Lectures 10 & 11

Question 1

For all x ≥ 1, we have ∣∑^∞(n=1) µ(n)/n∣ ≤ 1

When does the equality hold

Answer 1

When x = 2

Question 2

Show ∑_(n≤x) µ(n)/n = 1

Answer 2

∑_(n<2) µ(n)/n = µ(1)/1= 1.

Question 3

Give Legendre’s identity

Answer 3

[x]! = ∏_(p≤x) p^(α(p))

Where α(p) =∑^∞_(m=1) [x/(p^m)] .

Question 4

What does the following imply
* log[x]! = x log x − x + O(log x)

If x >= 2

Answer 4

∑_(n≤x) Λ(n) [x/n] = x log x − x + O(log x).

Question 5

What does the following equal
* ∑_(p≤x) [x/p] log p

For x ≥ 2

Answer 5

x log x + O(x)

Question 6

If a, b ∈ R such that ab = x, find
* ∑(q,d)(qd≤x) f(d)g(q)

Answer 6

= ∑(n≤a) f(n)G(x/n) + ∑(n≤b) g(n)F (x/n) − F(a)G(b)

Question 7

Give Chebyshev’s ψ-function

Answer 7

ψ(x) = ∑_(n≤x) Λ(n)

For x > 0

Question 8

Give Chebyshev’s θ-function

Answer 8

θ(x) = ∑_(p≤x) log p

For x > 0

Question 9

What are the bounds of
* ψ(x)/x − θ(x)/x

Answer 9

0 ≤ ψ(x)/x − θ(x)/x ≤ (log x)^2/(2 √x log 2)

Question 10

Give Abels identity

Answer 10

For any arithmetical function a(n), let A(x) = ∑_(n≤x) a(n), where A(x) = 0 if x < 1. Assume f has a continuous derivative on the interval [y, x], where 0 < y < x. Then we have

• ∑_(y<n≤x) a(n)f(n) = A(x)f(x) − A(y)f(y) − ∫^x_y A(t)f′(t)dt.

Question 11

∑_(y<n≤x) f(n)

Find this using Abel’s identity, then attain Euler’s summation formula

Answer 11

= [x]f(x) − [y]f(y) − ∫^x_y [t]f′(t)dt

Then by integration by parts

∫^x_y tf′(t)dt = xf(x) − yf(y) − ∫^x_yf(t)dt,