Euler's totient function
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| Euler's totient function | |
|---|---|
 | |
| Name | Euler's totient function |
| Domain | Positive integers |
| Codomain | Positive integers |
| Notation | φ(n) |
| Introduced | 18th century |
| Named after | Leonhard Euler |
Euler's totient function is a classical arithmetic function that counts the positive integers up to a given integer that are coprime to it. It plays a central role in the work of Leonhard Euler and appears in results connected to Pierre de Fermat, Carl Friedrich Gauss, Joseph-Louis Lagrange, and problems addressed by institutions such as the Royal Society and the Academy of Sciences (France). The function connects to major concepts studied by mathematicians like Adrien-Marie Legendre, Évariste Galois, Srinivasa Ramanujan, Harold Davenport, and modern researchers at universities such as University of Cambridge and Princeton University.
Definition and basic properties
For a positive integer n, the function equals the count of integers m with 1 ≤ m ≤ n that satisfy gcd(m,n)=1, a relation studied by Pierre de Fermat and Leonhard Euler. Basic properties include φ(1)=1 and the identity φ(n)
Computation and formulas
Explicit computation uses prime factorization, a technique refined by Eratosthenes and formalized via the Fundamental Theorem of Arithmetic later treated by Gauss. For n with prime factors p_i, formulas derive from inclusion–exclusion principles applied by Sophie Germain and Adrien-Marie Legendre. Computational methods tie to algorithms developed at Bell Labs and later at IBM and Microsoft Research, and are implemented in computer algebra systems used at Massachusetts Institute of Technology and Stanford University.
Multiplicativity and prime power formula
Multiplicativity states that φ(ab)=φ(a)φ(b) for gcd(a,b)=1, an algebraic property echoed in results by Évariste Galois and in group-theoretic contexts considered by William Rowan Hamilton and Arthur Cayley. The prime power formula φ(p^k)=p^k−p^{k−1} for prime p and integer k is central to proofs by Euler and appears in lectures at University of Göttingen and Harvard University. These formulas underpin work in cyclotomy pursued by Gauss and later by Leopold Kronecker.
Applications in number theory and cryptography
The function underlies Euler's theorem, a generalization of a result by Pierre de Fermat, which informs modern constructions like the RSA (cryptosystem) developed by Ron Rivest, Adi Shamir, and Leonard Adleman. It connects to group order arguments used by Arthur Cayley and to primality tests influenced by work at Bell Labs and Bellcore. Cryptographic protocols studied at MIT and Stanford University rely on properties of φ(n) for key generation, while analytic results involving φ(n) appear in research by G. H. Hardy, John Littlewood, and Atle Selberg.
Average order and distribution
Asymptotic behavior of φ(n) and its average order were studied by G. H. Hardy and Srinivasa Ramanujan, with later refinements by Paul Erdős, Pál Erdős collaborators, and Heini Halberstam. Results about normal order, typical size, and extremal bounds involve methods from analytic number theory developed by Ivan Vinogradov, Atle Selberg, and researchers at the Institute for Advanced Study. Distributional studies connect to conjectures and theorems linked to Riemann zeta function investigations pursued by Bernhard Riemann and later by Andrew Wiles in related arithmetical contexts.
Generalizations and related functions
Generalizations include the Jordan totient function introduced in correspondence among Camille Jordan and contemporaries, the Carmichael function studied by Robert Carmichael, and the Möbius-related inversion techniques credited to Augustin-Louis Cauchy and Johann Peter Gustav Lejeune Dirichlet. Related multiplicative functions appear in the work of Dirichlet, Leopold Kronecker, and Emil Artin, and connect to algebraic number theoretic constructs explored by Richard Dedekind and Helmut Hasse. Modern extensions are investigated at research centers like the Clay Mathematics Institute and in collaborations among scholars at University of Oxford and California Institute of Technology.