The Mo¨bius function on the set N of positive integers is defined by  1 if n 1 =  µ ( n ) = (−1) k if n is a product of k distinct primes  0 if n is divisible by the square of a prime. Let h and H...

1. 
   

   
   The
   
   Mo¨bius
   
   
   function
   
   on the set
   
   N
   
   of positive integers is defined by
   

   
   

^

1 if

n

1

=



µ

(

n

) =

(−1)


^k


if

n

is a product of

k

distinct primes

^

0 if

n

is divisible by the square of a prime.

1. 
   

   
   Let
   
   h
   
   and
   
   H
   
   be two functions from
   
   N
   
   to
   
   Z
   
   . Show that
   

   
   

H



(

n

) =

⁾




h

(

d

)

d

|

n

for all

n



∈

N



if and only if

n

h

(

n

) =

⁾




µ

(

d

)

H



(

\

for all

n

∈

N

.

d

|

n



^d

1. 
   

   
   Show that the number
   
   I
   
   
   
   _q
   
   
   (
   
   n
   
   ) of monic irreducible polynomials over
   

   
   

F



_q


of degree

n

is given by

I



(

n

)

<sup>1




)</sup>

_q


=


_n

d

|

n

µ

(

d

)

q

n


/


d





_.