Totative 📖 Wikipedia

In number theory, a totative of a given positive integer $n$ is an integer $k$ such that $0 < k \leq n$ and $k$ is coprime to $n$ . 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

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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_{\phi (n)}<n,

the mean square gap satisfies

\sum _{i=1}^{\phi (n)-1}(a_{i+1}-a_{i})^{2}<Cn^{2}/\phi (n)

for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.^[1]

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^ Montgomery, H.L.; Vaughan, R.C. (1986). "On the distribution of reduced residues". Ann. Math. 2. 123 (2): 311–333. doi:10.2307/1971274. JSTOR 1971274. Zbl 0591.10042.

Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B40. ISBN 978-0-387-20860-2. Zbl 1058.11001.

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