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.