Lecture 5

Question 1

Multiplicative functions

Answer 1

An arithemetical function f is called multiplicative if f ≠ 0 and if
* f(mn) = f(m)f(n),

whenever gcd(m, n) = 1.

Question 2

Completely Multiplicative functions

Answer 2

A multiplicative function f is called completely multiplicative if we also have
* f(mn) = f(m)f(n)

for all m, n.

Question 3

If f is multiplicative, what is f(1)

Answer 3

f(1) = 1

Question 4

When does f(p^a1_11. . . p^ak_k) = f(p^a1_1) . . . f(p^ak_k), for all primes pi and all integers ai ≥ 1

Answer 4

If and only if f is multiplicative

Question 5

When does f(p^a) = f(p)^a, for all primes p and all integers a ≥ 1

Answer 5

If and only if f is completely multiplicative

Question 6

When is the Dirichlet product f*g multiplicative

Answer 6

When f and g are multiplicative

Question 7

Is the Dirichlet convolution of two completely multiplicative functions always completely multiplicative

Answer 7

Not always

Question 8

Is the Dirichlet inverse of a multiplicative function multiplicative

Answer 8

Yes

Question 9

How can we define complete multiplicativity through the D inverse and mobius functiopn

Answer 9

Let f be a multiplicative function. Then f is a completely multiplicative function if and only if

• f^(−1)(n) = µ(n)f(n)

for all n ≥ 1.

Question 10

What is Liouville’s function

Answer 10

For n > 1, write n = p^a1_1. . . p^ak_k. Then Liouville’s function is defined as follows:

• λ(n) ∶=
  *1 if n = 1
  
  • (−1)^(a1+…+ak) if n > 1}

Question 11

Is Liouville’s function multiplicative

Answer 11

It is completely multiplicative

Question 12

How can we use Liouville’s function to find square numbers

Answer 12

For every n ≥ 1, we have

• ∑_(d∣n) λ(d) =
  
  • 1 if n is a square,
  • 0 otherwise.

Question 13

What is the divisor function (The sum of the α-th powers of the divisors of n)

Answer 13

For α ∈ R or α ∈ C and for any integer n ≥ 1, we define

• σα(n) = ∑_(d∣n) d^α

to be the sum of the α-th powers of the divisors of n

Question 14

Are the divisor functions completely multiplicative

Answer 14

No, in general only multiplicativw

Question 15

Inverse of divisor function

Answer 15

σ^(−1)α (n) = ∑(d∣n) d^α µ(d)µ(n/d)