194 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONSRemark 1.5. This zero-free region gives the PNT as above with an error term of
the form O(x exp(-c)log x)) for some absolute positive c; the Riemann Hypothesis
gives the (optimal) error term O(x^112 log x ).
Suppose one needs to show that for a given 7f, L(7r, s) does not vanish at s =
a 0 + it 0 for some t 0 E R and some a 0 :S; 1 but close to 1; the Hadamard/De
la Vallee-Poussin method is based on the possibility of constructing an auxiliary
Dirichlet series D with non-negative coefficients, convergent for s > 1, which is
divisible by L(7r, s + it 0 ) to an order larger than the order of the pole of D at s = 1.
In the case of Riemann's (, this is achieved with the product
D(s) = ((s)^3 ( (s + ito)^4 ((s + 2ito).
Indeed, from the trigonometric inequality 3+4 cos(t 0 )+cos(2t 0 ) ;;:: 0, one sees easily
that -D' ( s) / D( s) has non-negative Dirichlet coefficients, hence the coefficients of
D(s) are non-negative. Alternatively, one can also use
(1.12) D(s) = ((s)^3 ((s + ito)^2 ((s - ito)^2 ((s + 2ito)((s - 2ito).
Suppose that ((1 + it 0 ) = 0 (and t 0 -j. 0 since we know that ( does not vanish
on [O, 1]); then D(s) has a pole of order 3 at s = 1 (coming from the ( factors),
compensated by a zero of order at least 4, and thus D( s) has no poles on the
real axis, which contradicts Landau's Lemma. The extension of the non-vanishing
of ( ( s) to the zero-free region (1.11) follows from a more quantitative analysis
involving the logarithmic derivative of D(s), and is possible because 4 is strictly
larger than 3 (see below).
It is possible to extend the method to more general L-functions; the case
of Dirichlet characters is straightforward and was carried out by Landau, Gron-
wall and Titchmarsh (see [Dav] for instance), except for a new difficulty that
we discuss below. The case of general automorphic L-functions was treated by
Moreno (for d = 2), Perelli et al. (in a somewhat axiomatic setting), and by
Hoffstein/Ramakrishnan in general ([Mor2, Mor3, CMP, HR]); the proofs in this
generality are deeper, as they involve the analytic properties of Rankin/Selberg
L-functions. Indeed, the non-vanishing is obtained by considering the auxiliary
products:
(1.13) D(s) = ((s)L(7r 0 if-, s)^2 L(7r, s + it 0 )^2 L(ir, s - it 0 )^2
x L(7r 0 7f, s + 2it 0 )L(ir 0 if-, s - 2it 0 ),
if if-"I-7f 0 1-I 2ito or
(1.14) D(s) = ((s)L(7r, s + ito)^2 L(7r 0 7f, s + 2it 0 ) ,if if-= 7f 0 l-l^2 ito. These Dirichlet series have non-negative coefficients: these are
each the L-function of pairs L(II Q9 fr, s ), where II is the (self-dual) isobaric sum
representation (cf. Remark 1.3):
II= 1EB7f ® l· lito EB if- 0 1.1-ito (resp. II= 1EB7f ® l· lito ).
Again, if L(7r, 1 + it 0 ) = 0 then D(s) has no pole at s = 1, contradicting Lan-
dau's Lemma. To extend the non-vanishing to regions inside the critical strip, it
is more efficient to consider the logarithmic derivative D' ( s) / D ( s) combined with
Hadamard's factorization theorem and positivity arguments: a general version of
the de la Vallee-Poussin method is given in the following lemma of Goldfeld/ Hoff-
stein/ Lieman [GHL]: