In

number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, a totative of a given positive integer is an integer such that and is coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

to . Euler's totient function
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...

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
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician
In this page we keep the names in Hungarian order (family name first).
{{compact ToC , short1, side=yes
A
* Alexits György (1899–1 ...

conjectured that, writing the totatives of ''n'' as
:$0\; <\; a\_1\; <\; a\_2\; \backslash cdots\; <\; a\_\; <\; n\; ,$
the mean square gap satisfies
:$\backslash sum\_^\; (a\_-a\_i)^2\; <\; C\; n^2\; /\; \backslash phi(n)$
for some constant ''C'', and this was proven by Bob Vaughan
Robert Charles "Bob" Vaughan FRS
FRS may also refer to:
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and Hugh Montgomery.
See also

*Reduced residue system
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

References

*Further reading

*External links

* * Modular arithmetic {{Numtheory-stub