T2. 7. Arithmetic Functions

Question 1

Define arithmetic functions

Answer 1

A function from the natural numbers to complex

Question 2

Define the Dirichlet convolution of arithmetic functions

Answer 2

f*g(n) = SUM _d|n g(n/d)

Question 3

State the outcome:
11
1μ
What implication?

Answer 3

= 1
= 0

μ is the inverse of 1 under Dirichlet convolution

Question 4

State the mobius inversion formula

Answer 4

Define g(n) = SUM_d|n f(d)

f(n) = SUM_d|n g(d)μ(n/d)

Question 5

Give the SUM_1,∞ f*g/n^s

Answer 5

= SUM_1,∞ f(n)/n^s SUM_1,∞ g(n)/n^s

Question 6

What is the condition for an poly arithmetic function to be multiplicative

Answer 6

If f(mn) = f(m)f(n) whenever (m,n) = 1

Question 7

What is the condition for an poly arithmetic function to be completely multiplicative

Answer 7

If f(mn) = f(m)f(n) for all m,n

Question 8

How do multiplicative and completely multiplicative functions generalise the RZ formula?

Answer 8

SUM_n,∞ f(n)/n^s =

1. Check notes :)
2. PROD_p 1/1-f(p)p^-s

Question 9

How does μ generalise the RZ formula? Why?

Answer 9

It is completely multiplicative
SUM_n,∞ μ(n)/n^s = 1/ζ

Question 10

What remark can we make for f multiplicative and not the zero function?

Answer 10

f(1) = 1

Question 11

Give the literal and convolution definition of the Euler totient function

Answer 11

φ(n) = #{a ∈ N : a ≤ n and (a, n) = 1}

φ(n) = μ*N

Question 12

How does φ generalise the RZ formula? Why?

Answer 12

SUM_n,∞ μ(n)/n^s = 1/ζ