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  • ...eal part]] <math>1/2</math>. From the [[functional equation]] for the zeta function, it is easy to see that <math>\zeta(s)=0</math> when <math>s=-2,-4,-6,\ldot ...c{1}{2}</math>. Let <math>M(n)=\sum_{i=1}^n \mu(i)</math> be the [[Mertens function]]. It is easy to show that if <math>M(n)\le\sqrt{n}</math> for sufficiently
    2 KB (425 words) - 02:18, 29 June 2024
  • A (weak) '''multiplicative function''' <math>f: \mathbb{N} \to \mathbb{C}</math> is one such that <math>f(a\cdo '''Euler's totient function''': <math>\varphi (n) = \phi (n) : = \left| \{ k \in \mathbb{N} \ : \ k \le
    8 KB (1,401 words) - 16:49, 10 January 2025
  • The '''Möbius function''' is a multiplicative number theoretic function defined as follows: ==Multiplicity of the Function==
    5 KB (910 words) - 01:58, 1 March 2022
  • ''Proof'': For context, the Möbius Inversion Formula states that if <math>g(n)</math> and <math>f(n)</math> ar where <math>\mu(n)</math> denotes the Möbius Function. We begin with the following formula that states
    8 KB (1,438 words) - 13:50, 23 June 2022
  • ...the other by sums over divisors. Originally proposed by August Ferdinand Möbius in 1832, it has many uses in [[Number Theory]] and [[Combinatorics]]. ...f</math> be arithmetic functions and <math>\mu</math> denote the [[Möbius Function]]. Then it follows that
    2 KB (430 words) - 17:54, 15 March 2022
  • Let <math>f</math> be the unique function defined on the positive integers such that <cmath>\sum_{d\mid n}d\cdot f\le Very nice! Now, we need to show that this function is multiplicative, i.e. <math>f(pq) = f(p) \cdot f(q)</math> for <math>\tex
    7 KB (1,204 words) - 12:56, 14 September 2024