The Arithmetic of Quaternion Algebra

56 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
The difference (2) - (3) is
�
[f(x) − f ∗ �
(x)] =
A 2
SL(2,A)/SL(2,K)
[f ∗ (0) − f(0)τ(u).
It follows that the volume of SL(2, A)/SL(2, K) for the measure τ equals 1.
Historic note
The zeta function of a central simple algebra over a number field was introduced
by K. Hey in 1929, who showed his functional equation in the case where the
algebra is a field. M. Zorn noticed in 1933 the application of the functional
equation to the classification of quaternion algebra(§3). The results of K.Hey
were generalized by H.Laptin [1], M.Eichler [4], and H.Maass [2] to the notion
of L-functions with characters. Applying the adele technique to their study is
made by Fusijaki 1, and the formulation of the most generality of zeta functions
is due to R. Godement [1], [2]. One can find the development of their theories
in T.Tamagawa [3], H. shimizu [3]. The application of the functional equation
to the computationof Tamagawa numbers can be found in A.Weil [2].
Exercise
Riemann zeta function. Deduce from the functional equation (Theorem 2.2)
that of the Riemann zeta function ζ(s) = �
n≥n n−s , Res > 1, with the
known formula
ζ(s) = π −s/2 Γ(s/2)ζ(s)
is invariant by s ↦→ 1 − s, or again :
ζ(1 − s) = 2
cos(πs/2)Γ(s)ζ(s).
(2π) s
Prove then for every integer k ≥ 1, the numbers ζ(−2k) are zero, the numbers
ζ(1 − 2k) are nonzero and given by
ζ(1 − 2k) = 2(−1)k (2k − 1))!
(2π) 2k ζ(2k)
and
ζ(0) = − 1
2 .
We know Bernoulli numbers B2k are defined by the expansion
Demonstrate
x
e x − 1
= 1 − x/2 + �
k≥1
(−1) k+1 B2k
ζ(2k) = 22k−1
(2k)! B2kπ 2k .
x2k (2k)! .
Deduce the number ζ(1 − 2k) are rational and are given by the formula
ζ(1 − 2k) = (−1) k B2k
2k .