T2: 2. Chebyshev's Estimates Flashcards
1
Q
State the formula for the power of prime p in factorisation of n!
A
c_p(n) = SUM k=1,∞ ⌊n/p^k⌋
2
Q
Give Chebyshev’s initial statement
A
π(x) ≍ x/log x
x/logx «_space;π(x) «_space;x/logx
for all x ≥ 2
3
Q
State the Von-Mangoldt function
A
Λ(n) = { logp if n = p^k
0, otherwise
4
Q
State the setup for partial summation
A
Given a sequence of complex numbers {a_n} n=1,∞
and a function f: [1,x] → C, cont diff for some x:
define A(y) = SUMn≤y a_n
5
Q
State the partial summation formula
A
SUMn≤x a_n f(n) = A(x)f(x) - INT_1,x A(t)f’(t) dt
6
Q
Give Chebyshev’s functions
A
π(x) = SUM_p≤x 1
θ(x) = SUM_p≤x logp
ψ(x) = SUM_n≤x Λ(n)
7
Q
Give the relation between Chebyshev’s functions
A
ψ(x) ~ θ(x) ~ π(x)logx
