Math 412: Number Theory Lecture 11 M¨obius Inversion Formula Gexin Yu [email protected] College of William and Mary Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula Multiplicative functions Def: φ(n) is the number of elements in a reduced system of residues modulo n. (i.e., the number of coprimes to n in 1, 2,...,n .) { } Def: the sum of divisors of n: σ(n)= d n d and the number of | divisors of n: ⌧(n)= d n 1 | P For f φ,σ,⌧ , f (mnP)=f (m)f (n)if(m, n) = 1, namely, they are 2{ } multiplicative functions. Thm: if f is a multiplicative function, then F (n)= d n f (d)isalso multiplicative. | P And d n φ(d)=n. | P Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula Let F (n)= d n f (d). Then what is f in terms of F ? | P Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula Thm: µ(n)ismultiplicative. Thm: d n µ(d)=1if and only if n = 1, otherwise, it is 0. | P M¨obiusInversion Def: the M¨obius function, µ(n), is defined by 1, n =1 µ(n)= ( 1)r , n = p p ...p 8 − 1 2 r <>0, otherwise :> Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula Thm: d n µ(d)=1if and only if n = 1, otherwise, it is 0. | P M¨obiusInversion Def: the M¨obius function, µ(n), is defined by 1, n =1 µ(n)= ( 1)r , n = p p ...p 8 − 1 2 r <>0, otherwise Thm: µ(n)ismultiplicative.:> Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula M¨obiusInversion Def: the M¨obius function, µ(n), is defined by 1, n =1 µ(n)= ( 1)r , n = p p ...p 8 − 1 2 r <>0, otherwise Thm: µ(n)ismultiplicative.:> Thm: d n µ(d)=1if and only if n = 1, otherwise, it is 0. | P Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula M¨obiusInversion Mobius Inversion Formula: If F (n)= f (d), then f (n)= µ(d)F (n/d). d n d n X| X| Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula From σ(n)= d n d we have n = d n µ(n/d)σ(d). | | P P From ⌧(n)= d n 1wehave1= d n µ(n/d)⌧(d). | | P P Thm: if F (n)= d n f (d)andF is multiplicative, then f is also multiplicative. | P Applications of M¨obius Inversion n From n = d n φ(n), we have φ(n)= d n µ(d) d . | | P P Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula From ⌧(n)= d n 1wehave1= d n µ(n/d)⌧(d). | | P P Thm: if F (n)= d n f (d)andF is multiplicative, then f is also multiplicative. | P Applications of M¨obius Inversion n From n = d n φ(n), we have φ(n)= d n µ(d) d . | | P P From σ(n)= d n d we have n = d n µ(n/d)σ(d). | | P P Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula Thm: if F (n)= d n f (d)andF is multiplicative, then f is also multiplicative. | P Applications of M¨obius Inversion n From n = d n φ(n), we have φ(n)= d n µ(d) d . | | P P From σ(n)= d n d we have n = d n µ(n/d)σ(d). | | P P From ⌧(n)= d n 1wehave1= d n µ(n/d)⌧(d). | | P P Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula Applications of M¨obius Inversion n From n = d n φ(n), we have φ(n)= d n µ(d) d . | | P P From σ(n)= d n d we have n = d n µ(n/d)σ(d). | | P P From ⌧(n)= d n 1wehave1= d n µ(n/d)⌧(d). | | P P Thm: if F (n)= d n f (d)andF is multiplicative, then f is also multiplicative. | P Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula In general, one can have Dirichlet product (or Dirichlet Convolution): h f = h(d )f (d ). ⇤ 1 2 n=Xd1d2 Then we have g = µ f if and only if f =1 g. ⇤ ⇤ Thm: if f (n)= d n µ(d)F (n/d), then F (n)= d n f (d). | | P P Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula Thm: if f (n)= d n µ(d)F (n/d), then F (n)= d n f (d). | | P P In general, one can have Dirichlet product (or Dirichlet Convolution): h f = h(d )f (d ). ⇤ 1 2 n=Xd1d2 Then we have g = µ f if and only if f =1 g. ⇤ ⇤ Gexin Yu [email protected] Math 412: Number Theory Lecture 11 M¨obius Inversion Formula.