On the universality for functions in the

Selberg class

Jorn Steuding∗

August 2002

Abstract. We prove a universality theorem for functions in the Selberg class.

Keywords: universality, L-functions, Selberg class.

AMS subject classification numbers: 11M06, 11M41.

1 Introduction and statement of the main result

Let s = σ + it be a complex variable. The Riemann zeta-function is given by

ζ(s) =∞∑n=1

1

ns=∏p

(1−

1

ps

)−1

for σ > 1,

and by analytic continuation elsewhere, except for a simple pole at s = 1; here and

in the sequel p denotes always a prime number, and the product above is taken over

all primes. The Euler product gives a first glance on the close connection between

ζ(s) and the distribution of prime numbers. However, the Riemann zeta-function has

interesting function-theoretical properties beside. Voronin [32] proved a remarkable

universality theorem for ζ(s), namely that any non-vanishing continuous function g(s)

on the disc s ∈ C : |s| ≤ r with 0 < r < 14, which is analytic in the interior, can be

approximated uniformly by shifts of the Riemann zeta-function in the strip 12< σ < 1.

Reich [25] and Bagchi [1] improved Voronin’s result significantly. The strongest version

of Voronin’s theorem states: Suppose that K is a compact subset of the strip 12< σ < 1

with connected complement, and let g(s) be a non-vanishing continuous function on K

which is analytic in the interior of K. Then, for any ε > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max

s∈K|ζ(s+ iτ )− g(s)| < ε

> 0;

∗Institut fur Algebra und Geometrie, Fachbereich Mathematik, Johann Wolfgang Goethe-

Universitat Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany, steuding@math.uni-

frankfurt.de

1

here and in the sequel measA stands for the Lebesgue measure of the set A. The

theorem states that the set of translates on which ζ(s) approximates a given function

g(s) has a positive lower density!

Meanwhile, it is known that there exists a rich zoo of Dirichlet series with this

universality property; we mention only some significant examples: Voronin [32] proved

joint universality for Dirichlet L-functions to pairwise non-equivalent characters; Reich

[26] obtained universality for Dedekind zeta-functions; Bagchi [1] for certain Hurwitz

zeta-functions; Laurincikas [10], [11], [12] for certain Dirichlet series with multiplicative

coefficients, Lerch zeta-functions and Matsumoto zeta-functions; Matsumoto [15] for

Rankin-Selberg L-functions; Laurincikas and Matsumoto [13] for L-functions attached

to modular forms and, jointly with the author [14], for L-functions associated to new-

forms (resp. elliptic curves); Mishou [17] for Hecke L-functions. It is expected that

all functions given by a Dirichlet series and analytically continuable to the left of the

half plane of absolute convergence, which satisfy some natural growth conditions, are

universal. For a nice survey on this topic see [16].

The aim of this paper is to prove a universality theorem for functions in the Selberg

class; the proof makes use of the positive density method, introduced by Laurincikas

and Matsumoto [13]. Selberg [27] defined a general class of Dirichlet series having an

Euler product, analytic continuation and a functional equation, and formulated some

fundamental conjectures concerning them. This class of functions is of special interest

in the context of the generalized Riemann hypothesis; it is expected that for every

function in the Selberg class the analogue of the Riemann hypothesis holds, i.e. that

the nontrivial zeros lie on the critical line. For surveys on the Selberg class we refer to

[8] and [20].

The Selberg class S consists of Dirichlet series

F (s) :=∞∑n=1

a(n)

ns

satisfying the hypotheses:

• Ramanujan hypothesis: a(n) nε for every ε > 0;

• Analytic continuation: there exists a non-negative integer m such that (s −

1)mF (s) is an entire function of finite order;

• Functional equation: there are numbers Q > 0, λj > 0, µj with Reµj ≥ 0, and

some complex number ω with |ω| = 1 such that

ΛF (s) := Qsr∏j=1

Γ(λjs+ µj)F (s) = ωΛF (1− s);

2

• Euler product: F (s) satisfies

F (s) =∏p

Fp(s) with Fp(s) = exp

(∞∑k=1

b(pk)

pks

),

where b(pk) pkθ for some θ < 12.

By the latter axiom it is easily seen that the coefficients a(n) are multiplicative, and

that for each prime p

Fp(s) =∞∑k=0

a(pk)

pks,(1)

which converges absolutely for σ > 0; this is proved in [4].

The degree of F ∈ S is defined by

dF = 2r∑j=1

λj .

Despite of plenty of identities for the Gamma-function this quantity is well-defined; if

NF (T ) counts the number of zeros of F ∈ S in the region 0 ≤ σ ≤ 1, |t| ≤ T (according

to multiplicities), then one can show by contour integration that NF (T ) ∼ dFπT log T .

Denote by Sd all elements F ∈ S with degree dF equal to d. It is conjectured that

all F ∈ S have integral degree. All known examples of Dirichlet series in the Selberg

class are automorphic L-functions, and for all of them it turns out that the related

Euler factors Fp are the inverse of polynomials in p−s (of bounded degree). Examples

for functions in the Selberg class are the Riemann zeta-function and shifts of Dirichlet

L-functions L(s+ iθ, χ) attached to primitive characters χ and with θ ∈ R (degree 1);

normalized L-functions associated to holomorphic cuspforms and (conjecturally) nor-

malized L-functions attached to Maass waveforms (degree 2); Dedekind zeta-functions

to number fields K (degree [K : Q]).

