Analytic Number Theory - Iwaniec H.,Kowalski E.

where p runs over the non-trivial zeros of ((s). [Hint: Start as above, but move the line of integration far to the left instead of using the functional equation; the first term comes from the pole at s = 1 and the trivial zeros of ((s) at s = -2k, k ? 1 an integer.] Sometimes it is more convenient to work with the explicit formula in a Fourier form rather than the Mellin form (5.44). This can be derived from (5.44) by simple substitutions
L(AJ(n)+AJ(n))g(~n) n

=g(O)logq(f)+rh(

i7r)

4

where p = !+h runs over the zeros of L(J, s) in the strip 0 :S; a :S; 1 (the non-trivial and the trivial ones) with relevant multiplicity.

5.

110

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

5.6. The prime number theorem.

By Prime Number Theorem for an £-function L(!, s), we mean first the asymptotic behavior of the sum

1/;(f,x) = LAt(n)

(5.46)

n~x

which is essentially the sum of >-t(P) logp over primes. When L(f, s) = ((s), this amounts indeed to counting primes p ( x. Much of the arithmetic interest lies in the dependency of the error term in asymptotic approximations on extra parameters, notably on the conductor when f varies in a family. Therefore we are seeking results which are explicit in every possible parameter, although not the strongest ones. The strength of results depends on the depth of the zero-free region for L(f, s). The best we can hope for today is that L(f, s) has no zeros in the region (5.39) with at most one exception of a simple zero f3t in case off self-dual. By our definition (only for the purpose in this section) the exceptional zero is in the segment c 1- d41og(3q(f)) ( f3t < 1

(5.47)

where c is the absolute positive constant from (5.39). This definition may cause some discomfort for new enthusiasts of analytic number theory, because it is loose with respect to the constant c. However, the practice shows that one benefits from such a flexible concept. After all, c can be improved by new tools in the future, not to mention that we don't believe in the existence of the exceptional zero of any £--function. In full generality we do not yet have a strong bound for the individual coefficients of L(f, s). But a crude bound for A1 (n) on average will suffice. Specifically, we postulate the following: (5.48)

L IAt(n)l

2

«

2

2

xd log (xq(f))

n~x

for x ) 1, where the implied constant is absolute. EXERCISE

(5.49)

6. Derive from (5.48) that

1/;(f, x) =

L >-t(P) logp + 0( y'xd !og (xq(f)) 2

2

p~x

where the implied constant is absolute. Recall that the postulated zero-free region (5.39) was established in Theorem 5.10 under the assumption that the Rankin-Selberg convolution L(f 0 f, s) exists and has a simple pole at s = 1. We shall see that the same assumption is also sufficient for the second postulate (5.48). To this end we apply Proposition 5.7 for L(f0],s) rather than for L(f,s). Combining (5.28) and (5.27) (see also (5.11)) we get L' d2 -yU 0 f, a)« a_ 1 logq(f) for 1 < a ( 2, where the implied constant is absolute. This with a = 1 + (log 3x) - I yields (5.49) by the non-negativity of coefficients.

5.

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

111

From (5.49) we derive using Cauchy's inequality that

L

(5.50)

IAJ(n)l

«

dJXYlog(xq(J))

x
for x ) y ) 1 where the implied constant is absolute. Now we are ready to state and to prove the main Prime Number Theorem. THEOREM 5.13. Let L(J,s) be an L-function for which (5.S9} is a zero-free region with at most one exceptional real zero f3t in the segment {5.47), where c is a positive absolute constant. Suppose {5.48) holds with an implied constant absolute. Then we have 4

(5.51)

xf3J ( ( -cd- logx) 4) 1/;(J,x)=rx-{jf+O xexp y"'IgX+ logq(J) (dlogxq(J))

3

,

for x ) 1, the implied constant being absolute. By convention the term -xf3J / f3 f is not in (5. 51) if the exceptional zero does not exist.

REMARKS. The approximate formula (5.51) has meaning when the error term is smaller than the main term, and this is the case for X)

lc-

1 4

d log(dlogq)

where q = q(J) is the conductor, d is the degree and c is the relevant absolute positive constant. One can simplify the error term by compromising slightly the strength of the result. For example, we have

(5.52)

1/J(J,x) = rx-

~;

+ 0( Jq(f)xexp(-

2~4 ~))

where the implied constant is absolute. Indeed this holds true, because when the error term here is smaller than that in (5.51) the result is trivial. The conditions of Theorem 5.12 are satisfied if L(J 0 f, s) exists and has a simple pole at s = 1, but we do not restrict (5.51) to this case only. Indeed we shall establish the zero-free region (5.39) and we shall see the crude bound (5.48) directly without appealing to the Rankin-Selberg convolution L-function in many classical cases. PROOF OF THEOREM 5.12. First we smooth things out by writing 1/;(J, x) =

L At(n)4>(n) + O(dylxYlog(xq(J)))

where ¢(z) is a function support on [0, x + y], such that ¢(z) = 1 if 1 :( z :( x and l4>(z)i :( 1 elsewhere. The parameter y will be chosen later subject to 1 :( y :( x. For example, we take x¢(z) =min y' 1,1 + -y~

(z

for 0 :( z :( x

+y

and ¢(z) = 0 if z > x -

¢(s) = for s = a+ it, ~ :( a :( 2.

+ y.

z)

The Mellin transform of ¢(z) satisfies

x" ( X ) Jorx+y 4>(z)zs-rdz «~min 1, ~

112

5.

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

Now we can start with

LAt(n)4>(n)

=

1

1ri

2

J

L' --y;(f, s)tP(s)ds,

(2)

n

the integral converging absolutely. We shall be operating in the region

{s =a+ it I a;) 1- cJ/d4 log(q(f)(lt1

+ 3))}

where c1 = 2cj3, or c1 = cj3, depending on whether the exceptional zero is in the right half of the segment (5.47), or elsewhere (with possibility of non-existence). Let Z denote the boundary of this region. The point is that all the zeros of L(!, s) are distanced from Z by at least cj6d4 log(q(f)(ltl + 3)). Hence it follows by (5.27) and (5.28) that for s E Z,

,4

Moving the integration from the line Re (s) = 2 to Z we get by Cauchy's theorem " ' ~At(n)(n) = r¢(1)4>(f3t) +

1 1ri

2

n

J

L' --y;(f, s)¢(s)ds.

z

Here the presence of the exceptional zero term depends on whether we passed the points= f3t or not. For s = 1, or s = f3t we have

¢(s)

=

r

lo

zs- 1 dz

+ O(y) =

xs

s

+ O(y)

while the contour integral over Z is estimated by d where T arrive at

4

h~~min(1, I~Y)log (q(J)(Itl 2

= xjy

and a(T)

