Totative

Template:Short description In number theory, a totative of a given positive integer Template:Mvar is an integer Template:Mvar such that Template:Math and Template:Mvar is coprime to Template:Mvar. Euler's totient function φ(n) counts the number of totatives of n. The totatives under multiplication modulo n form the multiplicative group of integers modulo n.

Distribution

The distribution of totatives has been a subject of study. Paul Erdős conjectured that, writing the totatives of n as

$0<a_1<a_2\cdots <a_{\varphi (n)}<n,$

the mean square gap satisfies

$\sum _{i=1}^{\varphi (n)-1}(a_{i+1}-a_i{)}^2<C{n}^2/\varphi (n)$

for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.

See also

• Reduced residue system

References

• Template:Cite book

Further reading

• Template:Citation

External links

• Template:MathWorld
• Template:PlanetMath

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