# Mobius function

(0.001 seconds)

## 1—10 of 11 matching pages

##### 1: 27.5 Inversion Formulas
27.5.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)=\left\lfloor\frac{1}{n}\right\rfloor,$
27.5.3 $g(n)=\sum_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{|}n}g(d)% \mu\left(\frac{n}{d}\right).$
27.5.5 $\ln n=\sum_{d\mathbin{|}n}\Lambda\left(d\right)\Longleftrightarrow\Lambda\left% (n\right)=\sum_{d\mathbin{|}n}(\ln d)\mu\left(\frac{n}{d}\right).$
27.5.8 $g(n)=\prod_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\prod_{d\mathbin{|}n}% \left(g\left(\frac{n}{d}\right)\right)^{\mu\left(d\right)}.$
##### 3: 27.6 Divisor Sums
27.6.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)f(d)=\prod_{p\mathbin{|}n}(1-f(p)),$ $n>1$.
Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
27.6.4 $\sum_{d^{2}\mathbin{|}n}\mu\left(d\right)=|\mu\left(n\right)|,$
27.6.7 $\sum_{d\mathbin{|}n}\mu\left(d\right)\left(\frac{n}{d}\right)^{k}=J_{k}\left(n% \right),$
##### 4: 27.2 Functions
27.2.12 $\mu\left(n\right)=\begin{cases}1,&n=1,\\ (-1)^{\nu\left(n\right)},&a_{1}=a_{2}=\dots=a_{\nu\left(n\right)}=1,\\ 0,&\mbox{otherwise}.\end{cases}$
This is the Möbius function. …
##### 5: 27.4 Euler Products and Dirichlet Series
27.4.5 $\sum_{n=1}^{\infty}\mu\left(n\right)n^{-s}=\frac{1}{\zeta\left(s\right)},$ $\Re s>1$,
27.4.8 $\sum_{n=1}^{\infty}|\mu\left(n\right)|n^{-s}=\frac{\zeta\left(s\right)}{\zeta% \left(2s\right)},$ $\Re s>1$,
##### 6: 27.11 Asymptotic Formulas: Partial Sums
27.11.12 $\sum_{n\leq x}\mu\left(n\right)=O\left(xe^{-C\sqrt{\ln x}}\right),$ $x\to\infty$,
27.11.13 $\lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}\mu\left(n\right)=0,$
27.11.14 $\lim_{x\to\infty}\sum_{n\leq x}\frac{\mu\left(n\right)}{n}=0,$
27.11.15 $\lim_{x\to\infty}\sum_{n\leq x}\frac{\mu\left(n\right)\ln n}{n}=-1.$
##### 7: 27.7 Lambert Series as Generating Functions
27.7.3 $\sum_{n=1}^{\infty}\mu\left(n\right)\frac{x^{n}}{1-x^{n}}=x,$
##### 8: 27.10 Periodic Number-Theoretic Functions
It can also be expressed in terms of the Möbius function as a divisor sum:
27.10.5 $c_{k}\left(n\right)=\sum_{d\mathbin{|}\left(n,k\right)}d\mu\left(\frac{k}{d}% \right).$
##### 9: Bibliography R
• G. Rota (1964) On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340–368.
• ##### 10: 32.2 Differential Equations
They are distinct modulo Möbius (bilinear) transformations …in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\phi(z)$ are locally analytic functions. The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$. For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations $\mbox{P}_{\mbox{\scriptsize II}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …