Lithuanian Mathematical Journal, Vol. 50, No. 3, 2010, pp. 293–311

UNIVERSALITY FOR L-FUNCTIONS IN THE SELBERG CLASS H. Nagoshi a and J. Steuding b a

Department of Fundamental Studies, Gunma University, Tenjin-cho 1-5-1, Kiryu 376-8515, Japan b Department of Mathematics, Würzburg University, Am Hubland, 97074 Würzburg, Germany (e-mail: [email protected]; [email protected])

Received February 16, 2010

Abstract. We prove universality for L-functions L from the Selberg class satisfying some mild condition on the Dirichlet coefficients (which might be considered as a prime number theorem for L). This generalizes a previous universality theorem by the second author, where the L-function was assumed to have a polynomial Euler product satisfying the Ramanujan hypothesis. MSC: 11M41 Keywords: Selberg class, L-function, universality, value-distribution

1

INTRODUCTION

One of the most fascinating functions in mathematics is the Riemann zeta-function ζ(s). Since Riemann’s path-breaking memoir from 1859 it is well known that the distribution of prime numbers is intimately linked with the complex zeros of ζ(s), a relation that is not yet completely understood. Besides its relevance in number theory, the zeta-function obeys remarkable analytical properties. In 1975, Voronin [5] proved his famous universality theorem for ζ(s), which states that any nonvanishing analytic function can be approximated uniformly by certain purely imaginary shifts of the zeta-function in the critical strip: Let 0 < r < 14 and suppose that f (s) is a nonvanishing continuous function on the disk |s|  r that is analytic in the interior. Then, for any  > 0, there exists a positive real number τ such that       3  maxζ s + + iτ − f (s) < . 4 |s|r Moreover, the set of such approximating real numbers τ has positive lower density! This reminds us of Weierstrass’ approximation theorem on polynomial approximations to arbitrary continuous functions and the theorem of Mergelyan for the complex analogue; however, by Voronin’s theorem, a single function can approximate a huge class of target functions as close as possible! This astonishing approximation property is called universality. Voronin’s theorem has been improved and extended in various directions. In the meantime, many relatives of the zeta-function have been found that share the universality property in the above or a similar way; some c 2010 Springer Science+Business Media, Inc. 0363-1672/10/5003-0293 

293

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H. Nagoshi and J. Steuding

examples are given below. All these universal relatives have in common that they are representable by or are related to a Dirichlet series in some right half-plane. Although universality is a rather common phenomenon in analysis, the zeta-function and its relatives are the only explicit universal objects so far. (For more details, we refer to [1, 4].) In this paper, we are concerned with so-called L-functions that play an important role in arithmetic. We follow the axiomatic setting of Selberg. The Selberg class S is defined as the set of all Dirichlet series L(s) =

∞  a(n)

ns

n=1

satisfying the following axioms: • Ramanujan hypothesis: a(n)  n

(1.1)

for any  > 0, where the implicit constant may depend on . • Analytic continuation: there exists a nonnegative integer k such that (s − 1)k L(s) is an entire function of finite order. • Functional equation: L(s) satisfies a functional equation of the type ΛL (s) = ωΛL (1 − s),

where f 

ΛL (s) := L(s)Qs

Γ(λj s + μj )

j=1

with positive real numbers Q, λj and complex numbers μj and ω with Re μj  0 and |ω| = 1. • Euler product: L(s) has a product representation  L(s) = Lp (s), p

where the product is taken over all prime numbers p, and log Lp (s) =

∞  b(pk ) k=1

pks

(1.2)

with suitable coefficients b(pk ) satisfying   b pk  pkθ

1 for some θ < . 2

(1.3)

It is widely believed that the Selberg class contains all relevant L-functions. Examples of elements in the Selberg class are the Riemann zeta-function, Dirichlet L-functions to primitive characters, Dedekind zetafunctions, Hecke L-functions to number fields, L-functions associated with Hecke eigenforms, and Artin L-functions subject to the truth of Artin’s conjecture. In order to state our main theorem, we define the degree of L ∈ S by dL = 2

f  j=1

λj ;

Universality for L-functions in the Selberg class

295

although the data of the functional equation is not unique, the degree is well defined, which follows from the analogue of the Riemann–von Mangoldt formula for the number of complex zeros of L (see [4, (6.2)]). Moreover, let

1 1 σL := max , 1 − . (1.4) 2 dL Writing the complex variable as s = σ + it, the mean-square of L exists in the half-plane σ > σL , a fact we shall use in the course of our proof of universality. If it would turn out that the abscissa σm of the mean-square half-plane for L is strictly less than σL , as for certain Dedekind zeta-functions, then the method of proof (see also [4]) would allow an extension of the strip of universality D to σm < σ < 1 in the following universality theorem:  −s be a function of a complex variable s = σ + it in the Selberg class S . Theorem 1. Let L(s) = ∞ n=1 a(n)n Assume that 2 1   a(p) = κ, x→∞ π(x) lim

