1.1. The method of Chebyshev 3

by interchanging the order of summation in the double sum. Then

m≤N

n≤N/m

Λ(n) = log(N!) =

n≤N

log(n).

Any mapping f : N → C from the positive integers into the complex numbers

is called an arithmetic function. The functions Λ and log are examples of

arithmetic functions. Every arithmetic function f has a summatory function

F (x) =

n≤x

f(n).

Thus ψ is the summatory function of Λ. The summatory function

T(x)

def

=

n≤x

log(n)

of log is also important. The identity

m≤x

ψ

x

m

=

n≤x

Λ(n)

x

n

= T(x)

holds for nonnegative x since log(N!) = T(x) where N = [x]. This identity

is the starting point for the method of Chebyshev.

Proposition 1.1. The inequalities

log(2)x − log(4x) ≤ ψ(x) ≤ 2 log(2)x +

log2(x)

log(2)

hold for x ≥ 1.

Proof. The terms in the last sum in the computation

T(x) − 2 T

x

2

=

n≤x

log(n) − 2

m≤x/2

log(m)

=

n≤x

log(n) − 2

2m≤x

log(2m) + 2

2m≤x

log(2)

=

n≤x

(−1)n−1

log(n) + 2

x

2

log(2)

alternate in sign and increase in magnitude. So

T(x) − 2 T

x

2

− 2

x

2

log(2) ≤ log([x])

for x ≥ 1. Thus

log(2)x − log(4x) ≤ T(x) − 2 T

x

2

≤ log(2)x + log(x).

Substituting the expression

T(x) =

n≤x

ψ

x

n