Number Theory: An Introduction to Mathematics

4 The Riemann Hypothesis 377

Since this holds for arbitrarily smallδ>0, it follows that

lim

x→∞

α(x)≤A.

Thus there exists a positive constantMsuch that

0 ≤α(x)≤M for allx≥ 0.

On the other hand, ify=x−δ,wherex≥δ, then for|v|≤λδ

α(y−v/λ)≤eh(δ+v/λ)α(x)≤e^2 hδα(x)

and hence
∫λy

−∞

α(y−v/λ)kˆ(v)dv≤e^2 hδα(x)

∫λδ

−λδ

kˆ(v)dv+M

∫

|v|≥λδ

kˆ(v)dv.

We can chooseλ=λ(δ)so large that the second term on the right is less thanδC.
Then, lettingx→∞we obtain

AC≤e^2 hδC lim

x→∞

α(x)+δC.

Since this holds for arbitrarily smallδ>0, it follows that

A≤ lim

x→∞

α(x).

Combining this with the inequality of the previous paragraph, we conclude that
limx→∞α(x)=A.

Applying Theorem 9 to the special case mentioned before the statement of the
theorem, we obtainψ(ex)∼ex. As we have already seen in§2, this is equivalent to
the prime number theorem.

4 The Riemann Hypothesis

In his celebrated paper on the distribution of prime numbers Riemann (1859) proved
only two results. He showed that the definition ofζ(s)can be extended to the
whole complex plane, so thatζ(s)− 1 /(s− 1 )is everywhere holomorphic, and he
proved that the values ofζ(s)andζ( 1 −s)are connected by a certain functional
equation. This functional equation will now be derived by one of the two methods
which Riemann himself used. It is based on a remarkable identity which Jacobi (1829)
used in his treatise on elliptic functions.

Proposition 10For any t,y∈Rwith y> 0 ,

∑∞

n=−∞

e−(t+n)

(^2) πy
=y−^1 /^2

∑∞

n=−∞

e−n

(^2) π/y
e^2 πint. (7)
In particular,
∑∞
n=−∞
e−n
(^2) πy
=y−^1 /^2

∑∞

n=−∞

e−n

(^2) π/y

. (8)