The notion of a primitive function is very fruitful for studying the structure of the

Selberg class. A function 1 6≡ F ∈ S is called primitive if the equation F = F1F2 with

Fj ∈ S, j = 1, 2, implies F = F1 or F = F2. The central claim concerning primitive

functions is

Conjecture [Selberg [27]] Denote by aF (n) the coefficients of the Dirichlet series

representation of F ∈ S.

A: For all F ∈ S exists a positive integer nF such that

∑p≤x

|aF (p)|2

p= nF log log x+O(1);

3

B: for any primitive function F ,

∑p≤x

aF (p)aG(p)

p=

log log x+O(1) if F = G,

O(1) if F 6= G.

In a sense, primitive functions are expected to form an orthonormal system. Note that

the prime number theorem (see [31], §3.14) implies∑p≤x

1

p= log log x+ C +O

(exp

(−c√

log x))

,(2)

where c is some positive constant.

However, for our purpose we have to introduce a subclass. Denote by S the subset

of the Selberg class consisting of F ∈ S satisfying the following axioms:

• Euler product: for each prime p there exist complex numbers αj(p), 1 ≤ j ≤ m,

such that

F (s) =∏p

m∏j=1

(1−

αj(p)

ps

)−1

;

• Ramanujan-Petersson conjecture: for all but finitely many p we have

|αj(p)| = 1, 1 ≤ j ≤ m;

• Mean-square: there exist a positive constant κ such that

limx→∞

1

π(x)

∑p≤x

|a(p)|2 = κ,

where π(x) counts the prime numbers p ≤ x.

Note that Bombieri and Hejhal [2], resp. Bombieri and Perelli [3], defined similar

subclasses for their investigations on the value distribution of L-functions. In Chapter

5 we shall give a motivation for introducing the first two axioms in the context of

the Langlands program; the axiom on the mean square is closely related to Selberg’s

conjectures. If we assume additionally κ ∈ N in the axiom on the mean-square, we

may deduce Selberg’s Conjecture A. On the other side, a stronger version of Selberg’s

Conjecture A,

∑p≤x

|aF (p)|2

p= nF log log x+ cF + o

(1

log x

),

where cF is some constant depending on F , would imply the asymptotic formula on

the mean-square with κ = nF ; this is easily seen by partial integration.

Now we are in the position to state the main theorem:

4

Theorem 1 Suppose that F ∈ S. Let K be a compact subset of the strip

D :=s ∈ C : max

1

2, 1−

1

dF

< σ < 1

with connected complement, and let g(s) be a non-vanishing continuous function on K

which is analytic in the interior of K. Then, for any ε > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max

s∈K|F (s+ iτ )− g(s)| < ε

> 0.

Obvious examples for functions satisfying the conditions of the theorem above are

the Riemann zeta-function and Dirichlet L-functions L(s, χ) to primitive characters

χ. Further examples are Dedekind zeta-functions, Hecke L-functions, Rankin-Selberg

L-functions, Artin L-functions and L-functions associated to newforms; in Chapter 5

we shall give a further example.

Before we start with the proof we note some consequences on the value distribution

and functional independence. As in the case of the Riemann zeta-function (see [10])

one can show that i) the set

(F (s+ iτ ), F ′(s+ iτ ), . . . , F (n)(s+ iτ )) : τ ∈ R,

lies for fixed s ∈ D everywhere dense in Cn+1, and ii) if F1(z), . . . , FN(z) are continuous

functions on Cn+1, not all identically zero, then, for some s ∈ C,

N∑k=1

skFk(F (s), F ′(s), . . . , F (n)(s)) 6= 0.

2 Mean-square formulae and a limit theorem

First of all we shall prove a mean-square estimate for the coefficients of the Dirichlet

series of F .

Lemma 2 As x→∞,∑n≤x

|a(n)|2 x(log x)m2−1.

Proof. By the identity

∞∑n=1

a(n)

ns=∏p

m∏j=1

(1−

αj(p)

ps

)−1

=∏p

m∏j=1

(1 +

∞∑k=1

αj(p)k

pks

),

valid for σ > 1, and the unique prime factorization of the integers, we deduce

a(n) =∏pν‖n

∑k1+...+km=ν

m∏j=1

αj(p)kj ,

5

where pν‖n means that pν |n but pν+1 6 |n. Taking into account that |αj(p)| ≤ 1, we

find

|a(n)| ≤∏pν‖n

∑k1+...+km=ν

1 = dm(n),

say. Thus it is sufficient to find a mean-square estimate for the function dm(n); note

that dm(n) is a multiplicative arithmetic function (appearing as coefficients in the

Dirichlet series expansion of ζ(s)m). Consequently, we may write

dm(n)2 =∑d|n

g(d)

with some multiplicative function g. Since

dm(pν) = ](k1, . . . , km) ∈ Nm0 : k1 + . . .+ km = ν =(m+ ν − 1)!

ν!(m− 1)!,

we find g(1) = dm(1)2 = 1, g(p) = dm(p)2 − dm(1)2 = m2 − 1, and by induction

g(pν) = dm(pν)2 − dm(pν−1)2 ∼m2ν

ν!.

Hence we obtain∑n≤x

dm(n)2 ≤ x∑d≤x

g(d)

d≤ x

∏p≤x

(1 +

∞∑ν=1

g(pν)

pν

)

= x∏p≤x

(1 +

m2 − 1

p+∞∑ν=2

m2ν

ν!pν

)= x

∏p≤x

(1 +

m2 − 1

p

)+O(x).

