Last min T2 Flashcards
How do we define the meromorphic continuation of ζ? What pole and residue do we find?
Using partial summation of Riemann zeta.
Pole s=1, residue =1
Write the functional equation for ζ (completed Riemann zeta)
Z(s) = π^-s/2 Γ(s/2) ζ(s)
Z(s) = Z(1-s)
Define the meromorphic continuation of Γ(s). What poles and residues do we find?
Γ(s) = INT_0,∞ e^-t t^s-1 dt
Poles at s=-1,-2,… and residues (-1)^n/n!
Give two relation formulas for Γ(s)
Γ(s+1) = sΓ(s)
Γ(s)Γ(1-s) = π/sinπs
State a general form form for Γ(n), n in natural numbers
Γ(n) = (n-1)!
Give the zeroes of Γ(s) and ζ(s)
Γ(s) has no zeroes
ζ(s) has simple zeroes at s = -2, -4, … (trivial zeroes)
Any other zeroes have R(s) in (0,1)
If x is a zero, so is x, 1-x and 1-x
State the Riemann hypothesis
All non-trivial zeroes of the ζ(s) function have Re(s) = 1/2
Define an arithmetic function
A function which maps from the natural numbers to complex plane
State the divisor function d(n)
d(n) = #{d ∈ N : d | n}
State the Euler totient function ϕ(n)
ϕ(n) = #{a ∈ N : a ≤ n and (a, n) = 1}
State the Mobius function µ(n)
µ(n) = {1 if n=1
{(-1)^r if n=p_1,…p_r for distinct primes
State the convolution of arithmetic functions
f ∗ g(n) = SUM_d|n f(d) g(n/d)
State the convolutions:
1 ∗ 1
1 ∗ µ
= d
= I
State the Mobius inversion formula
f(n) = SUM_d|n g(d) µ(n/d)
State the sum of convolutions formula
SUM_1,∞
