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 Formu
Math 412: Number Theory Lecture 11 Möbius Inversion Formulagyu.people.wm.edu/Fall2016/Math412/nt-lec11-note.pdf · Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion
May 31, 2020
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Math 412: Number Theory
Lecture 11 Mobius Inversion Formula
Gexin Yu
College of William and Mary
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius 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) =P
d |n d and the number of
divisors of n: ⌧(n) =P
d |n 1
For f 2 {�,�, ⌧}, f (mn) = f (m)f (n) if (m, n) = 1, namely, they are
multiplicative functions.
Thm: if f is a multiplicative function, then F (n) =
Pd |n f (d) is also
multiplicative.
And
Pd |n �(d) = n.
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Let F (n) =
Pd |n f (d). Then what is f in terms of F?
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Mobius Inversion
Def: the Mobius function, µ(n), is defined by
µ(n) =
8><
>:
1, n = 1
(�1)
r , n = p
1
p
2
. . . pr
0, otherwise
Thm: µ(n) is multiplicative.
Thm:
Pd |n µ(d) = 1 if and only if n = 1, otherwise, it is 0.
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Mobius Inversion
Def: the Mobius function, µ(n), is defined by
µ(n) =
8><
>:
1, n = 1
(�1)
r , n = p
1
p
2
. . . pr
0, otherwise
Thm: µ(n) is multiplicative.
Thm:
Pd |n µ(d) = 1 if and only if n = 1, otherwise, it is 0.
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Mobius Inversion
Def: the Mobius function, µ(n), is defined by
µ(n) =
8><
>:
1, n = 1
(�1)
r , n = p
1
p
2
. . . pr
0, otherwise
Thm: µ(n) is multiplicative.
Thm:
Pd |n µ(d) = 1 if and only if n = 1, otherwise, it is 0.
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Mobius Inversion
Mobius Inversion Formula:
If F (n) =
X
d |n
f (d), then f (n) =
X
d |n
µ(d)F (n/d).
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Applications of Mobius Inversion
From n =
Pd |n �(n), we have �(n) =
Pd |n µ(d)
nd .
From �(n) =P
d |n d we have n =
Pd |n µ(n/d)�(d).
From ⌧(n) =P
d |n 1 we have 1 =
Pd |n µ(n/d)⌧(d).
Thm: if F (n) =
Pd |n f (d) and F is multiplicative, then f is also
multiplicative.
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Applications of Mobius Inversion
From n =
Pd |n �(n), we have �(n) =
Pd |n µ(d)
nd .
From �(n) =P
d |n d we have n =
Pd |n µ(n/d)�(d).
From ⌧(n) =P
d |n 1 we have 1 =
Pd |n µ(n/d)⌧(d).
Thm: if F (n) =
Pd |n f (d) and F is multiplicative, then f is also
multiplicative.
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Applications of Mobius Inversion
From n =
Pd |n �(n), we have �(n) =
Pd |n µ(d)
nd .
From �(n) =P
d |n d we have n =
Pd |n µ(n/d)�(d).
From ⌧(n) =P
d |n 1 we have 1 =
Pd |n µ(n/d)⌧(d).
Thm: if F (n) =
Pd |n f (d) and F is multiplicative, then f is also
multiplicative.
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Applications of Mobius Inversion
From n =
Pd |n �(n), we have �(n) =
Pd |n µ(d)
nd .
From �(n) =P
d |n d we have n =
Pd |n µ(n/d)�(d).
From ⌧(n) =P
d |n 1 we have 1 =
Pd |n µ(n/d)⌧(d).
Thm: if F (n) =
Pd |n f (d) and F is multiplicative, then f is also
multiplicative.
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Thm: if f (n) =
Pd |n µ(d)F (n/d), then F (n) =
Pd |n f (d).
In general, one can have Dirichlet product (or Dirichlet Convolution):
h ⇤ f =
X
n=d1
d2
h(d
1
)f (d
2
).
Then we have g = µ ⇤ f if and only if f = 1 ⇤ g .
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
Thm: if f (n) =
Pd |n µ(d)F (n/d), then F (n) =
Pd |n f (d).
In general, one can have Dirichlet product (or Dirichlet Convolution):
h ⇤ f =
X
n=d1
d2
h(d
1
)f (d
2
).
Then we have g = µ ⇤ f if and only if f = 1 ⇤ g .
Gexin Yu [email protected] Math 412: Number Theory Lecture 11 Mobius Inversion Formula
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