Definitions

Question 1

Divisibility

Answer 1

d divides n (d|n) whenever n = cd for some c.

Question 2

Common divisor

Answer 2

If d|a and d|b, then d is a common divisor of a and b.

Question 3

Greatest common divisor

Answer 3

The number d such that for integers a and b
(i) d ≥ 0
(ii) d|a and d|b
(iii) e|a and e|b ⇒ e|d
is called the greatest common divisor of a and b, denoted gcd(a, b), (a, b), or aDb.

Question 4

Prime, composite number

Answer 4

An integer n is called prime if n > 1 and if the only positive divisors of n and 1 are n itself. If n > 1 and is not a prime, then n is composite.

Question 5

Arithmetical function

Answer 5

A real or complex valued function defined on the positive integers is called an arithmetical function, f : N ⟶ C.

Question 6

Mobius function

Answer 6

For n > 1, write n = p₁ ͣ ¹… pₖ ͣ ᵏ. Then the Mobius function is defined as follows:
1 if n=1
µ(n) = (-1)ᵏ if a₁ = … = aₖ = 1
0 if some aᵢ ≥ 2

Question 7

Euler-Totient funciton

Answer 7

If n ≥ 1, the Euler-Totient function, φ(n), is defined as

φ(n) := #{k : k ≤ n and gcd(k, n) = 1}

Alternatively, 
              n
φ(n) :=   ∑   1
            k=1
       gcd(k,n)=1

Question 8

Dirichlet product / Dirichlet convolution

Answer 8

Given two arithmetical functions f and g, we define their Dirichlet product (or Dirichlet convolution) to be the arithmetical function h defined by the equation

h(n) = ∑ f(d) g(n/d)
d|n

Question 9

Identity function

Answer 9

The identity function I is defined as follows:

I(n) = [1/n] = 1 if n = 1
0 if n > 1

Question 10

Unit function

Answer 10

We define the unit function by u(n) = 1 for all n.

Question 11

Von-Mangoldt function

Answer 11

For any integer n ≥ 1, we define the Von-Mangoldt function Λ(n) by

Λ(n) = log p if n=pᵐ for some prime p and some m ≥ 1
0 otherwise

Question 12

Multiplicative function

Answer 12

An arithmetical function f is called multiplicative if f ≠ 0 and if f(mn) = f(m)f(n), whenever gcd(m, n) = 1.

Question 13

Completely multiplicative function

Answer 13

A multiplicative function f is called completely multiplicative if we have f(mn) = f(m)f(n) for all m, n.

Question 14

Liouville’s function

Answer 14

For n > 1 write n =p₁ ͣ ¹ … pₖ ͣ ᵏ. Then Liouville’s function is defined as

λ(n) := 1 if n =1
(-1) ͣ ¹ ⁺ ॱॱॱ ⁺ ͣ ᵏ if n > 1

Question 15

Divisor functions σ_α (n)

Answer 15

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

σ_α (n) = ∑ dᵅ
d|n

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

Question 16

Derivative of an arithmetical function

Answer 16

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

f’(n) = f(n) log n for n ≥ 1.

Question 17

Arithmetic mean

Answer 17

_ n
f = 1/n ∑ f(k)
k=1

Question 18

Big Oh notation

Answer 18

If g(x) > 0 for all x ≥ a, we write f(x) = O(g(x)) to mean that the quotient f(x) / g(x) is bounded for all x ≥ a. i.e. there exists a constant C > 0 such that |f(x)| ≤ Cg(x) for all x ≥ a. C is referred to as the implied constant.

Question 19

Vinogradov’s less than less notation

Answer 19

If f(x) = O(g(x)), then we can write this as f(x) ≪ g(x). Furthermore if g(x) ≪ f(x) ≪ g(x), we can write this as f(x) ≍ g(x).

Question 20

Little Oh notation

Answer 20

If lim(x⟶ ∞) f(x) / g(x) = 0, we write f(x) = o(g(x)).

