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  2. Euler's totient function - Wikipedia

    en.wikipedia.org/wiki/Euler's_totient_function

    In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd (n, k ...

  3. Euler function - Wikipedia

    en.wikipedia.org/wiki/Euler_function

    Euler function. Domain coloring plot of ϕ on the complex plane. In mathematics, the Euler function is given by. {\displaystyle \phi (q)=\prod _ {k=1}^ {\infty } (1-q^ {k}),\quad |q|<1.} Named after Leonhard Euler, it is a model example of a q -series and provides the prototypical example of a relation between combinatorics and complex analysis .

  4. Totient summatory function - Wikipedia

    en.wikipedia.org/wiki/Totient_summatory_function

    Totient summatory function. In number theory, the totient summatory function is a summatory function of Euler's totient function defined by: {\displaystyle \Phi (n):=\sum _ {k=1}^ {n}\varphi (k),\quad n\in \mathbf {N} } It is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n . The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22 ...

  5. Euler's theorem - Wikipedia

    en.wikipedia.org/wiki/Euler's_theorem

    In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, then is congruent to modulo n, where denotes Euler's totient function; that is. In 1736, Leonhard Euler published a proof of Fermat's little theorem [ 1] (stated by Fermat without proof ...

  6. Proof of the Euler product formula for the Riemann zeta function

    en.wikipedia.org/wiki/Proof_of_the_Euler_product...

    Proof of the Euler product formula. The method of Eratosthenes used to sieve out prime numbers is employed in this proof. This sketch of a proof makes use of simple algebra only. This was the method by which Euler originally discovered the formula. There is a certain sieving property that we can use to our advantage:

  7. Inclusion–exclusion principle - Wikipedia

    en.wikipedia.org/wiki/Inclusion–exclusion...

    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.

  8. Möbius inversion formula - Wikipedia

    en.wikipedia.org/wiki/Möbius_inversion_formula

    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 ...

  9. Euler's identity - Wikipedia

    en.wikipedia.org/wiki/Euler's_identity

    In mathematics, Euler's identity[ note 1] (also known as Euler's equation) is the equality where. is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula when evaluated for .