Mobius Function and Significance

Explanations 4

• The Mobius function is denoted as μ(n)\mu(n)μ(n) and counts the number of prime factors in n, when n has a square factor (obviously not 1), μ(n)\mu(n)μ(n) is 0, and when n has m…

  By joshua lee, Ray Li

• The Mo¨\ddot{\textbf{o}}o¨bius function μ(n)\mu(n)μ(n) is defined as follows: ∙\bullet∙ μ(n)=1\mu(n) = 1μ(n)=1 if nnn is a square-free positive integer with an even number of pr…

  By Ahaan Rungta

• Prove that, ∣∑d∣nμ(d)σ(d)d∣≥1n\left|\sum_{d|n}\dfrac{\mu(d)\sigma(d)}{d}\right|\geq\dfrac{1}{n}​d∣n∑​dμ(d)σ(d)​​≥n1​

  By amaterasu

• Prove that, for a multiplicative function fff, ∑d∣nμ(d)f(d)=∏p∣n(1−f(p)).\sum_{d|n}\mu(d)f(d)=\prod_{p|n}(1-f(p)).d∣n∑​μ(d)f(d)=p∣n∏​(1−f(p)).

  By amaterasu, Victor Wang