Editing Möbius function (section)

{{short description|Multiplicative function in number theory}}
{{About|the number-theoretic Möbius function|the combinatorial Möbius function|incidence algebra|the [[rational function]]s defined on the [[complex number]]s|Möbius transformation}}
{{use dmy dates|date=October 2024}}
{{More footnotes needed|date=October 2024}}
{{Infobox integer sequence
| image                 = 
| image_size            = 
| alt                   = 
| caption               = 
| named_after           = [[August Ferdinand Möbius]]
| publication_year      = 1832
| author                = [[August Ferdinand Möbius]]
| terms_number          = infinite
| con_number            =
| number                =
| parentsequence        =
| formula               =
| first_terms           = 1, −1, −1, 0, −1, 1, −1, 0, 0, 1
| largest_known_term    =
| OEIS                  = A008683
| OEIS_name             = Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.
}}
The '''Möbius function <math>\mu(n)</math>''' is a [[multiplicative function]] in [[number theory]] introduced by the German mathematician [[August Ferdinand Möbius]] (also transliterated ''Moebius'') in 1832.{{efn-lr|Hardy & Wright, Notes on ch. XVI: "... <math>\mu(n)</math> occurs implicitly in the works of Euler as early as 1748, but Möbius, in 1832, was the first to investigate its properties systematically". {{harv|Hardy|Wright|1980|loc=Notes on ch. XVI}}}}{{efn-lr|In the ''[[Disquisitiones Arithmeticae]]'' (1801) [[Carl Friedrich Gauss]] showed that the sum of the primitive roots (<math>\mod p</math>) is <math>\mu(p-1)</math>, (see [[#Properties and applications]]) but he didn't make further use of the function. In particular, he didn't use Möbius inversion in the ''Disquisitiones''.{{sfn|Gauss|1986|loc=Art. 81}} The ''[[Disquisitiones Arithmeticae]]'' has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.}}{{sfn|Möbius|1832|pp=105–123}} It is ubiquitous in elementary and [[analytic number theory]] and most often appears as part of its namesake the [[Möbius inversion formula]]. Following work of [[Gian-Carlo Rota]] in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted <math>\mu(x)</math>.