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It was introduced by L. Euler (1763).
 
It was introduced by L. Euler (1763).
  
====References====
 
<table>
 
<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Chandrasekharan,  "Introduction to analytic number theory" , Springer  (1968) {{MR|0249348}} {{ZBL|0169.37502}}</TD></TR>
 
</table>
 
 
 
 
====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. [[#References|[a1]]].
 
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. [[#References|[a1]]].
  
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$$\lim_{n\to\infty}\inf\phi(n)\frac{\ln\ln n}{n}=e^{-\gamma},$$
 
$$\lim_{n\to\infty}\inf\phi(n)\frac{\ln\ln n}{n}=e^{-\gamma},$$
  
where $\gamma$ is the [[Euler constant|Euler constant]], see also [[#References|[a1]]], Chapts. 18.4 and 18.5.
+
where $\gamma$ is the [[Euler constant]], see also [[#References|[a1]]], Chapts. 18.4 and 18.5.
 
 
====References====
 
<table>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}}</TD></TR>
 
</table>
 
  
====Comment====
 
 
D. H. Lehmer asked whether whether there is any composite number $n$ such that $\phi(n)$ divides $n-1$.  This is true of every [[prime number]], and Lehmer conjectured in 1932 that there are no composite numbers with this property: he showed that if any such $n$ exists, it must be odd, square-free, and divisible by at least seven primes.
 
D. H. Lehmer asked whether whether there is any composite number $n$ such that $\phi(n)$ divides $n-1$.  This is true of every [[prime number]], and Lehmer conjectured in 1932 that there are no composite numbers with this property: he showed that if any such $n$ exists, it must be odd, square-free, and divisible by at least seven primes.
  
 
====References====
 
====References====
 
<table>
 
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Chandrasekharan,  "Introduction to analytic number theory" , Springer  (1968) {{MR|0249348}} {{ZBL|0169.37502}}</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}}</TD></TR>
 
<TR><TD valign="top">[b1]</TD> <TD valign="top">  D.H. Lehmer, "On Euler's totient function", ''Bulletin of the American Mathematical Society'' '''38''' (1932) 745–751 Zbl 0005.34302.  DOI 10.1090/s0002-9904-1932-05521-5</TD></TR>
 
<TR><TD valign="top">[b1]</TD> <TD valign="top">  D.H. Lehmer, "On Euler's totient function", ''Bulletin of the American Mathematical Society'' '''38''' (1932) 745–751 Zbl 0005.34302.  DOI 10.1090/s0002-9904-1932-05521-5</TD></TR>
 
</table>
 
</table>
  
 
[[Category:Number theory]]
 
[[Category:Number theory]]

Revision as of 09:25, 10 November 2023


Euler's totient function

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 "totatives" of $n$). The Euler function is a multiplicative arithmetic function, 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).

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.

D. H. Lehmer asked whether whether there is any composite number $n$ such that $\phi(n)$ divides $n-1$. This is true of every prime number, and Lehmer conjectured in 1932 that there are no composite numbers with this property: he showed that if any such $n$ exists, it must be odd, square-free, and divisible by at least seven primes.

References

[1] K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) MR0249348 Zbl 0169.37502
[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
[b1] D.H. Lehmer, "On Euler's totient function", Bulletin of the American Mathematical Society 38 (1932) 745–751 Zbl 0005.34302. DOI 10.1090/s0002-9904-1932-05521-5
How to Cite This Entry:
Euler function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_function&oldid=54308
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article