# Lambda function

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In number theory, the Lambda function is a function on positive integers which gives the exponent of the multiplicative group modulo that integer.

The value of λ on a prime power is:

• ${\displaystyle \lambda (2)=1;~\lambda (4)=2;~\lambda (2^{n})=2^{n-2}{\mbox{ for }}n\geq 2;\,}$
• ${\displaystyle \lambda (p^{n})=p^{n-1}(p-1){\mbox{ for }}n\geq 1\,}$ if ${\displaystyle p\,}$ is an odd prime.

The value of λ on a general integer n with prime factorisation

${\displaystyle n=\prod _{i}p_{i}^{a_{i}}\,}$

is then

${\displaystyle \lambda (n)=\mathop {\mbox{lcm}} _{i}\{\lambda (p_{i}^{a_{i}})\}.\,}$

The value of λ(n) always divides the value of Euler's totient function φ(n): they are equal if and only if n has a primitive root.