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
Previous Page Next Page