Now a well-known formula due to Mertens gives the estimate of the lemma.

Furthermore, we need a mean-square estimate for the integral over F . Therefore

we apply

Lemma 3 (Potter [23]) Suppose that the functions

A(s) =∞∑n=1

an

nsand B(s) =

∞∑n=1

bn

ns

have a half plane of convergence, are of finite order, and that all singularities lie in a

subset of the complex plane of finite area. Further, assume the estimates∑n≤x

|an|2 xβ+ε and

∑n≤x

|bn|2 xβ+ε,

as x→∞, and that A(s) and B(s)

A(s) = h(s)B(1− s), where h(s) |t|c(a2−σ)

as |t| → ∞, and c is some positive constant. Then

limT→∞

1

2T

∫ T

−T|A(σ + it)|2 dt =

∞∑n=1

|an|2

n2σ

for σ > maxa2, 1

2(β + 1) − 1

c.

6

In view of the functional equation

F (s) = ωQ1−2σr∏j=1

Γ(λj(1− s) + µj)

Γ(λjs+ µj)F (1− s),

and by Stirling’s formula, we obtain for fixed σ

F (σ + it) |t|(12−σ) dF |F (1− σ + it)| as |t| → ∞.

Hence, we deduce by the Phragmen-Lindelof principle that F (s) is an entire function

of finite order, and satisfies the estimate

F (σ + it) |t|(1−σ)dF2 for 0 ≤ σ ≤ 1, as |t| → ∞.(3)

With regard to Lemma 2 application of Lemma 3 yields

Corollary 4

limT→∞

1

T

∫ T

0|F (σ + it)|2 dt =

∞∑n=1

|a(n)|2

n2σfor σ > max

1

2, 1−

1

dF

.

In order to prove Theorem 1 we need a limit theorem in the space of analytic

functions. Therefore, denote by H(D) the space of analytic functions on D equipped

with the topology of uniform convergence on compacta, and by B(S) the class of

Borel sets of a topological space S. Let γ = s ∈ C : |s| = 1 and Ω =∏p γp,

where γp = γ for each prime p. With product topology and pointwise multiplication

this infinite-dimensional torus Ω is a compact topological abelian group. Therefore

the probability Haar measure mH on (Ω,B(Ω)) exists. This gives a probability space

(Ω,B(Ω),mH). Let ω(p) stand for the projection of ω ∈ Ω to the coordinate space γp.

Then ω(p) : p ∈ P is a sequence of independent random variables defined on the

probability space (Ω,B(Ω),mH). Define for ω ∈ Ω

F (s, ω) =∏p

m∏j=1

(1−

αj(p)ω(p))

ps

)−1

.(4)

In [11] it was proved that the product converges for almost all ω ∈ Ω uniformly on

compact subsets of D. Further, it was shown that L(s, ω) is an H(D)-valued random

element on the probability space (Ω,B(Ω),mH) (functions in S form a subclass of

Matsumoto zeta-functions considered in [11]). Let P denote the distribution of the

random element L(s, ω), i.e.

P (A) = mH(ω ∈ Ω : F (s, ω) ∈ A) for A ∈ B(H(D)).

Then, by the axioms of S and Corollary 4, we obtain as a simple consequence of the

limit theorem in [11]

7

Theorem 5 The probability measure PT , defined by

PT (A) = limT→∞

1

Tmeasτ ∈ [0, T ] : F (s+ iτ ) ∈ A for A ∈ B(H(D)),

converges weakly to P , as T →∞.

For M > 0 define

DM =s ∈ C : max

1

2, 1−

1

dF

< σ < 1, |t| < M

.

Since DM ⊂ D we obtain, by the induced topology, that F (s, ω) is for s ∈ DM also an

H(DM )-valued random element on the probability space (Ω,B(Ω),mH). If Q denotes

the distribution of F (s, ω) on (H(DM ),B(H(DM))), we deduce from Theorem 5

Corollary 6 The probability measure QT , defined by

QT (A) = limT→∞

1

Tmeasτ ∈ [0, T ] : F (s+ iτ ) ∈ A for A ∈ B(H(DM)),

converges weakly to Q, as T →∞.

3 A denseness result

In view of the Euler product of F (s) and (4) we define for b(p) ∈ γ, s ∈ DM , and each

prime p

gp = gp(s) = gp(s, b(p)) = −m∑j=1

log

(1−

αj(p)b(p)

ps

).(5)

The key to prove the universality result is the following

Theorem 7 The set of all convergent series∑p gp(s) is dense in H(DM ).

For the proof we will need the following

Lemma 8 Let yp be a sequence in H(DM ) which satisfies

1o) if µ is a complex measure on (C,B(C)) with compact support contained in DM

such that

∑p

∣∣∣∣∫C yp dµ∣∣∣∣ <∞,

then ∫Csr dµ(s) = 0 for any r ∈ N ∪ 0;

8

2o) the series∑p yp converges on H(DM );

3o) for every compact K ⊂ DM

∑p

sups∈K|yp(s)|

2 <∞.

Then the set of all convergent series∑p

b(p)yp with b(p) ∈ γ,

is dense in H(DM ).

This lemma is a particular case of Theorem 6.3.10 of [10]. Further we recall some

statements on functions of exponential type. A function k(s) analytic in the closed

angular region | arg s| ≤ ϕ0 where 0 < ϕ0 ≤ π, is said to be of exponential type if

lim supr→∞

log |k(r exp(iϕ))|

r<∞ for |ϕ| ≤ ϕ0,

uniformly in ϕ. Later we shall use

Lemma 9 Let µ be a complex Borel measure on (C,B(C)) with compact support con-

tained in the half plane σ > σ0. Moreover, let

k(s) =∫C

exp(sz) dµ(z) for s ∈ C,

and k(s) 6≡ 0. Then

lim supr→∞

log |k(r)|

r> σ0.

This is Lemma 6.4.10 of [10].

Lemma 10 (Bernstein) Let k(s) be an entire function of exponential type, and let

ξm : m ∈ N be a sequence of complex numbers. Moreover, let λ, η and ω be real

positive numbers such that

1o) lim supy→∞log |k(±iy)|

y≤ λ,

2o) |ξm − ξn| ≥ ω|m− n|,

3o) limm→∞ξmm

= η,

4o) λη < π.

9

Then

lim supm→∞

log |k(ξm)|

|ξm|= lim sup

r→∞

log |k(r)|

r.

This lemma is a version of Bernstein’s theorem; for the proof see [10], Theorem 6.4.12.

Now we are in the position to give the

Proof of Theorem 7. Define gp = gp(s) = gp(s, 1). First we prove that the set of all

convergent series∑p>N

b(p)gp(s) with b(p) ∈ γ(6)

is dense in H(DM ). Let b(p) : b(p) ∈ γ be a sequence such that the series

∑p

b(p)gp with gp = gp(s) =

gp if p > N,

0 if p ≤ N,(7)

converges in H(DM ). We show that such a sequence b(p) exists. By the Taylor

expansion of the logarithm,

gp(s) =a(p)

ps+ rp(s) with rp(s) p−2σ.

The series∑p rp(s) converges uniformly on compact subsets of DM . Moreover we see,

as in the proof that L(s, ω, f) is a random element, that the series

∑p

ω(p)a(p)

ps

converges uniformly for almost all ω ∈ Ω on compact subsets of DM . Consequently,

there exists a sequence b(p) : b(p) ∈ γ such that the series

∑p

b(p)a(p)

ps

converges in H(DM ). This proves, together with the convergence of∑p rp(s), that (7)

converges in H(DM ).

Now let fp = fp(s) = b(p)gp(s). To prove the denseness of the set of all convergent

series (6) it is sufficient to show that the set of all convergent series∑p

b(p)fp with b(p) ∈ γ(8)

is dense in H(DM ). For this we will verify the hypotheses of Lemma 8. Obviously,

hypotheses 2o and 3o are fulfilled. To prove hypothesis 1o let µ be a complex Borel

measure with compact support contained in DM such that∑p

∣∣∣∣∫C fp(s) dµ

∣∣∣∣ <∞.(9)

10

Define

hp(s) =b(p)a(p)

ps,

then ∑p

|fp(s)− hp(s)| <∞

uniformly on compact subsets of DM . By the Ramanujan-Petersson conjecture, we

may define angles θp ∈ [0, π2] by

|a(p)| =

∣∣∣∣∣∣m∑j=1

αj(p)

∣∣∣∣∣∣ = m cos φp for prime p.(10)

In view of (9),∑p

cosφp|ρ(log p)| <∞,(11)

where

ρ(z) =∫C

exp(−sz) dµ(s).

Now we apply Lemma 10 with k(s) = ρ(s). By the definition of ρ(s) we have

|ρ(±iy)| ≤ exp(My)∫C|dµ(s)|

for y > 0. Therefore,

lim supy→∞

log |ρ(±iy)|

y≤M,

and the condition 1o of Lemma 10 is valid with α = M . Fix a number η with 0 < η < πM

,

and define

A =n ∈ N : ∃ r ∈

((n−

1

4

)η,

(n+

1

4

)η

]with |ρ(s)| ≤ exp(−r)

.

Further, fix a number φ with 0 < φ < min

1,√κm

, and define Pφ =

p ∈ P : cosφp > φ. Then (11) yields∑p∈Pφ|ρ(log p)| <∞.(12)

Now ∑p∈Pφ|ρ(log p)| ≥

∑n6∈A

∑p

′ |ρ(log p)| ≥∑n6∈A

∑p

′ 1

p,

11

where∑′n denotes the sum over all primes p ∈ Pφ satisfying the inequalities(

n−1

4

)η < log p ≤

(n +

1

4

)η.

Thus, in view of (12),∑n6∈A

∑p∈Pφα<p≤β

1

p<∞,(13)

where α = exp((n− 1

4

)η), β = exp

((n+ 1

4

)η). Let πφ(x) = ]p ≤ x : p ∈ Pφ,

then we obtain for α ≤ u ≤ β∑α<p≤u

cos2 φp ≤∑p∈Pφα<p≤u

1 + φ2∑p6∈Pφα<p≤u

1

= (1− φ2)(πφ(u)− πφ(α)) + φ2(π(u)− π(α).

