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.
