Also note that Theorem 5.16 follows directly from the above theorem and

Exercise 5.8.

Although the above estimate on the error term in the approximation of

π(x) by li(x) is pretty good, it is conjectured that the actual error term is

much smaller:

Conjecture 5.18. For all x ≥ 2.01, we have

|π(x) ’ li(x)| < x1/2 log x.

Conjecture 5.18 is equivalent to a famous conjecture called the Riemann

hypothesis, which is an assumption about the location of the zeros of a

certain function, called Riemann™s zeta function. We give a very brief,

high-level account of this conjecture, and its connection to the theory of the

distribution of primes.

For real s > 1, the zeta function is de¬ned as

∞

1

ζ(s) := . (5.11)

ns

n=1

5.5 The prime number theorem . . . and beyond 89

Note that because s > 1, the in¬nite series de¬ning ζ(s) converges. A

simple, but important, connection between the zeta function and the theory

of prime numbers is the following:

Theorem 5.19 (Euler™s identity). For real s > 1, we have

(1 ’ p’s )’1 ,

ζ(s) = (5.12)

p

where the product is over all primes p.

Proof. The rigorous interpretation of the in¬nite product on the right-hand

side of (5.12) is as a limit of ¬nite products. Thus, if p1 , p2 , . . . is the list of

primes, we are really proving that

r

(1 ’ p’s )’1 .

ζ(s) = lim i

r’∞

i=1

Now, from the identity

∞

p’s )’1 p’es ,

(1 ’ =

i i

e=0

we have

r

(1 ’ p’s )’1 = 1 + p’s + p1 + · · ·

’2s

· · · 1 + p’s + p’2s + · · ·

r r

1

i

i=1

∞ ∞

(pe1 · · · per )s

···

= r

1

e1 =0 er =0

∞

gr (n)

= ,

ns

n=1

where

1 if n is divisible only by the primes p1 , . . . , pr ;

gr (n) :=

0 otherwise.

Here, we have made use of the fact (see §A5) that we can multiply term-wise

in¬nite series with non-negative terms.

Now, for any > 0, there exists n0 such that ∞ 0 n’s < (because

n=n

the series de¬ning ζ(s) converges). Moreover, there exists an r0 such that

gr (n) = 1 for all n < n0 and r ≥ r0 . Therefore, for r ≥ r0 , we have

∞ ∞

gr (n)

n’s < .

’ ζ(s) ¤

s

n n=n

n=1 0

90 The distribution of primes

It follows that

∞

gr (n)

lim = ζ(s),

ns

r’∞

n=1

which proves the theorem. 2

While Theorem 5.19 is nice, things become much more interesting if one

extends the domain of de¬nition of the zeta function to the complex plane.

For the reader who is familiar with just a little complex analysis, it is easy

to see that the in¬nite series de¬ning the zeta function in (5.11) converges

absolutely for complex numbers s whose real part is greater than 1, and that

(5.12) holds as well for such s. However, it is possible to extend the domain

of de¬nition of ζ even further”in fact, one can extend the de¬nition of ζ in

a “nice way ” (in the language of complex analysis, analytically continue)

to the entire complex plane (except the point s = 1, where there is a simple

pole). Exactly how this is done is beyond the scope of this text, but assuming

this extended de¬nition of ζ, we can now state the Riemann hypothesis:

Conjecture 5.20 (Riemann hypothesis). For any complex number s =

x + yi, where x and y are real numbers with 0 < x < 1 and x = 1/2, we

have ζ(s) = 0.

A lot is known about the zeros of the zeta function in the “critical strip,”

consisting of those points s whose real part is greater than 0 and less than

1: it is known that there are in¬nitely many of them, and there are even

good estimates about their density. It turns out that one can apply standard

tools in complex analysis, like contour integration, to the zeta function (and

functions derived from it) to answer various questions about the distribution

of primes. Indeed, such techniques may be used to prove the prime num-

ber theorem. However, if one assumes the Riemann hypothesis, then these

techniques yield much sharper results, such as the bound in Conjecture 5.18.

Exercise 5.21. For any arithmetic function a, we can form the Dirichlet

series

∞

a(n)

Fa (s) := .

ns

n=1

For simplicity we assume that s takes only real values, even though such

series are usually studied for complex values of s.

(a) Show that if the Dirichlet series Fa (s) converges absolutely for some

real s, then it converges absolutely for all real s ≥ s.

5.5 The prime number theorem . . . and beyond 91

(b) From part (a), conclude that for any given arithmetic function a,

there is an interval of absolute convergence of the form (s0 , ∞),

where we allow s0 = ’∞ and s0 = ∞, such that Fa (s) converges

absolutely for s > s0 , and does not converge absolutely for s < s0 .

(c) Let a and b be arithmetic functions such that Fa (s) has an interval

of absolute convergence (s0 , ∞) and Fb (s) has an interval of absolute

convergence (s0 , ∞), and assume that s0 < ∞ and s0 < ∞. Let

c := a b be the Dirichlet product of a and b, as de¬ned in §2.6.

Show that for all s ∈ (max(s0 , s0 ), ∞), the series Fc (s) converges