By partial summation, the axiom on the mean square of the coefficients of F ∈ S yields∑p≤x

cos2 φp =1

m2

∑p≤x

|a(p)|2 ∼κ

m2π(x),(14)

as x→∞. Hence,

πφ(u)− πφ(α) ≥

(κm2 − φ2

1− φ2+ o(1)

)(π(u)− π(α))

for u ≥ α(1 + δ), as n→∞. Thus, we obtain by partial sumation

∑p∈Pφα<p≤β

1

p=

∫ β

α

dπφ(u)

u≥

(κm2 − φ2

1− φ2+ o(1)

) ∫ β

α

dπ(u)

u

≥

(κm2 − φ2

1− φ2+ o(1)

) ∑α(1+δ)<p≤β

1

p,(15)

as n→∞. With regard to (2)

∑α(1+δ)<p≤β

1

p=

(1

2−

log(1 + δ)

η

)1

n+O

(1

n2

).

This gives in (15)

∑p∈Pφα<p≤β

1

p≥

κm2 − φ2

1− φ2

(1

2−

log(1 + δ)

η+ o(1)

)1

n+O

(1

n2

),

as n→∞. Consequently,∑n6∈A

1

n<∞.(16)

12

Let A = ak : k ∈ N with a1 < a2 < . . .. Then (16) implies

limk→∞

ak

k= 1.(17)

By the definition of the set A there exists a sequence ξk such that(ak −

1

4

)η < ξk ≤

(ak +

1

4

)η and |ρ(ξk)| ≤ exp(−ξk).

Hence, from (17) it follows that

limk→∞

ξk

k= η and lim sup

k→∞

log |ρ(ξk)|

ξk≤ −1.

Applying Lemma 10, we obtain

lim supr→∞

log |ρ(r)|

r≤ −1.(18)

However, by Lemma 9, if ρ(z) 6≡ 0, then

lim supr→∞

log |ρ(r)|

r> 0,

contradicting (18). Therefore ρ(z) ≡ 0, and by differentiation∫Csr dµ(s) = 0 for r = 0, 1, 2, . . . .

Thus also hypothesis 1o of Lemma 8 is satisfied. Therefore, we obtain by Lemma 8 the

denseness of all convergent series (8), and hence of all convergent series (6).

Let y(s) ∈ H(DM ), K be a compact subset of DM and ε > 0. Fix N such that

sups∈K

∑p>N

∞∑ν=2

1

νpνσ

<ε

4m.(19)

By the denseness of all convergent series (6) in H(DM ) we see that there exists a

sequence b(p) : b(p) ∈ γ such that

sups∈K

∣∣∣∣∣∣y(s)−∑p≤N

gp(s)−∑p>N

b(p)gp(s)

∣∣∣∣∣∣ < ε

2.(20)

Setting

b(p) =

1 if p ≤ N,

b(p) if p > N,

then (19) and (20) imply

sups∈K

∣∣∣∣∣y(s)−∑p

gp(s)

∣∣∣∣∣ = sups∈K

∣∣∣∣∣∣y(s)−∑p≤N

gp(s)−∑p>N

gp(s)

∣∣∣∣∣∣≤ sup

s∈K

∣∣∣∣∣∣y(s)−∑p≤N

gp(s)−∑p>N

b(p)gp(s)

∣∣∣∣∣∣+ sups∈K

∣∣∣∣∣∣∑p>N

b(p)gp(s)−∑p>N

gp(s)

∣∣∣∣∣∣≤

ε

2+ 2m sup

s∈K

∑p>N

∞∑ν=2

1

νpνσ

< ε.

Since y(s), K and ε are arbitrary, the theorem is proved.

13

4 The support of the measure QT

Now we identify the measure QT , defined in Corollary 6.

Lemma 11 The support of the measure QT is the set

SM = ϕ ∈ H(DM ) : ϕ(s) 6= 0 for s ∈ DM , or ϕ(s) ≡ 0.

In order to prove this lemma we make use of the following two lemmas.

Lemma 12 (Hurwitz) Let fn(s) be a sequence of functions analytic on DM such

that fn(s) → f(s) uniformly on DM , as n → ∞. Suppose that f(s) 6≡ 0, then an

interior point s0 of DM is a zero of f(s) if, and only if, there exists a sequence sn in

DM such that sn → s0, as n→∞, and fn(sn) = 0 for all n large enough.

A proof of Hurwitz’ theorem can be found in [31], section 3.4.5.

Lemma 13 Let Xn be a sequence of independent H(G)-valued random elements,

where G is a region in C, and suppose that the series∑∞n=1 Xn converges almost

everywhere. Then the support of the sum of this series is the closure of the set of

all ϕ ∈ H(G) which may be written as a convergent series

ϕ =∞∑n=1

ϕn with ϕn ∈ SXn,

where SXn is the support of the random element Xn.

This is Theorem 1.7.10 of [10].

Now we can give the

Proof of Lemma 11. The sequence ω(p) is a sequence of independent random

variables on the probability space (Ω,B(Ω),mH). Define xp(s) = gp(s, ω(p)), then

xp(s) is a sequence of independent H(DM )-valued random elements. The support of

each ω(p) is the unit circle γ, and therefore the support of the random elements xp(s)

is the set

ϕ ∈ H(DM ) : ϕ(s) = yp(s, b) with b ∈ γ,

where yp(s, b) = gp(s, b). Consequently, by Lemma 13, the support of the H(DM )-

valued random element

logL(s, ω, f) =∑p

xp(s)

14

is the closure of the set of all convergent series∑p fp(s) in the notation of section 3.

