Euler function

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The arithmetic function whose value at is equal to the number of positive integers not exceeding and relatively prime to . The Euler function is multiplicative, that is and for . The function satisfies the relations

It was introduced by L. Euler (1763).


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


The function can be evaluated by , where the product is taken over all primes dividing , cf. [a1].

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

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


[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8
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Euler function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article