View source for Möbius function

An [[Arithmetic function|arithmetic function]] of natural argument: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642802.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642803.png" /> is divisible by the square of a prime number, otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642805.png" /> is the number of prime factors of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642806.png" />. This function was introduced by A. Möbius in 1832.

The Möbius function is a [[Multiplicative arithmetic function|multiplicative arithmetic function]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642807.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642808.png" />. It is used in the study of other arithmetic functions; it appears in inversion formulas (see, e.g. [[Möbius series|Möbius series]]). The following estimate is known for the mean value of the Möbius function [[#References|[2]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m0642809.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428010.png" /> is a constant. The fact that the mean value tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428011.png" /> implies an asymptotic law for the [[Distribution of prime numbers|distribution of prime numbers]] in the natural series.


The Möbius function is related to the Riemann zeros via the formula

\begin{equation} \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{2n}}{(2n)! \zeta(2n+1)}\int_{-\infty}^{\infty}g(x) e^{-x(2n+1/2)} \, dx,\end{equation}
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,   "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Walfisz,   "Weylsche Exponentialsummen in der neueren Zahlentheorie" , Deutsch. Verlag Wissenschaft.  (1963)</TD></TR></table>



====Comments====
The multiplicative arithmetic functions form a [[Group|group]] under the convolution product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428012.png" />. The Möbius function is in fact the inverse of the constant multiplicative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428013.png" /> (defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064280/m06428015.png" />) under this convolution product. From this there follows many  "inversion formulas" , cf. e.g. [[Möbius series|Möbius series]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Clarendon Press  (1979)</TD></TR></table>