# Euler function

From Encyclopedia of Mathematics

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).

#### References

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

#### Comments

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.

#### References

[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 |

**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