Exercises 37

(31) Calculate 1 ∗ λ, 1 ∗ and d ∗ λ where is the indicator function of the

squares.

(32) Calculate id ∗ id, φ ∗ σ and d ∗ φ.

(33) Calculate f ∗ f for f totally multiplicative.

(34) Use the Binomial Theorem to show that 1 ∗ μ = e.

(35) Show that the sum of the primitive n-th roots of unity equals μ(n).

(36) Deduce the formula for T in terms of ψ by means of 1 ∗ Λ = log.

(37) Show that 1∗(μ· log) = −Λ.

(38) Show that

n≤x

log(rad(n)) = x log(x) + O(x)

where rad is the radical.

(39) a) Show that the number of rationals in the interval [0, 1] written in

lowest terms and with denominator ≤ x is asymptotic to 3x2/π2.

b) Show that the chance that two randomly chosen large integers be

coprime is

6/π2.

c) A lattice point (j, k) occults the lattice point (m, n) if both lie on the

same ray from the origin and (j, k) lies closer to the origin. Show that

the proportion of lattice points in Z × Z visible from the origin is

6/π2.

(40) Show that

d|n

2ω(d)

=

d(n2),

and generalize (Dirichlet).

(41) Establish the estimate

n≤x

d(gcd(m, n)) −

σ(m)

m

x ≤ d(m)

for any fixed positive integer m.

(42) Show that

n≤x

σ(n)

n

=

π2

6

x + O(log(x))

as x → +∞.

(43) Let P

+(n)

denote the largest prime factor of n. Show that the infinite

series

P

+(n)≤x

μ(n)

n

converges absolutely for every x ≥ 1 and that the sum is always positive.

Find the limit of the sum as x → +∞.