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In number theory, a totative of a given positive integer n is an integer k such that 0 < kn 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.


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

the mean square gap satisfies

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

See also[edit]


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

Further reading[edit]

  • Sándor, Jozsef; Crstici, Borislav (2004), Handbook of number theory II, Dordrecht: Kluwer Academic, pp. 242–250, ISBN 1-4020-2546-7, Zbl 1079.11001 

External links[edit]