is $\phi(1)=1$ and $\phi(mn)=\phi(m)\phi(n)$ for $(m,n)=1$. The function $\phi(n)$ satisfies the relations

$$\sum_{d|n}\phi(d)=n,$$

$$c\frac{n}{\ln\ln n}\leq\phi(n)\leq n,$$

$$\sum_{n\leq x}\phi(n)=\frac{3}{\pi^2}x^2+O(x\ln x).$$

It was introduced by L. Euler (1763).

References

[1] K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) MR0249348 Zbl 0169.37502


Comments

The function $\phi(n)$ can be evaluated by $\phi(n)=n\prod_{p|n}(1-p^{-1})$, where the product is taken over all primes dividing $n$, cf. [a1].

For a derivation of the asymptotic formula in the article above, as well as of the formula

$$\lim_{n\to\infty}\inf\phi(n)\frac{\ln\ln n}{n}=e^{-\gamma},$$

where $\gamma$ is the Euler constant, see also [a1], Chapts. 18.4 and 18.5.

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 MR0568909 Zbl 0423.10001
How to Cite This Entry:
Euler function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_function&oldid=11814
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article