Lecture 6

Question 1

What is a generalised convolution

Answer 1

Let F ∶ (0,+∞) → C such that F(x) = 0 for 0 < x < 1 and let α be an arithemtical function. Also let G ∶ (0,+∞) → C such that G(x) = 0 for 0 < x < 1 and
G(x) = (α ○ F)(x) = ∑_(n≤x) α(n)F (x/n) .

Question 2

If F(x) = 0 for all nonintegral x, what property holds for a generalised convolution

Answer 2

(α ○ F)(m) = (α ∗ F)(m)

Question 3

What restrictions do we require on α and β for α ○ (β ○ F) = (α ∗ β) ○ F to hold

Answer 3

They need to be arithmetical functions

Question 4

What is the identity function

Answer 4

I(n) = [1/n]

Question 5

What operation makes the identity function a left identity

Answer 5

The composition operation, ○

Question 6

State the generalised inversion formula

Answer 6

Let α be an arithemtical function with Dirichlet inverse α^(−1). Then the equation

• G(x) = ∑_(n≤x) α(n)F (x/n)

Implies

• F(x) = ∑_(n≤x) α^(−1)(n)G(x/n) .

And Vice versa

Question 7

What is the generalised Mobius inversion formula

Answer 7

If α is a completely multiplicative
function, we have
* G(x) = ∑(n≤x) α(n)F (x/n)
if and only if
* F(x) = ∑(n≤x) µ(n)α(n)G(x/n)

Question 8

What is the derivative of an arithmetical function

Answer 8

For any arithmetical function f, we define its derivative f′ to be the arithmetical function given by the equation

• f′ (n) = f(n) log n

Question 9

Give properties of arithmetical derivatives

* (f+g)’, (f*g)’ , (f^(-1))’

Answer 9

a) (f + g)′ = f′ + g′.
b) (f ∗ g)′ = f′ ∗ g + f ∗ g′.
c) (f^(−1))′ = −f′ ∗ (f ∗ f)^(−1), provided that f(1) ≠ 0.

Question 10

What is the Selberg identity

Answer 10

Λ(n) log n + ∑(d∣n) Λ(n)Λ(n/d) = ∑(d∣n) µ(d) log^2(n/d) .

For n ≥ 1