Lecture 7

Question 1

What is the Big Oh Notation

Answer 1

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. That is, there exists a constant C > 0 such that

• ∣f(x)∣ ≤ Cg(x)

for all x ≥ a. The constant C is often referred to as the implied constant

Question 2

What does An equation of the form
f(x) = h(x) + O(g(x)) mean

Answer 2

An equation of the form
f(x) = h(x) + O(g(x))

Question 3

What does f(t) = O(g(t)) imply

for t ≥ a

Answer 3

∫^x_a f(t)dt = O (∫^x_a g(t)dt)

for x ≥ a.

Question 4

Vinogradov’s “less than less than” Notation

Answer 4

If f(x) = O(g(x)), then we can write this as
* f(x) ≪ g(x)

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

Question 5

Little Oh Notation

Answer 5

If
* lim(x→∞) f(x)/g(x) = 0

we write f(x) = o(g(x))

Question 6

Define Asymptotic

Answer 6

• lim_(x→∞) f(x)/g(x) = 1

we say that f(x) is asymptotic to g(x) as x → ∞, and we write

• f(x) ∼ g(x) as x → ∞

Question 7

What is Euler’s summation formula

Answer 7

If f has a continuous derivative f′ on the interval [y, x], where 0 < y < x, then

• ∑_(y<n≤x) f(n) = ∫^x_y f(t)dt + ∫^x_y (t − [t])f′(t)dt + f(x)([x] − x) − f(y)([y] − y)

Question 8

Define the Riemann-zeta function for s>1

Answer 8

• ζ(s) =∑^∞(n=1) 1/(n^s)

For s>1

Question 9

Define the Riemann-zeta function for f 0 < s < 1

Answer 9

ζ(s) = lim_(x→∞)(∑_(n≤x) (1/(n^s)) − x^(1−s)/(1 − s))

Question 10

Define Euler’s constant

Answer 10

γ = lim_(n→∞) (∑^n_(k=1) 1/k − log n) .

Question 11

∑_(n≤x) 1/n

Give the asymptotic formula

Answer 11

log x + γ + O (1/x)

where γ is Euler’s constant

Question 12

∑_(n≤x) 1/(n^s)

Give the asymptotic formula if s > 0, s ≠ 1.

Answer 12

x^(1−s)/(1 − s) + ζ(s) + O(x^(−s))

Question 13

∑_(n≤x) 1/(n^s)

Give the asymptotic formula if s > 1

Answer 13

O(x^(1−s))

Question 14

∑_(n≤x) n^α

Give the asymptotic formula if α ≥ 0

Answer 14

x^(α+1)/(α + 1)+ O(x^α)