36 1. Arithmetic Functions
(21) a) Use integration by parts to show that
n+1
n
f(u) du =
f(n)
2
+
f(n + 1)
2
−
n+1
n
u − n −
1
2
f (u) du.
Here n is an integer and f a continuous function on the interval [n, n+1]
with f piecewise continuous there.
b) Use part a) to establish the Euler-Maclaurin summation formula.
(22) Prove the estimate
∞
n=−∞
e−πn2u
=
1
√
u
+ O(1)
as u →
0+.
This series comes from the theory of elliptic theta functions,
and satisfies a functional equation that yields a far more precise estimate.
(23) † Establish the estimate
n≤x
logm
x
n
= m!x +
O(m!logm(x))
uniformly in positive integers m.
(24) Show without integration that T(n) = n log(n) + O(n) by subdividing
the interval [1,n] between successive powers of 2.
(25) Show that
p≤x
1 +
1
p
∼ c log(x)
as x → +∞, for some positive constant c.
(26) Show that
p≤x
log2(p)
p
=
1
2
log2(x)
+ O(log(x))
as x → +∞.
(27) Show that the real number 1 is a point of accumulation of the sequence
of ratios (pk+1/pk)k=1
∞
of successive primes.
(28) Show that the series
p
1
p(log log(p))2
converges.
(29) A divisor d of an integer n is called a block divisor if it is coprime to
its complementary divisor n/d. In group theory such divisors are called
Hall divisors. Count the number of block divisors of n.
(30) Show that the multiplicative arithmetic functions under Dirichlet con-
volution constitute a subgroup of the group of all arithmetic functions
that have a Dirichlet inverse.
