Lecture 14

Question 1

Apply Shapiros theorem to the Von Magnoldt function

Answer 1

∑(n≤x) Λ(n)/n = log x + O(1)

Question 2

∑(n≤x) ψ (x/n)

Find the answer using Shapiros theorem

Answer 2

x log x − x + O(log x)

Question 3

∑_(n≤x) θ (x/n)

Find the answer using Shapiros theorem

Answer 3

x log x + O(x)

Question 4

∑(p≤x) 1/p

Give an aymptotic for this divergent sum

Answer 4

log log x + A + O (1/log x)

for all x greater than or equal to 2

Question 5

lim_(x→∞)(M(x)/x − H(x)/x log x)

Give the value of the limit and the formula for H(x)

Answer 5

• Limit equals 0
• H(x) = ∑_(n≤x) µ(n) log n