+3))1dsl

= 1- cJ/d 4 log(q(J)T). 13

1/:{f, x) = rx-

«

d.Jx"(T)log3 (q(J)T)

Gathering the above results we

x {3; + 0(d (xr2 + x"(T)) log (xq(J))) 4

1

3

where Tis at our disposal subject to 1 ( T ( x. We choose T = exp(~v'logx) getting (5.51 ). We still have to address the question of the exceptional zero term. An explanation about its contribution to the obtained formula is needed if the exceptional zero does exist and it lies in the left half of the segment (5.47), but in this case the term -xf3t / f3t is consumed by the existing error term, so it can be 0 ignored. This completes the proof of Theorem 5.12. If one is willing to employ more zeros of L(f, s), then one can derive along the above lines stronger approximations to 1/;(f, x). EXERCISE 7. Assume that L(J, s) satisfies the Ramanujan-Petersson conjecture. Derive (using Perron's formula (5.111)) the following approximate expansion

(5.53)

1/;(f,x)=rx-

L bi<:;T

xP- 1

(X

-p-+0 T(logx)log(xdq(J))

)

.

.~

5.

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

ll3

where p = {3 + iJ runs over the zeros of L(J, s) in the critical strip of height up to T, with any 1 :( T :( x, and the implied constant is absolute. EXERCISE 8. Assume that L(J, s) satisfies the Ramanujan-Petersson conjecture and Theorem 5.13 holds for f. Prove that r :( d. 5.7. The Grand Riemann Hypothesis. The Grand Riemann Hypothesis (GRH for short) refers to the following conjectural statement about zeros of £-functions: GRAND RIEMANN HYPOTHESIS. Let L(J, s) be an £-function. Then all zeros of L(J,s) in the critical strip 0 < Re(s) < 1 are on the critical line Re(s) = ~· REMARKS. Needless to say we believe that every £-function (subject to our definition in Section 5.1) satisfies the Grand Riemann Hypothesis. Yet, proving this even for one £-function would be an achievement on a historical scale for human beings. Note that an £-function may have zeros on the line Re (s) = 0 (certainly some trivial zeros, probably not the genuine ones), but as we already proved not on Re ( s) = 1. The GR.H also ensures that the trivial zeros in the open critical strip are all on the critical line as well; do they exist is not clear, most likely they do not. It is very important to consider only £-functions with Euler products. Indeed it is known that Dirichlet series without an Euler product (still civilized ones) have many zeros even in the half-plane of absolute convergence. Arithmetic geometers should not be concerned, because their £-functions originate from an Euler product of some kind. Of course, this easy start makes it harder to investigate £-functions further in analytic aspects. For example, it was only recently that analytic continuation of the Hasse-Weil £-functions of elliptic curves was established by way of modularity. One should not underestimate the analytic continuation of an Euler product beyond the region of absolute convergence, for one often implements deep arithmetic resources in one way or another. As envisioned by Riemann, this venture into the critical strip illuminates prime numbers and relevant arithmetic objects built on them by revealing their dual companions - the zeros of £-functions. lVIysterious as they are today, these zeros (most likely complex transcendental numbers) will eventually be cracked for further analysis. However, for the time being, as far as the GR.H goes, we are 011ly sure they are fine tuned to yield surprisingly strong estimates. Yes, the uniformity of estimates in terms of involved parameters which can be derived from the GR.H are astonishing. On a negative note this superb uniformity could be a reminder how slim prospects are to prove the GR.H in our lifetime. No tools of current analytic number theory are capable of treating the questions for which the GR.H provides easy answers. In this section we present a few traditional analytic consequences of the GR.H, paying attention to the uniformity in terms of the conductor. First we state a few equivalent facts which help to grasp the analytic meaning of the Riemann hypothesis. PROPOSITION 5.14. Let ~ :( n < 1. The following statements are equivalent for an £-function: ( 1) There are neither zeros nor poles of ( s - 1)" L(J, s) in u > n, where r is a non-negative integer. (2) The inverse, the logarithmic derivative and the logarithm of (s -1)' L(J, s), the latter normalized so that log L(J, s) ---> 0 as u ___, +oo, are holomorphic in u > n.

5.

114

CLASSICAL ANALYTIC THEORY OF L-FUNCTIONS

{3) Let J1 f ( n) denote the coefficients in the Dirichlet series expansion for the inverse L(J,s)- 1 Then (5.54)

the implied constant depending on f and E: > 0. {4) Let r)! 0 be the order of the pole of L(f, s) at s = 1. Then

1/;(J,x) = rx +O(x'*),

(5.55)

the implied constant depending on f and

E:

> 0.

The Riemann hypothesis asserts that the above statements are true with n = ~. What is interesting (perhaps somewhat surprising for a beginner) is that the estimate (5.55) is self-improving by passage through zeros. Indeed, first (5.55) implies that the zeros are on the line Re (s) = ~' whence one derives by the expansion (5.53) with T = x using the crude bound (5.26) the following THEOREM 5.15. Assuming the Riemann Hypothesis for L(f, s) and the Ramanujan-Petersson Conjecture for L(f, s) we have

1/;(J, x) = rx

(5.56)

I

+ O(x2 (logx) log(xdq(J)),

for x)! 1, the implied constant being absolute. The Riemann hypothesis shows that L(f, s), the inverse L(f, s)- 1 , the logarithmic derivative Jf: (!, s) and the logarithm log L(f, s) can be well approximated by extremely short partial sums of the corresponding Dirichlet series uniformly in Res = u )! n > ~. Clearly this is not possible for s on the critical line, where the zeros are. We begin by considering

-

~ (f,s)

=

L

A~~n).

n

Let ¢(y) be a continuous function on [0, oo[ whose Mellin transform ~(w) satisfies

w(w

(5.57)

+ 1).j,(w) «

1, for w such that - ~ ( Re (w) ( ~-

Suppose ci>(w) has a simple pole at w = 0 with residue 1 (normalization). For example, ¢(y) = e-Y with ¢(w) = f(w), or ¢(y) = max(1- y,O) with .j,(w)

w-l(w

+ 1)-1.

PROPOSITION

(5.58)

-

. 5.16. For~

~ (!, s) =

L

< Re (s) (

~ we have

A~~n) ¢(i)

n

-r.j,(1-s)Xl-s+L¢(p-s)XP-•+o( logq(J,s)) P (2u- 1)VX for any X)! 1, where the implied constant depends only on that in (5.57).

5. CLASSICAL ANALYTIC THEORY OF L-FUNCTIONS

115

PROOF. By now it should be a standard argument for a reader to derive the formula (5.58) by contour integration with the error term being equal exactly -1. 27rl

J

L' --L(J,s+w)rj>(w)Xwdw.