By Theorem 7 the set of these series is dense in H(DM ). The map

h : H(DM )→ H(DM ), f 7→ exp(f)

is a continuous function sending logL(s, ω, f) to L(s, ω, f) and H(DM ) to SM \ 0.

Therefore, the support SL of L(s, ω, f) contains SM \ 0. On the other hand, the

support of L(s, ω, f) is closed. By Lemma 12 it follows that SM \ 0 = SM . Thus,

SM ⊂ SL. Since the Ramanujan-Petersson hypothesis is satisfied, the functions

exp(gp(s, ω(p))) =m∏j=1

(1−

αj(p)ω(p)

ps

)

are non-zero for s ∈ DM , ω ∈ Ω. Hence, L(s, ω, f) is an almost surely convergent

product of non-vanishing factors. If we apply Lemma 12 again, we conclude that

L(s, ω, f) ∈ SM almost surely. Therefore SL ⊂ SM . The lemma is proved.

Now we are in the position to give the

Proof of Theorem 1. Since K is a compact subset of D, there exists a number M

such that K ⊂ DM .

First we suppose that g(s) has a non-vanishing analytic continuation to H(DM ).

Denote by Φ the set of functions ϕ ∈ H(DM ) such that

sups∈K|ϕ(s)− g(s)| < ε.

By Lemma 11 the function g(s) is contained in the support SL of the random element

F (s, ω). Since by Corollary 6 the measure QT converges weakly to Q, as T →∞, and

the set Φ is open, it follows from the properties of weak convergence and support that

lim infT→∞

νT

(sups∈K|F (s+ iτ )− g(s)| < ε

)≥ Q(Φ) > 0.(21)

Now let g(s) be as in the statement of the theorem. Here we have to apply a

well-known approximation result for polynomials (a proof can be found in [33]):

Lemma 14 (Mergelyan) Let K be a compact subset of C with connected comple-

ment. Then any continuous function g(s) on K which is analytic in the interior of K

is approximable uniformly on K by polynomials in s.

Thus, there exists a sequence pn(s) of polynomials such that pn(s)→ g(s) as n→∞

uniformly on K. Since g(s) is non-vanishing on K, we have pm(s) 6= 0 on K for

sufficiently large m, and

sups∈K|g(s)− pm(s)| <

ε

4.(22)

15

Since the polynomial pm(s) has only finitely many zeros, there exists a region G1

whose complement is connected such that K ⊂ G1 and pm(s) 6= 0 on G1. Hence there

exists a continuous branch log pm(s) on G1, and log pm(s) is analytic in the interior

of G1. Thus, by Lemma 14, there exists a sequence qn(s) of polynomials such that

qn(s)→ log pn(s) as n→∞ uniformly on K. Hence, for sufficiently large k

sups∈K|pm(s)− exp(qk(s))| <

ε

4.

Thus and from (22) we obtain

sups∈K|g(s)− exp(qk(s))| <

ε

2.(23)

From (21) we deduce

lim infT→∞

νT

(sups∈K|F (s+ iτ )− exp(qk(s))| <

ε

2

)> 0.

This proves in connection with (23) the theorem.

5 Langlands program and power L-functions

In this final chapter we shall give a motivation for introducing the subclass S and give

a further application of our universality theorem.

The Langlands program tries to unify number theory and representation theory.

These two disciplines are linked by L-functions associated to automorphic representa-

tions and the relations between the analytic properties and the underlying algebraic

structures; for an introduction to the Langlands program see [6] and [18]. For the

sake of simplicity we now deal only with Q (and not with an arbitrary number field

K). Denote by A the adele ring of Q. Further, let π be an automorphic cuspidal

representation of GLm(Q), i.e. an irreducible unitary representation of GLm(A) which

appears in its regular representation on GLm(Q)\GLm(A). Then π can be factored into

a direct product π = ⊗pπp with each πp being an irreducible unitary representation of

GLm(Qp) if p <∞, where Qp is the field of p-adic numbers, and of GLm(R) if p =∞.

For all but a finite number of places p the representation πp is unramified. We define

the L-function associated to π by

L(s, π) =∏p

m∏j=1

(1−

αj(p)

ps

)−1

,(24)

and the completed L-function by

Λ(s, π) = L(s, π∞)L(s, π) with L(s, π∞) =m∏j=1

Γ

(s − αj(∞)

2

);

16

here the numbers αj(p), 1 ≤ j ≤ m are determined from the local representations

πp, p ≤∞. Jacquet [7] showed that any Λ(s, π) satisfies after a suitable normalization

the functional equation

Λ(s, π) = επNs−1

2π Λ(1− s, π),

where π is the congradient representation of π, Nπ ∈ N is the conductor of π and επ is

of modulus 1, and is the sign of the functional equation.

For m = 1 one obtains simply the Riemann zeta-function and the Dirichlet L-

functions whereas for m = 2 one gets L-functions associated to modular forms. It is

expected that all zeta-functions arising in number theory are but special realizations

of L-functions to automorphic representations constructed above. On the other side it

is expected that all functions in the Selberg class are automorphic L-functions. M.R.

Murty [19] proved that, assuming Selberg’s conjecture, i) if π is any irreducible cus-

pidal automorphic representation of GLm(A) which satisfies the Ramanujan-Petersson

conjecture, then L(s, π) is primitive (in the sense of the Selberg class), and ii) if K is

a Galois extension of Q with solvable group G, and χ is an irreducible character of G

of degree m, then there exists an irreducible cuspidal automorphic representation π of

GLm(A) such that L(s, χ) = L(s, π).

