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Thus, it is often called Euler's phi function or simply the phi function. In 1879, J. J. Sylvester coined the term totient for this function, [14] [15] so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's. The cototient of n is defined as n − φ(n).
The summatory of reciprocal totient function is defined as ():= = ()Edmund Landau showed in 1900 that this function has the asymptotic behavior (+ ) + + ()where γ is the Euler–Mascheroni constant,
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
Here φ denotes Euler's totient function. ... {1, 5, 7, 11}. The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced ...
The Carmichael lambda function of a prime power can be expressed in terms of the Euler totient. Any number that is not 1 or a prime power can be written uniquely as the product of distinct prime powers, in which case λ of the product is the least common multiple of the λ of the prime power factors. Specifically, λ(n) is given by the recurrence
Euler's totient or phi function, φ(n) is an arithmetic function that counts the number of positive integers less than or equal to n that are relatively prime to n. That is, if n is a positive integer, then φ(n) is the number of integers k in the range 1 ≤ k ≤ n which have no common factor with n other than 1.
An average order of φ(n), Euler's totient function of n, is 6n / π 2; An average order of r(n), the number of ways of expressing n as a sum of two squares, is π; The average order of representations of a natural number as a sum of three squares is 4πn / 3;
For example, if one starts with Euler's totient function φ, and repeatedly applies the transformation process, one obtains: φ the totient function; φ ∗ 1 = I, where I(n) = n is the identity function; I ∗ 1 = σ 1 = σ, the divisor function; If the starting function is the Möbius function itself, the list of functions is: μ, the Möbius ...