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: (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: (n)= d n d and the number of | divisors of n: ⌧(n)= d n 1 | P For f ,,⌧ , 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 (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

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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 (n)= d n d we have n = d n µ(n/d)(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 (n), we have (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 (n), we have (n)= d n µ(d) d . | | P P

From (n)= d n d we have n = d n µ(n/d)(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 (n), we have (n)= d n µ(d) d . | | P P

From (n)= d n d we have n = d n µ(n/d)(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 (n), we have (n)= d n µ(d) d . | | P P

From (n)= d n d we have n = d n µ(n/d)(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