This motivates the special form of the Euler product in the definition of the subclass

S; the axiom on the mean square was already discussed in the context of the Selberg

conjectures. It only remains to consider the Ramanujan-Petersson conjecture. The

Ramanujan τ -function is defined by the power series

∆(z) :=∞∑n=1

τ (n)qn = q∞∏n=1

(1− qn)24 with q = exp(2πiz), Re z > 0;

∆(z) is a normalized cuspform of weight 12 to the full modular group SL2(Z). Ramanu-

jan [24] conjectured that τ (n) is multiplicative and satisfies the estimate |τ (p)| ≤ 2p112 .

This was proved by Mordell and Deligne [5], respectively. Petersson [22] extended this

to the coefficients of modular forms of level N . It is expected that this holds for all L-

functions of arithmetical nature; after a suitable normalization of the coefficients (such

that the L-function satisfies a functional equation with point symmetry at s = 12):

Conjecture [Ramanujan-Petersson] With the notation from above, if πp is unram-

ified for p <∞, then

|αj(p)| = 1 for 1 ≤ j ≤ m,

and if π∞ is unramified, then Reαj(∞) = 0 for 1 ≤ j ≤ m.

With view to all these widely believed conjectures it might be possible that the subclass

S coincides with the Selberg class.

17

We conclude with an application of our universality theorem to some typical L-

functions in the Langlands setting. Symmetric power L-functions became important

by Serre’s reformulation of the Sato-Tate conjecture [28]. However, before we can give a

definition of symmetric power L-functions we have to recall some facts from the theory

of modular forms. Let f(z) be a normalized cusp form of weight k and level N . In

particular, f(z) has a Fourier expansion

f(z) =∞∑n=1

a(n)nk−1

2 exp(2πinz).

By the theory of Hecke operators, the coefficients turn out to be multiplicative, and

we may attach to f its L-function

L(s, f) =∞∑n=1

a(n)

ns=∏p|N

(1−

a(p)

ps

)−1 ∏p6 |N

(1−

a(p)

ps+

1

p2s

)−1

;

both series and product converge absolutely for σ > 1. Hecke proved that L(s, f) has

an analytic continuation to an entire function and satisfies a functional equation of

Riemann type. With regard to Deligne’s celebrated proof of the Ramanujan-Petersson

conjecture [5] we may define an angle θp ∈ [0, π] by setting

a(p) = 2 cos θp

for each prime p; this should be compared with the angles defined by (10). Now let k

be an even positive integer. For any non-negative integer m the symmetric m-th power

L-function attached to f is given by

Lm(s, f) :=∏p6 |N

m∏j=0

(1−

exp(iθp(m− 2j))

ps

)−1

for σ > 1. It can be shown that

L0(s, f) = ζ(s), L1(s, f) = L(s, f) and L2(s, f) =ζ(2s)

ζ(s)L(s, f ⊗ f),

where L(s, f⊗f) is the Rankin-Selberg convolution L-function. Shimura [30] obtained

the analytic continuation and functional equation in case of m = 2; for m > 2 this

is an open problem. Serre [28] conjectured that if p ranges over the set of prime

numbers, then the angles θp are uniformly distributed with respect to the Sato-Tate

measure 2π

sin2 θ dθ (in analogy to a similar conjecture on elliptic curves due to Sato and

Tate). Furthermore, Serre proved that the non-vanishing of Lm(s, f) on the abscissa of

convergence σ = 1 for all m ∈ N would imply the Sato-Tate conjecture for newforms,

namely

limx→∞

1

π(x)]p ≤ x : α < θp < β =

2

π

∫ β

αsin2 θ dθ.

18

However, in the case of L-functions associated to modular forms it would be sufficient

to have an analytic continuation to σ ≥ 1 for proving the Sato-Tate conjecture; see [21].

If this is known to hold for all Lr(s, f), r ≤ 2(m+ 1), then one can deduce asymptotic

formulae for the 2mth-power moments of cos θp. Taking deep results of Hecke, Ogg,

Shahidi and Shimura into account, M.R. Murty and V.K. Murty [20] proved that if

Lr(s, f) has an analytic continuation up to σ ≥ 12

for all r ≤ 2(m+ 1), then

limx→∞

1

π(x)

∑p≤x

(2 cos θp)2r = 1

r+1

(2rr

)for r ≤ m+ 1,

limx→∞

1

π(x)

∑p≤x

(2 cos θp)2r+1 = 0 for r ≤ m.

This implies Lm(s, f) ∈ S for m = 0, 1 unconditionally, and for m ≥ 2 conditionally

(depending on the analytic continuation). Thus, Theorem 1 yields the universality of

Lm(s, f) for m = 0, 1 unconditionally, and for m ≥ 2 if all Lm(s, f) have analytic con-

tinuation throughout C. By the powerful methods of the Langlands program Shahidi

[29] obtained analytic continuation to σ ≥ 1 for m ≤ 4 (in particular cases more is

known, see [9]).

Acknowledgements. The author is very grateful to professors R. Garunkstis, A.

Laurincikas, K. Matsumoto for several discussions on universality and to Prof. M.R.

Murty for submitting material on Artin L-functions and the Selberg class.

