Exercises 39

b) Find a constant c(θ) such that

n≤x

[x/n]−θ

= c(θ)x + O(1)

holds uniformly for θ ≥ 0. Uniformity means that the constants C and

x0 that are implicit in the O-term do not depend on the parameter θ.

(53) Show that

n≤x

log(φ(n)) = x log(x) + (c − 1)x + O(log(x))

where

c =

p

1

p

log 1 −

1

p

.

Find the average order of log(φ(n)). Calculate the local average and

choose the length h of the interval in terms of x so as to minimize the

error term.

(54) a) Use the identity

n≤x

x

n

− ψ

x

n

− 2γ = D(x) − T(x) − 2γ[x]

and the Dirichlet bound in the divisor problem to show that ψ(x) =

O(x) independently of the Chebyshev method. This idea is due to Nina

Spears. Her approach may be refined to yield close upper and lower

numerical bounds for ψ(x)/x. Though unfortunately only at the cost of

extensive computation.

Note that this establishes the bound ϑ(x) = O(x) and Proposition

1.9 without reliance on the method of Chebyshev.

b) Show that

x/Kp≤x

log(p)

p

= log(K) + O(1)

where K ≥ 1 is an arbitrary constant, and the error is uniform in K.

c) Apply b) to show that ϑ(x) x without reliance on the method of

Chebyshev (H. N. Shapiro).

(55) † Find an asymptotic estimate for the sum

n≤x

d(n)

log(n) + 2γ

,

with a bound for the error term.