(-~)

Indeed this is obtained by starting with the above integral on the line Re (w) = = - ~. The simple poles at w = 0, 1 - s, p - s produce the corresponding terms in (5.58). Now on the line Re (w) = -~ we have Re (s + w) =a-~' and we deduce by (5.26) and (5.27) that ~ and moving it to the line Re (w)

L'

-L(J' s + w)

«

(2a -1)- 1 1ogq(J, s + w).

Integrating this against the bound from (5.57), we get the error term in (5.58). 0 The sum over the zeros in (5.58) is estimated directly (without cancellation) by means of (5.26), (5.27) and (5.57) giving (5.59)

_!:!_(!, s) L

=" AJ(n) ¢(!!._)r¢(1- s)Xl-s + o(logq(J, s) x~-a) X 2a 6

-1

n5

n

for any X~ 1, where the implied constant depends only on that in (5.57). Next we are going to estimate the sum of AJ(n). Essentially any crude bound, but independent of the conductor, like IAJ(n)l :( n, would suffice for our purpose. Nevertheless, since we already assumed the GRH, there is no point in ignoring the Ramanujan-Petersson conjecture for the local roots, which yields IAJ(n)l :( dA(n). From this we get

LA~~n)¢(i)

«dXl-a +dlogX.

n

Moreover, the polar term in (5.59) is -r.j,(1- s)X 1 -s = r(s- 1)- 1 + O(r X 1 -a + r log X). Hence (5.59) simplifies to - L' (f,s) = _r_ +O(Iogq(J,s) x~-a +dXl-a +dlogX). L s- 1 2a- 1 Finally choosing X= log 2 q(!, s) we obtain THEOREM 5.17. Assume the Riemann Hypothesis and the Ramanujan-Petersson conjecture for L(f, s). We have 2 L' (f,s) = -r- + 0 ( --(logq(!,s)) d --L s -1 2a -1

2

a

+ dloglogq(!,s) )

for any s with ~ < a :( ~, the implied constant being absolute.

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116

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

CoROLLARY 5.18. For a ) ~ +(log log log q)/(loglog q) we have L' --(! s)- -r- +0 ( -log -q- ) L

'

- s - 1

log log q

where q = q(f, s) is the analytic conductor and the implied constant is absolute.

.·

To get an estimate for the logarithm of the £-function we integrate the logarithmic derivative along the horizontal line log L(f, s) =log L(f, ~+it)-

5/4

1

L' -(!, n + it)dn.

" L Applying (5.59) we deduce the following estimates

THEOREM 5.19. Assume the Riemann Hypothesis and the Ramanujan-Petersson conjecture for L(f, s). We have !og(pr(s)L(f,s))

d(!ogq(f s)j2- 2"

« (2a -1 )Iog' 1ogq (! ,s ) +dloglogq(f,s)

for any s with ~ < a :( ~, the implied constant being absolute (recall Pr(s) ( S - 1)"( S + 1) -r. As immediate consequences of Theorem 5.19 we have lower and upper bounds for L(f, s) in the half-plane u ) ~ + c:

q(f, s)-"

«

Pr(s)L(f, s)

«

q(f, s)"

with implied constants depending only on c. The upper bound can be extended up to the critical line by the convexity principle. In particular, we obtain COROLLARY 5.20. For s with Re(s) = ~ and any (5.60)

L(f, s)

«

E

> 0 we have

q(f, s )"

where the implied constant depends only on

E.

The last estimate received considerable attention (it is known as the Lindeli:if Hypothesis), because it is sufficient for solving many fundamental problems of arithmetical nature due to its uniformity in the conductor. We just proved that it follows from the Riemann hypothesis. One may think that the Lindeli:if hypothesis should be easier to establish, but there is a popular opinion that the Riemann hypothesis must be solved first, the point being that the Riemann hypothesis is linked to more natural mathematical structures. Clearly the Lindelof conjecture implies the following estimate for sums of coefficients of the £-function (5.61)

L

AJ(n) = xPj(!ogx)

+ o(A (xq(f))")

where Pj(X) is the polynomial of degree r- 1 given by PJ(!og x) =res L(f, s)xs-r. s=l

Some unconditional results, weaker yet useful, are given in Chapter 4 (see Corollary 4.9 for ((s)L(s, X4) and Exercise 7 of Chapter 4 for ((s) 2 for instance).

j

5.

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

117

5.8. Simple consequences of GRH.

\Ve illustrate some of the simple arithmetic consequences of GRH, besides the Lindeliif Hypothesis, which has important applications of its own. The first problem is to estimate the order of vanishing of an £-function at a given point on the critical line, in terms of the conductor of f. \Ve consider the case of the central point s = 1/2. Note that (5.27) implies ord L(J,s) :( m(1,J) s=l/2

«

logq(J)

unconditionally, which we can improve slightly. PROPOSITION 5.21. Let L(f, s) be an entire £-function satisfying GRH. We have ord L(J, s)

(5.62)

s=l/2

«

log q(J)

log(~ log q (f))

with an absolute implied constant (recall that log q(f) > d). PROOF. We apply the explicit formula (5.45) to a test function g(y) supported on [-2Y,2YJ with 0 :( g(y) :( 2 and g(y) ;;:, 1 if IYI :( Y such that its Fourier transform h(t) is non-negative on !!!.. By positivity we can drop all the zeros p = ~ + h f= ~ in (5.45) getting mh(O) :( -2

LRe (AJ(n))n-~ g(logn) + O(logq(J)),

where m is the multiplicity of the zero p = ~- On the left side we have h(O) ::=: Y. Since we have IAJ(n)l :( dA(n)n by (5.2), the sum over prime powers non the right side is bounded by

2d

L

1

y

A(n)n2 «de ,

where the implied constant is absolute. Choosing 2Y =log(~ log q(f)) we get

m« with an absolute implied constant.

logq(f) log( j log q (f))

-...,.,:c:...:.:.::..c_ _

0

Proposition 5.21 is optimal although the estimate of the sum over primes is quite crude, which might suggest some improvement is possible. The issue is that this sum is extremely short with the choice of Y, in fact n :( e2Y :( ~ log q(f) and it is conceivable that At(n) does not change sign in this range. If we take L(J, s)k for k -> +oo instead of L(J, s), we see that (5.62) cannot hold without the factor However, in a number of cases, it is possible to do better; see Proposition 5.34 and Exercise 13 below.

l

The problem of detecting sign changes of AJ(n) for very small n is related to another classical application of the Grand Riemann Hypothesis: how many coefficients (in terms of the conductor q(J)) of L(f, s) at primes are required to distinguish L(J, s) among £-functions of a certain class?

5.