References

[1] B. Bagchi, The statistical behaviour and universality properties of the Riemann

zeta-function and other allied Dirichlet series, Ph.D.Thesis, Calcutta, Indian Sta-

tistical Institute, 1981

[2] E. Bombieri, D.A. Hejhal, On the distribution of zeros of linear combinations

of Euler products, Duke Math. J. 80 (1995), 821-862

[3] E. Bombieri, A. Perelli, Distinct zeros of L-functions, Acta Arith. 83 (1998),

271-281

[4] J.B. Conrey, A. Ghosh, On the Selberg class of Dirichlet series: small degrees,

Duke Math. J. 72 (1993), 673-693

[5] P. Deligne, La Conjecture de Weil I, II, Publ. I.H.E.S. 43 (1974), 273-307; 52

(1981), 313-428

[6] S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer.

Math. Soc. 10 (1984), 177-219

19

[7] H. Jacquet, Principal L-functions of the linear group, Proc. Symp. Pure Math.

Vol 33, AMS part II (1979), 63-86

[8] J. Kaczorowski, A. Perelli, The Selberg class: a survey, Number theory in

progress. Proceedings of the international conference in honor of the 60th birth-

day of Andrej Schinzel, Zakopane 1997. Vol. 2: Elementary and analytic number

theory. De Gruyter 1999, 953-992

[9] H.H. Kim, F. Shahidi, Symmetric cube L-functions for GL2 are entire, Ann.

Math. 150 (1999), 645-662

[10] A. Laurincikas, Limit theorems for the Riemann zeta-function, Kluwer Acad-

emic Publishers, Dordrecht 1996

[11] A. Laurincikas, On the limit distribution of the Matsumoto zeta-function I,II,

Acta Arith. 79 (1997), 31-39; Liet. Matem. Rink. 36 (1996), 464-485 (in Russian)

[12] A. Laurincikas, The universality of the Lerch zeta-functions, Liet. Matem. Rink.

37 (1997), 367-375 (in Russian); Lith. Math. J. 37 (1997), 275-280

[13] A. Laurincikas, K. Matsumoto, The universality of zeta-functions attached

to certain cusp forms, Acta Arith. 98 (2001), 345-359

[14] A. Laurincikas, K. Matsumoto, J. Steuding, The universality of L-

functions associated to newforms, (to appear)

[15] K. Matsumoto, The mean values and universality of Rankin-Selberg L-functions,

in: Number Theory, Proceedings of the Turku Symposium on Number theory in

memory of Kustaa Inkeri (1999), M. Jutila, T. Metsankyla (Eds.), Walter de

Gruyter, Berlin, New York 2001, 201-221

[16] K. Matsumoto, Probabilistic value-distribution theory of zeta-functions, Sugaku

53 (2001), 279-296 (in Japanese)

[17] H. Mishou, The universality theorem for L-functions associated with ideal class

characters, Acta Arith. 98 (2001), 395-410

[18] M.R. Murty, A motivated introduction to the Langlands program, Advances in

Number theory, F. Gouvea and N. Yui, eds., Clarendon Press Oxford 1993, 37-66

[19] M.R. Murty, Selberg’s conjectures and Artin L-functions, Bull. Amer. Math.

Soc. 31 (1994), 1-14

[20] M.R. Murty, V.K. Murty, Non-vanishing of L-functions and applications,

Birkhauser 1997

20

[21] V.K. Murty, On the Sato-Tate conjecture, Number theory related to Fermat’s

Last Theorem, (ed. N. Koblitz), Birkhauser 1982, 195-205

[22] H. Petersson, Konstruktion der samtlichen Losungen eienr Riemannschen

Funktionalgleichung durch Dirichletreihen mit Eulerscher Produktentwicklung II,

Math. Ann. 117 (1940/41), 39-64

[23] H.S.A. Potter, The mean values of certain Dirichlet series I, Proc. London

Math. Soc. 46 (1940), 467-468

[24] S. Ramanujan, On certain arithmetical functions, Trans. Camb. Phil. Soc. 22

(1916), 159-184

[25] A. Reich, Universelle Wertverteilung von Eulerprodukten, Nach. Akad. Wiss.

Gottingen, Math.-Phys. Kl. (1977), 1-17

[26] A. Reich, Wertverteilung von Zetafunktionen, Arch. Math. 34 (1980), 440-451

[27] A. Selberg, Old and new conjectures and results about a class of Dirichlet series,

Proceedings of the Amalfi Conference on Analytic Number Theory, Amalfi 1989,

Salerno (1992), 367-385

[28] J.-P. Serre, Abelian l-adic representations and elliptic curves, Research Notes

in Mathematics, AK Peters 1968

[29] F. Shahidi, Symmetric power L-functions for GL(2), in: Elliptic curves and

related objects, ed. H. Kisilevsky and M.R. Murty, CRM Proceedings and Lecture

Notes, Vol. 4 (1994), 159-182

[30] G. Shimura, On modular forms of half integral weight, Ann. Math. 97 (1973),

440-481

[31] E.C. Titchmarsh, The theory of functions, Oxford 1932

[32] S.M. Voronin, Theorem on the ’universality’ of the Riemann zeta-function, Izv.

Akad. Nauk SSSR, Ser. Matem., 39 (1975 (in Russian); Math. USSR Izv. 9 (1975),

443-445

[33] J.L. Walsh, Interpolation and approximation by rational functions in the complex

domain, Amer. Math. Soc. Coll. Publ. vol. 20, 1960

21