2 1. Arithmetic Functions
with weight 1. This gives the weighted counting function
ϑ(x)
def
=
p≤x
log(p)
introduced by P. L. Chebyshev. To obtain nice formulas that are easier to
analyze, it is advantageous also to count prime powers
pk
with weight log(p).
The von Mangoldt function Λ given by
Λ(pk)
= log(p) when the argument
is a prime power, and zero otherwise, serves this purpose. The weighted
counting function
ψ(x)
def
=
pk≤x
log(p) =
n≤x
Λ(n)
was also introduced by Chebyshev. Since prime powers with exponent higher
than one are quite sparse, ψ(x) mainly counts primes with weight log(p).
The functions ψ(x) and ϑ(x) are thus approximately equal, and both are
closely related to the counting function π(x) of the primes. In particular it
will turn out that the Prime Number Theorem may equally well be expressed
as one of the asymptotic relations ψ(x) x or ϑ(x) x. These formulations
are often more convenient.
The integers n = pm in the interval 0 n N that are divisible by
a prescribed prime p are given by the integer solutions m of the inequality
0 m N/p. The largest integer less than or equal to a real number x is
denoted by [x]. It is called the integer part of x or the Gauss bracket. Clearly
[N/p] is the number of integers n as above. The same reasoning shows that,
of these integers, exactly
[N/pk]
are divisible by
pk.
This observation allows
us to write down the prime factorization
N! =
pk
p[N/pk]
of the factorial, due to A.-M. Legendre. The product is taken over all prime
powers, but has only finitely many factors different from 1 because
[N/pk]
=
0 when
pk
N. The importance of the identity lies in the fact that the left-
hand side does not contain the primes explicitly, and is susceptible of being
estimated analytically. Taking the logarithm on both sides of the Legendre
identity yields
n≤N
Λ(n)
N
n
=
pk
log(p)
N
pk
= log(N!).
Now
pk
log(p)
N
pk
=
pk
log(p)
mpk≤N
1 =
m≤N
pk≤N/m
log(p)
Previous Page Next Page