118

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

PROPOSITION 5.22. Let L(f, s) and L(g, s) be two £-functions with the same degree d and gamma factor. Assume that L(f 0 ], s) and L(f 0 g, s) exist and the latter is entire, that GRH holds for both and that the local roots at ramified primes for L(J 0 ], s) and L(f 0 g, s) are of modulus:( 1. There exists a prime p :( C(dlogq(f)q(g)) 2 unramified for f and g such that the local roots of L(J,s) and L(g, s) at p are different. Here C is some absolute positive constant. PROOF. The point is that by assumption the £-function L(J 0 ], s) has a pole at s = 1 whereas we assume that L(f 0 g, s) is entire. Assume the local roots of f and g coincide for all primes p :( 2X which are unramified for f and g. Then At09 (n) =A !®f(n) for n :( 2X such that (n, q(f)q(g)) = 1. By the explicit formula (5.44) applied to a function of the type cp(n/ X) where ¢) 0 is smooth, compactly supported on [1, 2], and ¢ of 0, we have L_At®!J(n)cp(n/X)

«

Vxlogq(f0g),

L_A 10](n)¢(njX) = .jy(O)X + O(Vxlogq(J 0 /)) where ~(0) J <.p(t)dt > 0. These estimates are smooth versions of the prime number theorem for the Rankin-Selberg convolution £-functions. We used GRH to estimate the sum over zeros. On the other hand, the two left sides coincide by assumption up to the contribution from ramified primes. This contribution is small, precisely it is

L

«

\A 10f(n)\ + \AJ 09 (n)\

«

d 2 (1ogq(J)q(g)) log X.

n(;2X,(n,q(f)q(g) )#I

Therefore c/Y(O)X « Vxlogq(J0])q(f0g)+d2 (1ogq(J)q(g))logX, or X « (dlogq(J)q(g)) 2 where the implied constant is absolute (recall that the definition of the Rankin-Selberg convolution implies the bound q(J 0 g) :( (q(J)q(g))d). 0 REMARK. The best general unconditional result is much weaker. Suppose L(f, s) has degree d with roots \ni (p) \ < p 114 and that it admits L (f 0 /, s). Then for every E: > 0 there exists C > 0, depending on E: and on f with the following property: If L(g, s) has the same gamma factor as L(J, s) (so the same degree), if L(f 0 g, s) is entire and if ,\ 9 (p) = >-t(P) for all p :( Cq(g)d/2+•, then g =f. To prove this, notice that if X) 1 is the largest integer for which >-t(P) = >. 9 (p) for p :( X, then by multiplicativity we have >-t(n) = ,\9 (n) for n I N(X) where N (X) is the product of primes :( X. For h = f, or h = g, one can factor L(f 0

h, s)

= Lb(f 0

h, s)H(f, h, s)

where Lb (f 0 h, s) is the Dirichlet series restricted to squarefree integers and H (J, h, s) converges absolutely for u > ~. In particular, L b (J 0 h, s) is entire if h = g and has a pole at s = 1 for h = f, and satisfies the same convexity bound as the Ra.nkin-Selberg £-function. We get by complex integration on the line u = ~+O, with 0 > 0 small enough

L ,\f0h¢(nj X)= o(f, g)X ?(log X)+ O(x~+'q(J 0 g)!+•) n~l

. I ,

5. CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

119

where T is the order of the pole of L(f ®/, s) at s = 1 and P jc 0 is the polynomial of degree T - 1 given by P(logX) = resL'(J®h,s)Xs~l. s=l

Comparing the formula for h =

f

and that for h = g yields the result.

5.9. The Riemann zeta function and Dirichlet £-functions. The Riemann zeta function ((s) is a self-dual £-function, with conductor 1, gamma factor r(s) = 7r~sf 2 r(s/2) and root number 1. Its Rankin-Selberg square is also ((s). More generally, let X be a primitive Dirichlet character modulo q. The Dirichlet £-function L(s, x) is an £--function of degree 1 with conductor q, gamma factor

r(s)

=

where 8 = 0 if x(-1) = 1 and 8 c(X) = T(x)/ .fij, where

T(X) =

11'~s/2rC; 8), = 1 if x(-1) = -1, and its root number is

L

x(x)e(~)

x (mod q)

is the Gauss sum associated to x (see Section 3.4). If xis non-trivial, then L(s, x) is entire, otherwise it has a simple pole at s = 1 with residue 1. Any L(s, x) satisfies trivially the Ramanujan-Petersson conjecture. The analytic conductor is given by q(x, s) = q(it+8\+3) :=: q(\tl +3) and q(x) = (2+8)q :=: q; see Theorem 4.15. A Dirichlet £-function L(s, x) is self-dual if and only if x is quadratic. In this case c(x) = 1 by Theorem 3.3. Any two Dirichlet characters x 1 and x 2 admit a Rankin-Selberg convolution given simply by L(Xl 0 X2, s) = L(X3, s), where X3 is the primitive character that induces the product XlX2· We have X3 = XlX2 if, for instance, the conductors q; of Xi are coprime, but not in general. In particular, the Rankin-Selberg square of L(s, x) is ((s) and the finite product (5.10) is

Fixing a modulus q and considering all primitive characters modulo q gives an example of a "family'' of £-functions. Fixing X ) 2 and considering all primitive quadratic characters of conductor q ,::; X gives another family. By (5.22) we deduce the convexity bound for Dirichlet £-functions. THEOREM 5.23. Let X modulo q be a primitive Dirichlet character. We have (5.63)

L(s,x)

«

(q\s\)1

for Res = ~, the implied constant being absolute.

The following exercise provides very simple estimates which are quite handy anyway.

120

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

5.

EXERCISE 9. Let for 0 :( a :( 1 we have

x modulo q be a

non-trivial Dirichlet character. Show that

(5.64) for k ;:;, 1, the implied constant depending only on k. 1 Ln,-;;x x(n)l :( min(x, q).J

[Hint: Use the bound

Theorem 5.8 becomes: THEOREM 5.24. Let x modulo q be a primitive Dirichlet character, N(T, x) the number of zeros p = {3 + ir of L(s, x) in the critical strip 0 :( {3 :( 1 with hi :( T. We have

N(T,x) = !'._log qT +0(q(T+3)), 1T 2?Te with an absolute implied constant. In addition to the general explicit formula of Theorem 5.11, it is sometimes more practical to work with the following approximate formula for 1/J(x,x)

= LA(n)x(n).

PROPOSITION 5.25. For any character we have (5.65)

1/J(x,x)=lixx-

xP;1+0(f(logxq)2)

L L(p.x)=O IIm(p)I(T

where lix = 1 if X is absolute.

=

Xo and lix

=0

otherwise, 1 :( T :( x and the implied constant

PROOF. This is a special case of (5.53) for primitive characters. However, (5.65) also holds as it is for x not primitive. Indeed, suppose x*(mod q*) with q* I q is the primitive character which induces x(mod q). Then 11/J(x,x)-1/J(x,x*)l:(

L

logp«(logxq) 2 •

pu(x,piq

Moreover, the zeros of L(s, x) agree with these of L(s, x*) except for the ones coming from the product

IT

(1- p-s)

Piq,pjq•

which are p = 2?Til/logp. Hence these zeros contribute at most

L L [xP; 1[ « Plq

(logxq)2.

