A PRIMER OF ANALYTIC NUMBER THEORY: From Pythagoras to ...

13.3 Siegel Zeros and the Class Number 323
every term in �(t, F). Then, (13.10) implies that
�(s, F) ≥ 1 1
−
s − 1 s .
When F(x, y) is the principal class, throw out everything except the summand
(x, y) = (1, 0) in the first integral. Because F(1, 0) = 1, (13.10) implies that
�(s, F) ≥ 1 1
−
s − 1 s +
� ∞
exp(−2�t/ � s dt
|d|)t
t
1
in this case. We now sum these inequalities over all h equivalence classes to
get
�(s, d) ≥
h
s(s − 1) +
� ∞
1
exp(−2�t/ � |d|)t
s dt
, (13.14)
t
where we used (13.9) to sum the expressions �(s, F) and where we put the h
expressions 1/(s − 1) − 1/s over a common denominator. Now, (13.11) says
that the left side above is
�(s, d) = h
+ O(1) as s → 1.
s − 1
So, particularly when s approaches 1 from below, the function is negative,
because then 1/(s − 1) →−∞. The GRH would then say it must be negative
for all s between 1/2 and 1; it cannot change signs. Otherwise, it is
certainly negative for � < s < 1, where � is the rightmost counterexample
to GRH. Now is a good time to point out that any possible counterexamples
to GRH must come from L(s, �d) in the factorization (13.9). The term
2|d| s/2 (2�) −sƔ(s) is positive for s > 0, and we proved in Section 8.1 that
�(s) has no real zeros on the interval 0 < s < 1.
Hecke’s idea was simply to investigate the consequences for (13.14) of the
inequality �(s, d) < 0. We have
0 >�(s, d) ≥
h
s(s − 1) +
� ∞
which, on rearranging the terms, says that
� ∞
h > s(1 − s) exp(−2�t/ � s dt
|d|)t
t .
1
1
exp(−2�t/ � s dt
|d|)t
t ,
If we integrate over an even smaller range, √ |d| ≤t, the integral is yet smaller,
and we then change the variables in the integral with u = t/ √ |d|, which