Definitions Flashcards
Divisibility
d divides n (d|n) whenever n = cd for some c.
Common divisor
If d|a and d|b, then d is a common divisor of a and b.
Greatest common divisor
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.
Prime, composite number
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.
Arithmetical function
A real or complex valued function defined on the positive integers is called an arithmetical function, f : N ⟶ C.
Mobius function
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
Euler-Totient funciton
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
Dirichlet product / Dirichlet convolution
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
Identity function
The identity function I is defined as follows:
I(n) = [1/n] = 1 if n = 1
0 if n > 1
Unit function
We define the unit function by u(n) = 1 for all n.
Von-Mangoldt function
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
Multiplicative function
An arithmetical function f is called multiplicative if f ≠ 0 and if f(mn) = f(m)f(n), whenever gcd(m, n) = 1.
Completely multiplicative function
A multiplicative function f is called completely multiplicative if we have f(mn) = f(m)f(n) for all m, n.
Liouville’s function
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
Divisor functions σ_α (n)
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.
Derivative of an arithmetical function
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.
Arithmetic mean
_ n
f = 1/n ∑ f(k)
k=1
Big Oh notation
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.
Vinogradov’s less than less notation
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).
Little Oh notation
If lim(x⟶ ∞) f(x) / g(x) = 0, we write f(x) = o(g(x)).
Asymptotic
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 ⟶ ∞.
Riemann-zeta function
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⟶∞
Chebyshev’s ψ-function
For x > 0, Chebyshev’s ψ-function is defined by the formula
ψ(x) = ∑ Λ(n)
n≤x
Chebyshev’s θ-function
For x > 0, Chebyshev’s θ-function is defined by the equation
θ(x) = ∑ log p
p≤x
M(x)
If x ≥ 1, we define
M(x) = ∑ µ(n)
n≤x
H(x)
If x ≥ 1, we define
H(x) = ∑ µ(n) log n
n≤x
Abscissa of absolute convergence
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 = -∞.
Gamma function
For σ > 0, we have the integral representation
∞
Γ(s) = ∫ xˢ⁻¹ e⁻ˣ dx
0
Abscissa of convergence
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 = -∞.
Dirichlet character
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
Principal / trivial character
The principal (or trivial) character modulo k is given by
χ₀(n) = 1 if (n, k) = 1
0 if (n, k) > 1
Holomorphic Function
A function on an open set U which is differentiable at every point of U is called a holomorphic function on U.
Principal part of a Laurent series
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.
Isolated singularity
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.
Pole of order N
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)