;:gl~ (T

The above corrections are absorbed by the error term in (5.65) completing the proof. 0

r 5.

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

121

By (5.65) we derive a strong approximation to

7jJ(x,q,a)

A(n)

= n~x

n::::::a

(mod q)

in terms of zeros of £-functions. First by the orthogonality of characters we have 1

L

7jJ(x,q,a) =
x(a)I/J(x,x)

X (mod q)

Applying (5.65) we obtain (5.66)

7jJ(x;q,a)=
L

x(a)

x (mod q)

L xP; 1 +o(f(logxq) 2) L(p.x)=O IIm(p)I(T

for q;?. 1 , (a, q) = 1 and 1 :( T :( x, with an absolute implied constant. Theorem 5.10 gives the zero-free region for Dirichlet £-functions, except for the possible exceptional zero f3x for a real primitive character X· Because of its importance we include here the slightly different traditional proof. THEOREM 5.26. There exists an absolute constant c > 0 such that for any primitive Dirichlet character x modulo q, L(s, x) has at most one zero in the region a;?.1-

(5.67)

c . logq(Jtl + 3)

The exceptional zero may occur only if x is real, and it is then a simple real zero, say f3x, with c 1 - - - :(

(5.68)

1og3q

f3x < 1.

PRooF. Assume X is non-trivial (the case of ((s) is similar, but it needs a change of vocabulary because our definition of £-functions does not include (( s +it) with t ol 0 for the reason that it has a pole at s = 1 - it oil). Then consider the £-function of degree 8

L(f, s) = ((s) 3L(s +it, x) 4 L(s + 2it, X2) for some t E IR where x2 is the primitive character modulo q2 I q which induces By the trigonometric inequality of de Ia Vallee Poussin

x2 •

(5.69) 3Jzl + 4Re(z) + Re(z 2 ) = lzJ(3 + 4cos8 + cos28) = 2jzj(1 + cos(8)) 2 ;?. 0 for z = Jzl exp(i&), it follows that Re (At(n)) ;?. 0 for all n;?. 1 such that (n, q) = 1. In fact, a simple computation using the definition of x 2 shows that this remains valid for all n ;?. 1. · Let p = (3 + h be a zero of L(f, s) with (3;?. ~· If X is not real, or if"' ol 0, then taking t ="'it follows that L(f, s) has a pole of order 3 at s = 1 while (3 is a real zero of order 4 of L(f, s). Hence by Lemma 5.9 we have

(3 < 1 _

c2

logq(f, p)

<1_

c

log(q(hl + 3))

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

5.

122

with an absolute positive constant c. If ;x 2 = 1, p = {3 is real we take t = 0, then L(f, s) has a pole of order 4 at s = 1. By Lemma 5.9 again, L(f, s) can have at most four real zeros with multiplicity satisfying the above inequality, which means that {3 must be a simple zero of L(s, x). 0 REMARK. One can also recover that {3 < 1 by Theorem 2.1. As an application of Theorem 5.13, one derives the following prime number theorem. THEOREM 5.27. Let X modulo q be a primitive Dirichlet character. We have (5.70)

L n,;;x

x(n)A(n) = 5xx- xf3x + f3x

o(xexp(~ogx

)(logq) 4 )

log x +log q

for any x ) 1, with some absolute effective constant c > 0, the implied constant being absolute. For any q) 1 and (a, q) = 1 we have X ;\'(a) X/3' ( ( -cJogx ) 4) A(n) = -(-)- - ( - ) - + 0 :rexp y!ogX (logq) 'P q 'P q f3x log x + log q

(5.71) n~x

n=a (mod q)

for x) 1, where X (mod q) is the possible exceptional real character modulo q and f3x the corresponding exceptional zero. Have in mind that there could be one or two real primitive characters modulo q, for example, if q = Sr with r odd squarefree we have two such characters xsXr and X4X8Xr·. Nevertheless, at most one of these may give an exceptional zero. In fact, the exceptional characters are very rare, and the exceptional zero itself can be somewhat controlled. THEOREM 5.28. (1) (Landau) Let x1 and X2 be distinct real primitive characters modulo ql and q2. Assume X 1 and X2 have real zeros {31 and {32 respectively. There exists an absolute constant c > 0 such that (5.72)

(2) (Siegel) For any primitive real Dirichlet character x modulo q, we have

/3 <

(5.73)

X "

1- c(c:)

q'

for any c: > 0, with a constant c(c:) > 0 depending only on c:. If c: < ~, this constant is ineffective, meaning it is not numerically computable. PROOF. (1) Consider the L-function of degree 4 (5.74)

L(f,s) = ((s)L(s,xl)L(s,x2)L(s,xlX2)

which has conductor at most (q1 q2 ) 2 . Since the x1 , x2 are non-trivial, and distinct, L(f, s) has a pole of order 1 at s = 1. Moreover,

L'

-y(f, s) =

L n)l

(1

+ XI(n))(1 + X2(n))A(n)n-s

r

5. CLASSICAL ANALYTIC THEORY OF L-FUNCTIONS

123

has non-negative coefficients. By Lemma 5.9, there exists c > 0 such that L(J, s) has at most one real zero with (3;:;, 1-c/logq(J). Since q(J) :( 9(q 1 q2) 2, this gives

(5.72).

(2) For the proof of (5.73), we follow the nice argument of Goldfeld [Gol]. Let Xb x2 be (for the moment) arbitrary primitive real characters modulo q1 and q2 and consider again the £-function (5.74). We have AJ(n) ;:;, 0. Let TJ(x) be a function on [0, +oo[ such that 0 :( TJ :( 1, TJ(x) = 1 for 0 :( x :( 1 and TJ(X) = 0 for x ;:;, 2. Consider the sums " : \ 1 (n)

(n) ;?1

Z((J,x)=~---;;]3"7:;:

(since AJ(l) = 1)

n

for ~ :( (3 :( 1 and x ;:;, 1. By contour integration we have

Z((J, x) =

2 ~i j L(J, s + (J)x•ry(s)ds = L(J, (3) (2)

1

+ £(1, X1)L(1, X2)L(1, X1X2)ry(1- (3)x -/3 +

2 ~i

J

L(f, s + (J)iJ(s)x•ds.

(~-/3)

1

By (5.63), the last integral is« q1 q2x'i-13. Thus we have I

L(J, (3)

+ £(1, xi)£(1, ·X2)L(l. XIX2)ry(1- (3)x 1-/3;? 1 + O(qlq2x2-/3).

Now a'3sume that (31;:;, ~is a zero of L(s,x1). Then itfollows I

£(1, X1)L(1, X2)L(1, X1X2)ry(1- fJ1)x 1 -13' ;? 1 + O(q 1 q2x'i-/3 1 ). \Ve take x = c(q 1 q2) 4 with c sufficiently large, so the term on the left is> ~- From (5.64) we have L(1, xi) «log q1 and £(1, X1X2) « log q1 q2. Moreover, ry(1- (Jr) « (1- (Jr)- 1 since ry(s) has a simple pole at s = 0 with residue 1. Gathering these estimates we get our basic lower bound (5.75) We will use (5.75) to prove (5.73), nsing the hypothetical zero (3 1 of L(s, xt) as a tool to control other characters. Let 0 < E :( ~. If all L( s, x) for x real modulo q do not vanish on the real segment s ;:;, 1- E, then (5.73) is obvious (just take c(c) =c). Suppose otherwise that for some x1 modulo q1, the £-function L(s, xr) has a zero s = (3 1 ;:;, 1 -E. Then for any other real primitive character X2 modulo q2 , we can write (5.75) as L( 1, X2)

»

q2 4 " (log q2 ) - 2

where the implied constant depends only onE, since q1 , x1 and fJ1 are fixed once E is chosen. On the other hand, if (3 2 is a real zero of L(s,x2) we have by the mean-value theorem

for some a with (3 2 :(a :( 1 by (5.64). Combining those two bounds gives 1- (32 » q2 42 , hence (5.73) after re-defining E. 0

5.

124

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

t

REMARK. Essentially the bound (5.73) was first proved withE= by Landau E > 0 shortly afterwards by Siegel [Siel]. The ineffectiveness of the Siegel bound (5. 73) is apparent in applications. For instance, it is not possible to use it to solve the class number one problem for

[La2] (also not effective), and with any

1

imaginary quadratic fields. Yes, by Siegel's lower bound £(1, x) :;p !l2 ~e and by the Dirichlet Class Number Formula (2.31), or (22.59), we get

w..[E

1

h(IQ(~)) = -~L(1,x) ::$> !l2~c

(5.76)

27r for -ll a negative fundamental discriminant, X being the corresponding Kronecker symbol. Hence h tends to infinity as does Ill[. But when computing all ll with h(IQ( ~)) = 1 it is not possible to know when to stop. In Chapters 22 and 23, we use much more sophisticated ideas of Goldfeld involving £-functions of elliptic curves to prove an effective bound, although much weaker (see Theorem 23.2).

Siegel's estimate implies the Siegel-Walfisz theorem about primes in arithmetic progressions. COROLLARY 5.29. Let q ;? 1 and A > 0. We have (5.77)

rr(x;q,a)

=

x ) Li(x) (
(5.78)

7jJ(x;q,a)

=


+O((lo:x)A)'

for any x ;? 2. Moreover, for any primitive Dirichlet character X modulo q have

> 2, we

LX(P) « Jqx(logx)~A,

(5.79)

p~x

L x(n)Jl(n) « Jq:r(logx)~A

(5.80)

n~x

for any A

>

0. The implied constants depend only on A. but are ineffective.

PROOF. If q ~ (logx)A+ 1 one gets (5.78) from the prime number formula (5.71) by employing the bound (5.73). For larger q there is nothing to prove, because the formula (5.78) is trivial. Then (5.77) follows from (5.78) by partial summation. Similarly one derives (5.79) from (5.52) by employing the bound (5.73) The case of (5.80) is a bit harder. One could follow the proof of (5.51) using L ~ 1 instead of L' / L, but there are serious complications which occur near multiple zeros or clusters of zeros (so far we cannot rule out this scenario). Therefore we are going to derive (5.80) directly from (5.79). First we write

L x(n)J.L(n) = 1- L x(m)J.L(m) L

x(p)

m~x

where Pm stands for the largest prime divisor of m. Let r ;:;, 0 be the number of prime divisors of m. Then the summation conditions imply m 1 /r ~ Pm < xjm, so m < xr/(r+ 1 l. Hence, using (5.79) the inner sum over p satisfies

L...,XP

" ' () P

Jqx(l ogx)~A «-m

m

Jqx(r+1)A ,;;--m logx

«

T(m)Jqx , m(iogx)A

J

I

I

5. CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

125

because (r + 1)A « 2r = T(m). Finally, summing over m trivially we obtain (5.80) with extra factor (log x )2 . This completes the proof by resetting A. 0 For Dirichlet characters, the problem discussed in Proposition 5.22 becomes that of finding the smallest prime p with x(p) =/= 1 for a non-trivial character x modulo q > 2, say Pmin(x). One derives directly from various estimates for short character sums the following bounds:

using respectively the estimates of Poly a- Vinogradov, Burgess, Lindelof hypothesis, Riemann hypothesis. Actually we only get (logq) 4 by (5.56), but with a little extra effort one can squeeze from the GRH the bound (logq) 2 . One can also do better unconditionally using Burgess's estimate enhanced by combinatorial arguments (see e.g. [Hl]). Linnik proved (by the large sieve inequality) that the smallest quadratic non-residue for a primitive real character modulo q ~ N, q prime, is ~ N' with at most finitely many exceptions, the number of exceptions depending only on E > 0. See Section 7.4 for the proof and an analogue for elliptic curves. The Riemann zeta function is further discussed in Chapters 10, 24, 25, and Dirichlet characters in Chapters 17, 18. 5.10. £-functions of number fields. Let K JQ be a number field of degree d. The Dedekind zeta function

(K(s)

=2_)Na)-s =IT (1- (Np)-s)- 1 a;I'O

defines a self-dual £-function of degree d = [K: Q) with conductor D = ldKI, the absolute value of the discriminant of K, and root number +1. The gamma factor is (5.81)

'Y(s) = rr -ds/2r (s)r1+r2 r (S-+-1)r2 2 2

where r 1 is the number of real embeddings of K and r 2 the number of pairs of complex embeddings so that d = r 1 + 2r2. The zeta function has simple pole at s = 1 with residue 2r 1(2rr 2 hR res (K(s) = rn , s=1 wvD

t

where h is the class number of K, R the regulator and w the number of roots of unity. For a proof, see e.g. [La). The Ramanujan-Petersson conjecture is obviously satisfied for (K(s). The analytic conductor is

and q((K) = 2r 1+r 23r2D ~ 3dD. There exists an absolute constant c D > cd- 1 (see Exercise 10 below) so we have logq((K) :=::log D.

> 1 such that

5.

126

CLASSICAL ANALYTIC THEORY OF L-FUNCTIONS

REMARK. Dedekind zeta functions factor as products of Artin £-functions (see Section 5.13). Let EjQ be the Galois closure of K, G = Gal(E/Q) and H = Gal(E/K). Let rc;H be the permutation representation of G on the coset space G j H. This is the representation induced by the trivial representation of H, hence by the invariance of Artin £-functions under induction we have L(rc;H,s) = (K(s). One can decompose rc;H in terms of irreducible representations of G, rc;H = tfJpnpp, getting the factorization (5.82)

(K(s)

=IT L(s,ptP. p

In case of a failure of the Artin conjecture (a possibility we cannot yet rule out) this is not a factorization in terms of £-functions in the sense of Section 5.1. If KJQ is abelian, then E = K, H = 1 and rc;H = tfJX where x runs over a group of Dirichlet characters so that

(K(s)

=IT L(s,x) X

is a product of £-functions for distinct primitive Dirichlet characters of conductors whose product is D = ldKI and exactly one character is trivial. We deduce from the above and the results of the previous sections: THEOREM 5.30. Let KJQ be a number field of degree d. We have

for 0 :( a :( 1, the implied constant depending only on d and

E.

THEOREM 5.31. Let KJQ be a number field of degree d. Let NK(T) be the number of zeros p = (3 + h of (K(s) in the critical strip 0 :( (3 :( 1 with hi :( T. We have

T

DTd

rr

27re

NK(T) = - l o g -)d (

+ O(logDTd)

forT ;:;, 2, the implied constant being absolute.

One can use the analytic properties of £-functions to deduce information on the discriminant of number fields. For instance, the explicit formula yields quite good lower bounds forD= ldKI· This powerful method was implemented by Odlyzko [Od], Poitou [Poi], Serre [Se7] and others. Here is a simple result. THEOREM 5.32. Assume GRH for the Dedekind zeta functions of number fields. Then we have (5.83)

liminf

as d = [K : Q] = r 1

~{logD + (d- r2)%{i) + r2%w};? log1r,

+ 2r2

---+

+oo.

l

l

5. CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS PROOF.

127

Apply the explicit formula (5.44) to (x(s) with 1

2

¢(x) = x-2e-a(logx) ' for some a > 0. Although this test function is not of compact support, it decays very fast at infinity, so ( 5.44) holds. The Mellin transform is ,2 ¢(!+it)= /¥IT -e-:ra o

a

2

for t E R Note that this is positive (compare with Proposition 5.55) and that ¢(x) = x- 1 ¢(x- 1 ). Since (x(s) is self-dual we get 1 2

o

1 2

n

P

1+oo ¢(x)dx + -1. J1'-(s)¢(s)ds.

1 2 0

LAx(n)¢(n) +- L¢(p) = -logD +-

1

1

(2)

By positivity, we derive 1 -logD + -11+= ¢(x)dx + - 1 . 2 0 2ITt 2

(5.84)

o

2ITt

J (~)

1' -(s)¢(s)ds;;::, 0. 1 o

By (5.81) and the Fourier inversion formula we have (5.85)

1 ---:

2ITt

J

d (d-r2) -(s)¢(s)ds = --logiT +-

1'

0

1

2

2IT

1r'

it . -(-41 +)¢(-1 + tt)dt 0

IR f

(~) r2

+ 2IT

2

r

2

r' 3 "t 1 jiR f(4 + 'i-)¢(2 + it)dt. 0

When a--> 0, the function (2IT)- 1 ~(~ +it) converges to the Dirac delta at 0, hence

r rf' 2IT jiR f(4 + 1 2IT 1

r' 1 •t 1 r' 1 jiR f(4 + '2 )¢(2 + it)dt--> f(:j) 3

0

tt

0

1

2 )¢(2

.

+ tt)dt-->

r'

3

f(;j)

Therefore, dividing (5.84) by d, then letting d--> +oo, and then letting a derive (5.83).

-->

0, we 0

Similar arguments can be applied unconditionally with slightly weaker result using test functions as constructed in Proposition 5.55. EXERCISE 10. Show that there exists an absolute constant c > 1 such that D = ldxl > cd for any number field K/Q of degree d) 2, and, in particular, D;::, 3 if K =/= Q so K has at least one ramified prime. [Hint: Proceed as in Theorem 5.51 below.]

One cannot apply directly Theorem 5.10 because the Rankin-Selberg square of

(x(s) involves convolutions of Artin £-functions of K which are not known to be entire, although they are conjectured to be. However, one can adapt the traditional arguments (as in the proof of Theorem 5.26) using (x(s) 3 (x(s + it) 4 (x(s + 2it)

5.

128

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

and derive the zero-free region, then the prime number theorem for K. We phrase it in terms of logNp if a = pk for some k ) 1, AK(a) = { O otherwise. Thus these are coefficients in the Dirichlet series

THEOREM 5.33. Let KJIQ be a number field of degree d. There exists an absolute constant c > 0 such that (g(s) has no zero in the region

a

> 1-

"'

c

--=-----=---,-,-,-------,-, d2 log 3)d

D(it[ +

except possibly a simple real zero (3 < 1. We have

for x ) 2, where c > 0 is an absolute constant, and the term -xf3 /(3 should be removed if there is no exceptional zero. Be aware that the zero-free region of Theorem 5.33 is not necessarily the best accessible. Indeed by (5.82), one can factorize (K(s) and if one knows that the factors are L-functions, then zero-free regions for them produce one for (K which can be much better because of a smaller conductor. For instance, if K = IQ(Jlp) is the field of p-th roots of unity, we have

(K(s) = ((s)

IT L(s,x) x#J

where the product is over non-trivial Dirichlet characters modulo p. Thus by Theorem 5.26 and Theorem 5.28 there exists an absolute constant c > 0 such that (K(s) has at most one simple real zero (3 in the region a

whereas d = p- 1 and D

c

> 1 - ,-------,,..-,----;:-:-

"'

= ldKI = pP-

logp([ti

2

+ 3)

so Theorem 5.33 is much weaker.

Confirming that the zeros of real Dirichlet characters are the most difficult to deal with, Stark [St3] has shown that if KJIQ does not contain a quadratic subfield, then there is no exceptional zero of (K ( s). For Dedekind zeta functions, Proposition 5.21 can be improved. PRoPOSITION

5.34. Let Kj!Q be a number field. Assume GRH holdsfor(K(s).

We have ord (K(s) <::: I

s=~

log 3D D I og og 3

where D = [dK[ and the implied constant is absolute.

5.

CLASSICAL ANALYTIC THEORY OF £-FUNCTIONS

129

PROOF. Let m be the order of vanishing at ~. We argue as in the proof of (5.62). In applying the explicit formula (5.45) an additional term arises from the simple pole of (x(s) at s = 1, hence we get

mh(O)

'I + L h(-) = P"~ 2rr

j+oo g(y)eYI dy 2

_

00

-2 LA~ g(logNa) + O(log3D). vNa 0 Both sums over the zeros and the prime ideals can be dropped by positivity. Since h(O) ;:o: Y and the integral is O(ey) we infer that mY « ey +log 3D. Taking Y = log log 3D yields the result. 0 Generalizing Dirichlet characters to number fields are Heeke characters (Grossencharakters). Those al'e defined thoroughly in Section 3.8 for imaginary quadratic fields, but here we use the basic terminology in the general case (for characters of weight 0). See e.g. [La] for details. Let KJQ be a number field and~ a primitive Heeke Grossencharakter modulo (m. fl) where m is a non-zero integral ideal inK and fl is a set of real infinite places where ~ is ramified. The Heeke £-function is defined by

for Re(s) > 1. Heeke showed that L(~,s) is an £-function of degree d = [K: Q] which is entire if~ =/1 and which coincides with (x(s) for~= 1. The conductor is ll = ldxiNx;Qm, where Nx;Q is the norm and the gamma factor is given by -d

(s)r,+r2-lfll (s + 1)r2+lfll

12 '"Y(~,s)=rrsr2

r-2-

,

so q(~) ~ 4dldx1Nx;Qm = 4dfl. The root number can be explicitly written as a normalized Gauss sum for ~. One gets the following generalization of Theorem 5.26. THEOREM 5.35. Let KJQ be a number field,~ a Heeke Grossencharakter modulo (m. fl). There exists an absolute effective constant c > 0 such that L(~, s) has at most a simple real zero in the region c a>1. dlog ll(ltl + 3)

The exceptional zero can occur only for a real character and it is < 1. REMARK. This section highlights the fact that £-functions of various types are naturally defined over any number field K, using Dirichlet series over non-zero integral ideals, and Euler products over primes ideals. Thus there are Artin £functions over K, automorphic £-functions over K, £-functions of varieties over K, whereas in a sense of analytic number theory we like to see these as £-functions over Q. This view is possible because any £-function of degree d over K, where [K : Q] = J, is an £-function over Q, as defined in Section 5.1, of degree df. For instance, (x(s) is of degree 1 over K. Thus it is easy for the reader to see what results there are for £-functions over number fields, with uniformity in the

5.

130

CLASSICAL ANALYTIC THEORY OF L-FUNCTIONS

number field. Sometimes, however, the proof has to be adapted, as for Theorem 5.33, because the Rankin-Selberg £-functions have a different form depending on whether the £-functions are seen over K or over IQl. The reader should have no difficulty supplying new proofs if necessary. It is worth pointing out that for all known £-functions, the conductor of an £-function of degree d over K takes the form q = /dK /d N K/Qf for some non-zero integral ideal f of K. Thus the conductor and the analytic conductor contain the dependency in the discriminant of the ba..-;e

~K.

Analytic number theory flourished beautifully in the last two centuries in the garden of £-functions over the rationals. The extension to number fields is fruitful as well, but it needs to be more explored. To encourage the newcomers we end this section with a few cute applications of the theory over the Gaussian domain Z[i]. There are four units in Z[i], namely 1, -1, i, -i. Every ideal of Z[i] is principaL Consider the "angle" characters

~k(a)=(,:,)

=eikargo

=

L "

c:,t/n/-s

has conductor D = 4 and it satisfies the self-dual functional equation

A(s, ~k) = 7r-sr(s +

W)L(s, ~k) = A(1- s, ~k)-

AII characters are complex except the trivial character for k = 0 giving the Dedekind zeta function L(s,~0 ) = ((s)L(s,x 4 ). Therefore there is no exceptional zero, and the Prime Number Theorem (5.52) gives

L lrrl(x

(,:,t

=2DkLi(x)+0(\k/xe-cv'IQgX)

where c > 0 and the implied constant are absolute. As an exercise we propose to derive from the above formula the following equidistribution property of Gaussian primes in sectors. THEOREM 5.36. Let 0 < (3- n ,:;;:

~

and x ;) 2. Then

/{nEZ[i] \\n\,;;;x, n
.

··.····1:_-.··.·.j

·]

.1

ik

if a = (n) oJ 0 for any k 0 (mod 4). This is a primitive Heeke character of conductor (1). The associated £-function

L(s,~k) =

~

> 0 and the implied constant are absolute.

[Hint: Approximate the characteristic function of [n, (3] modulo 2n by a smooth function, periodic of period 2n, and symmetric with respect to x - t ±x±n. Expand this function into Fourier series and apply for x = argn. By the symmetry the nonzero Fourier terms are on frequencies k 0 (mod 4). At each frequency apply the Prime Number Theorem. J

=

See Section 11.8 for an application of this result to show that the Wei! bound for the number of points on a curve over a finite field is optimal in horizontal sense.

,,

5. CLASSICAL ANALYTIC THEORY OF L-FUNCTIONS

131

EXERCISE 11. Playing with argn and In I derive the equidistribution of Gaussian primes in regular domains of C.

EXERCISE

that pq = a 2

12. Prove that there are infinitely many pairs of primes p, q such with 0 < b < (3loga) 2

+ b2

We recommend W. Duke [Du4] for a powerful treatment of a variety of problems by means of Heeke characters. 5.11. Classical automorphic £-functions.

By classical automorphic forms we mean the forms on G£(2), either holomorphic or eigenfunctions of the Laplace operator (the Maass forms). Let f be a primitive holomorphic cusp form, of weight k ~ 1, level q, with nebentypus x (see Chapter 14 for definitions). Let

f(z)

=

L AJ(n)n(k-l)f2e(nz) n)l

be its normalized Fourier expansion at the cusp oo. Then

n

p

is an £-function of degree 2 with conductor q and gamma factor given by

with ck = 2(J-k}/ 2 .j7r by the Legendre duplication formula. Therefore the analytic conductor is

+ k2 1 1+ 3)(1s + k~ 1 + 3) < q(lsl + lkl + 3) 2 , q( k21 + 3)( k~l + 3) :=:: qk2. 1

q(J, s) = q(ls q(J) =

The root number is iki)(J) where ry(J) satisfies W f = ryj. It is known that L(f, s) satisfies the Ramanujan-Petersson conjecture by work of Deligne [Del] for k ~ 2 and Deligne - Serre [DeSe] for k = 1. The dual form J satisfies

>.J(n) = x(n)>.J(n), if (n,q) = 1. Have in mind then f can be self-dual for non-trivial nebentypus. For example, if ~ is a class group character of an imaginary quadratic field K = IQl( ..fi5), then the form f with coefficients given by

AJ(n) =

L

~(n)

Na=n

is self-dual of weight k = 1, oflevel q = IDI and nebentypus X= XD, the Kronecker symbol. For these facts, see Section 14.7 and Proposition 14.13, and the references there.

132

5.

CLASSICAL ANALYTIC THEORY OF L-FUNCTIONS

Similarly, let