ANALYTIC NUMBER THEORY: HOMEWORK 1
Exercise 1. For any two integers k, n let (k, n) denote their gcd.
(1) Show that there is a multiplicative arithmetic function g, satisfying that
for any arithmetic function f
n
X
f ((k, n)) =
X
k=1
d|n
(2) Use this to deduce the identity
n
f (d)g( ).
d
Pn
k=1 (k, n)µ((k, n))
= µ(n).
Exercise 2. Prove that for x > 2
X d(n)
1
= log2 (x) + 2γ log(x) + O(1),
n
2
n≤x
where γ is Euler’s constant.
Exercise 3. For s > 0 and integer q > 1, find an asymptotic formula for
X 1
,
ns
n≤x
(n,q)=1
with an error term that goes to zero as x → ∞. (note: the formula will look
different for s < 1, s = 1 and s > 1).
Exercise 4. For x ≥ 2 we define
x
Z
Li(x) =
2
Prove that
Li(x) =
and that
Z
2
x
+
log(x)
x
Z
2
dt
.
log(t)
x
2
dt
−
,
log2 t log 2
dt
x
).
2 = O(
log t
log2 x
Exercise 5. Recall that PNT is equivalent to ψ(x) ∼ x. In this exercise you will
show that if ψ(x)/x has a limit it must be 1.
(1) Prove the asymptotic formula 1
X x
ψ( ) = x log x + O(x)
n
n≤x
x
x
(2) Assuming Chebyshev’s bound c1 log(x)
≤ π(x) ≤ c2 log(x)
, show that |ψ(x)/x|
is uniformly bounded.
x x
1Hint: consider the sum P
n≤x ψ( n )[ n ].
1
2
ANALYTIC NUMBER THEORY: HOMEWORK 1
(3) Let δ = lim sup ψ(x)/x. Given > 0 choose N so that ψ(x) > ( + δ)x for
all x > N . Show that for x > N ,
X
ψ(x) ≤ (δ + )x log x + xψ(N ),
n≤x
and deduce that δ ≥ 1.
(4) Use a similar argument to show that γ = lim inf ψ(x)/x satisfies γ ≤ 1.
Exercise 6. Given an arithmetic function f , we define the corresponding Dirichlet
series
∞
X
f (n)
.
L(f, s) =
ns
n=1
(1) Assume that f is a multiplicative function and show that when the series
absolutely converges it has an Euler product
!
∞
Y X
L(f, s) =
f (pν )p−νs .
p
(2) Suppose that f (n) =
P
k=0
d|n g(d) and show that L(f, s) = L(g, s)ζ(s).
0
(3) Show that L(Λ, s) = − ζζ (s).