Question 21

Asymptotic

Answer 21

If lim(x ⟶ ∞) f(x) / g(x) = 1, we say that f(x) is asymptotic to g as x ⟶ ∞, and we write f(x) ∼ g(x) as x ⟶ ∞.

Question 22

Riemann-zeta function

Answer 22

Let ζ(s) denote the Riemann-zeta function which is defined by the equation
∞
ζ(s) = ∑ n⁻ˢ if s > 1
n=1

ζ(s) = lim ( ∑ n⁻ˢ - (x¹⁻ˢ) / (1 - s) ) if 0 < s < 1
x⟶∞

Question 23

Chebyshev’s ψ-function

Answer 23

For x > 0, Chebyshev’s ψ-function is defined by the formula

ψ(x) = ∑ Λ(n)
n≤x

Question 24

Chebyshev’s θ-function

Answer 24

For x > 0, Chebyshev’s θ-function is defined by the equation

θ(x) = ∑ log p
p≤x

Question 25

M(x)

Answer 25

If x ≥ 1, we define

M(x) = ∑ µ(n)
n≤x

Question 26

H(x)

Answer 26

If x ≥ 1, we define

H(x) = ∑ µ(n) log n
n≤x

Question 27

Abscissa of absolute convergence

Answer 27

Suppose the series ∑ |f(n)/nˢ| does not converge for all s or diverge for all s. Then there exists a real number 𝜎ₐ, called the abscissa of absolute convergence, such that the series ∑ f(n)/nˢ converges absolutely if 𝜎 > 𝜎ₐ but does not converge absolutely if 𝜎 < 𝜎ₐ.
If the series absolutely converges everywhere, we define 𝜎ₐ = +∞, and if the series absolutely converges nowhere, we define 𝜎ₐ = -∞.

Question 28

Gamma function

Answer 28

For σ > 0, we have the integral representation
∞
Γ(s) = ∫ xˢ⁻¹ e⁻ˣ dx
0

Question 29

Abscissa of convergence

Answer 29

If the series ∑ f(n)/nˢ does not converge everywhere or diverge everywhere, then there exists a real number 𝜎_c, called the abscissa of convergence, such that the series converges for all s in the half-plane 𝜎 > 𝜎_c and diverges for all s in the half-plane 𝜎 < 𝜎_c.
If the series converges everywhere, we define 𝜎_c = +∞, and if the series converges nowhere, we define 𝜎_c = -∞.

Question 30

Dirichlet character

Answer 30

Let k be a positive integer. Then a function χ : Z ⟶ C is called a Dirichlet character modulo k if

(i) χ(mn) = χ(m) χ(n) for all m, n ∈ Z
(ii) χ(n + k) = χ(n) for all n ∈ Z
(iii) χ(1) = 1
(iv) χ(n) = 0 whenever (n, k) > 1

Question 31

Principal / trivial character

Answer 31

The principal (or trivial) character modulo k is given by

χ₀(n) = 1  if (n, k) = 1
           0 if (n, k) > 1

Question 32

Holomorphic Function

Answer 32

A function on an open set U which is differentiable at every point of U is called a holomorphic function on U.

Question 33

Principal part of a Laurent series

Answer 33

Suppose
∞
∑ aₙ(z - a)ⁿ
n=-∞
is a Laurent series convergent in an annulus
A = {z ∈ C : R < |z - a| < S} for R < S. Then
-1
∑ aₙ(z - a)ⁿ
n=-∞
is referred to as the principal part of the Laurent series.

Question 34

Isolated singularity

Answer 34

Let U be a domain on which f is holomorphic. If a is a point not in U but such that the punctured disc D’(a, r) = {z ∈ C : 0 < |z - a| < r} is a subset of U for some r > 0, then we say that f has an isolated singularity at a.

Question 35

Pole of order N

Answer 35

If there is an integer N ∈ N such that b_(-N) ≠ 0 but b₋ₘ = 0 if
m > N, then we say that f has a pole of order N at a. The principal part is finite and equals
b_(-N) / (z - a)^N + … + b₋₁ / (z - a)