5 Generalizations and Analogues 385ζK(s)=∏
P∈P( 1 −|P|−s)−^1holds in this open half-plane. Furthermore the definition ofζK(s)may be extended
so that it is nonzero and holomorphic in the closed half-planeRs≥1, except for a
simple pole ats=1. By applying Ikehara’s theorem we can then obtain theprime
ideal theorem, which was first proved by Landau (1903):
πK(x)∼x/logx,whereπK(x)denotes the number of prime ideals ofRwith norm≤x.
It was shown by Hecke (1917) that the definition of the Dedekind zeta-function
ζK(s)may also be extended so that it is holomorphic in the whole complex plane,
except for the simple pole ats=1, and so that, for some constantA>0 and non-
negative integersr 1 ,r 2 (which can all be explicitly described in terms of the structure
of the algebraic number fieldK),
ZK(s)=AΓ(s/ 2 )r^1 Γ(s)r^2 ζK(s)satisfies the functional equation
ZK(s)=ZK( 1 −s).Theextended Riemann hypothesisasserts that, for every algebraic number fieldK,ζK(s)=0forRs> 1 / 2.The numerical evidence for the extended Riemann hypothesis is favourable, although
in the nature of things it cannot be tested as extensively as the ordinary Riemann
hypothesis. The extended Riemann hypothesis implies error bounds for the prime ideal
theorem of the same order as those which the ordinary Riemann hypothesis implies
for the prime number theorem. However, it also has many other consequences. We
mention only two.
It has been shown by Bach (1990), making precise an earlier result of Ankeny
(1952), that if the extended Riemannhypothesis holds then, for each primep,thereisa
quadratic non-residueamodpwitha<2log^2 p. Thus we do not have to search far in
order to find a quadratic non-residue, or to disprove the extended Riemann hypothesis.
It will be recalled from Chapter II that ifpis a prime andaan integer not divisible
byp,thenap−^1 ≡1modp. For each primepthere exists aprimitive root,i.e.an
integerasuch thatak≡1modpfor 1≤k<p−1. It is easily seen that an even
square is never a primitive root, that an odd square (including 1) is a primitive root
only for the primep=2, and that−1 is a primitive root only for the primesp= 2 ,3.
Assuming the extended Riemann hypothesis, Hooley (1967) has proved a famous
conjecture of Artin (1927): if the integerais not a square or−1, then there exist
infinitely many primespfor whichais a primitive root. Moreover, ifNa(x)denotes
the number of primesp≤xfor whichais a primitive root, then
Na(x)∼Aax/logx forx→∞,