12 1. Arithmetic Functions
The last inequality yields asymptotic estimates for the summatory functions
of log(m) and 1/m. The first of these is a weak version of Stirling’s formula.
Proposition 1.7. The estimates
n
m=1
log(m) = n log(n) n +
log(n)
2
+ 1 + Rn
and
n
m=1
1
m
= log(n) + γ +
1
2n
+ Sn
hold for all positive integers n with |Rn| 3/16 and |Sn|
3/(16n2).
Proof. The partial summation formula of Proposition 1.5 gives
n
m=1
log(m) = n log(n)
n
1
[u]
du
u
= n log(n) n + 1 +
1
2
log(n) +
n
1
S(u)
du
u
when f(n) 1 and g(x) = log(x). The last integral is bounded by 3/16
in absolute value by Proposition 1.6. Using the partial summation formula
again yields
n
m=1
1
m
=
1
n

n
1
[u]
du
u2
= 1 + log(n) +
1
2n

1
2


1
S(u)
du
u2
+

n
S(u)
du
u2
.
The last term is bounded by
3/(16n2)
by Proposition 1.6.
The real number γ = 0.5772 . . . is known as the Euler-Mascheroni con-
stant. It is unknown whether this is irrational. Note that Proposition 1.7
yields the version T(x) = x log(x) x + O(log(x)) of Stirling’s formula that
is most commonly applied in analytic number theory.
Proposition 1.8 (Euler-Maclaurin summation formula). If A B are
integers and f a continuous function on the interval [A, B] with f piecewise
continuous there, then
B
n=A
f(n) =
B
A
f(u) du +
f(A) + f(B)
2
+
B
A
S(u)f (u) du
with S(u) = u [u] 1/2 the sawtooth function.
Previous Page Next Page