Euler’s product formula and the density of primes
Euler derived a beautiful formula which equates a familiar infinite series with an infinite product defined over all prime numbers. This formula became the basis for much of the subsequent study of the
distribution of prime numbers.
The infinite series is familiar from calculus: by the “integral test” we know that when s > 1 the series
ζ(s) = 1 +
1
1
1
1
1
+ s + s + s + s ···
s
2
3
4
5
6
(1)
converges. [Verify this!] Moreover, if p > 1 then for any positive s we also have a convergent geometric
series of the form
1
1
1
= 1 + s + 2s + · · ·
(2)
1 − p−s
p
p
Euler’s insight was this. Each term in ζ(s) is uniquely factorable as
1
1
1
= s e1 × · · · × s e
ns
(p1 )
(pk ) k
(3)
where the pj are distinct primes and the ej > 0. But by the distributive property (and unique factorization, again) the infinite product of all of the series in (2) is the sum of all of the terms, each appearing
exactly once. That is,
∞
Y
X
1
1
=
.
(4)
s
n
1 − p−s
n=1
p a prime
[Check the details!]
Note that this gives another proof that there are infinitely many primes. Indeed, if there were
only finitely many, then thre would be only finitely many terms in the product on the right-hand side
of (4), and so we would obtain a finite limit as we let s → 1. However, the left-hand side becomes
infinite as s → 1. [Why?] It would take over a century to elapse and the appearance of the remarkable
GFB Riemann before this idea was fully exploited. When Riemann took up this thread, he fashioned a
beautiful tapestry which includes, among other things, the famous Prime Number Theorem.
This theorem states that if π(x) denotes the number of primes less than x then
Z x
du
π(x) ∼
.
(5)
2 log u
If we apply L’Hospital’s Rule to the right-hand side of this we obtain the result
π(x) ∼
x
.
log x
(6)
[Check this!] This is a less accurate, but asymptotically equivalent form. (See the table below.)
Either form tells us that the density of primes in the neighborhood of x — or, somewhat more
loosely, the probability of finding a prime near x — is approximately 1/ log x. We can go back to the
Euler product formula and give a heuristic explanation as to why this might be so.
Heuristically, Euler’s product formula tells us that
X 1
Y
1
≈
,
n
1 − p−1
n≤x
p≤x
and since the left-hand side is asymptotic to log x [why?] we obtain that
X
log(log x) ∼ −
log(1 − p−1 ).
p≤x
Now when |t| < 1 we have that
log(1 − t) = −t + c(t) · t2 ,
1
where 0 < |c(t)| < 1 [why?] and thus
log(log x) ∼
X1
X1
+ constant ∼
.
p
p
(7)
p≤x
p≤x
On the other hand,
Z
log(log x) =
1
log x
du
=
u
Z
e
x
1 dt
·
.
t log t
(8)
Let’s step back and see what the heuristics (7) and (8) are trying to tell us. If we think of sums (and
integrals) as averaging processes, then each is a calculation of a weighted average (or expected value) of
the function 1/t. The first assigns a density of 1 to primes and 0 everywhere else. The second assigns
a continuous density of 1/ log t. Thus we might be led to speculate that the density of primes in the
neighborhood of x is 1/ log x.
For detailed mathematical analysis of Riemann’s famous paper, which gave birth to analytic number
theory, read Harold Edwards’ Riemann’s Zeta Function. For a broader sweep, including a biography of
Riemann, see John Derbyshire’s Prime Obsession.
Rn
n
π(n)
dt/ log t
rel. err. n/ log n
rel. err.
2
106
107
108
109
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
78498 7.8626 · 104
−1.64 · 10−3 7.2382 · 104
5
664579 6.6492 · 10
−5.09 · 10−4 6.2042 · 105
6
5761455 5.7622 · 10
−1.31 · 10−4 5.4287 · 106
7
50847534 5.0849 · 10
−3.34 · 10−5 4.8255 · 107
8
455052511 4.5506 · 10
−6.82 · 10−6 4.3429 · 108
9
4118054813 4.1181 · 10
−2.81 · 10−6 3.9481 · 109
10
37607912018 3.7608 · 10
−1.01 · 10−6 3.6191 · 1010
11
346065536839 3.4607 · 10
−3.16 · 10−7 3.3407 · 1011
12
3204941750802 3.2049 · 10
−9.60 · 10−8 3.1021 · 1012
13
29844570422669 2.9845 · 10
−3.00 · 10−8 2.8953 · 1013
14
279238341033925 2.7924 · 10
−4.00 · 10−9 2.7143 · 1014
15
2623557157654233 2.6236 · 10
7.60 · 10−9 2.5547 · 1015
16
24739954287740860 2.4740 · 10
2.80 · 10−9 2.4127 · 1016
234057667276344607 2.3406 · 1017
6.40 · 10−9 2.2858 · 1017
2220819602560918840 2.2208 · 1018
9.90 · 10−9 2.1715 · 1018
21127269486018731928 2.1127 · 1019
1.28 · 10−8 2.0681 · 1019
This table was compiled using Maple (ver. 9.5).
2
7.79 · 10−2
6.64 · 10−2
5.78 · 10−2
5.10 · 10−2
4.56 · 10−2
4.13 · 10−2
3.77 · 10−2
3.47 · 10−2
3.21 · 10−2
2.99 · 10−2
2.79 · 10−2
2.63 · 10−2
2.48 · 10−2
2.34 · 10−2
2.22 · 10−2
2.11 · 10−2