(1.5)

px

where κ is some positive constant (depending on L), the summation is over all primes p  x, and π(x) counts the number of those primes. Let K be a compact set in the strip D := {s ∈ C: σL < σ < 1} with connected complement, and let g(s) be a nonvanishing continuous function on K that is analytic in the interior of K. Then, for any  > 0, lim inf T →∞



  1 meas τ ∈ [0, T ]: maxL(s + iτ ) − g(s) <  > 0, s∈K T

where meas denotes the Lebesgue measure. In a previous universality theorem for elements from the Selberg class by the second-named author (see [4, Section 6.6]), the L-function in question was additionally assumed to have a polynomial Euler product satisfying the Ramanujan hypothesis, some condition that is satisfied by all known examples of L-functions in the Selberg class and is widely conjectured to be true for all L ∈ S ; however, any proof that this is indeed true seems to be out of reach by the present day methods. Theorem 1 asserts that, irrespective of the truth of this conjecture, any L-function in the Selberg class satisfying (1.5) has the universality property. There are several standard applications of universality, e.g., any universal L-function L(s) is functionally independent and, in particular, does not satisfy any algebraic differential equation. For this and other consequences of universality, we refer to [1, 4]. 2

A DENSENESS RESULT

section is In the sequel, let L(s) and K be as in Theorem 1, and let the numbers b(pk ) be as in (1.2). This −s considered c(n)n devoted to proving Proposition 1 below. As mentioned in Section 1, the L-function ∞ n=1 in [4, Section 6.6] is assumed to have a polynomial Euler product satisfying the Ramanujan hypothesis. This assumption gives the estimate c(p)  1

(see [4, Lemma 2.2]). For our L-function L(s) = a(p)  p Lith. Math. J., 50(3):293–311, 2010.

for all primes p

∞

n=1 a(n)n

−s ,

(2.1)

which belongs to a larger class, we have

for all  > 0 and all primes p

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by (1.1), but a stronger estimate like (2.1) is not ensured. In order to handle this weaker situation and the numbers b(pk ) (k  2), we prepare the numbers σ1 , σ2 , and δ as follows. We fix real numbers σ1 and σ2 satisfying   1  σL < σ1 < min Re s and max Re s < σ2 < 1. (2.2) s∈K s∈K 2 Define the strip D1 by D1 := {s ∈ C: σ1 < σ < σ2 }.

(2.3)

Noting that 1 − σ2 > 0 and σ1 > 21 , we fix a small positive real number δ satisfying 1 − σ2 − 2δ > 0

Axiom (1.1) of the Selberg class S gives   a(p)  Cpδ

and

1 σ1 − δ > . 2

for all primes p,

(2.4)

(2.5)

where C is some positive constant that may depend on δ . Let γ denote the unit circle {s ∈ C: |s| = 1}. For a simply connected region D in C, we denote by H(D) the space of holomorphic functions on D, and we equip it with the topology of uniform convergence on compacta. For a topological space S , let B(S) denote the class of Borel sets of S . Lemma 1. (i) We have a(p) = b(p)

for all primes p. (ii) For any  > 0, we have the inequality   k    b p   2k − 1 pk /k for all primes p and all k ∈ N. Proof. Assertion (i) is well known (see, e.g., [4, p. 112]). Assertion (ii) is obtained from [3, p. 122, Exercise  8.2.9] and axiom (1.1).  Lemma 2. Let D be a simply connected region in C. Let {fm } be a sequence in H(D) that satisfies the following assumptions: (a) if μ is a complex Borel measure on (C, B(C)) with compact support contained in D such that  ∞      fm (s) dμ(s) < ∞,  

m=1 C

then



sr dμ(s) = 0

C

(b) the series

∞

m=1 fm

converges in H(D);

for every r = 0, 1, 2, . . . ;

Universality for L-functions in the Selberg class

297

(c) for any compact K ⊂ D, ∞  m=1

 2 maxfm (s) < ∞. s∈K

Then the set of all convergent series ∞ 

am fm

with am ∈ γ

m=1

is dense in H(D). Proof.

 This is [1, Theorem 6.3.10]. 

Lemma 3. Let μ be a complex Borel measure on (C, B(C)) with compact support in the half-plane σ > σ0 . Moreover, let  f (z) = esz dμ(s). C

If f (z) ≡ 0, then lim sup r→∞

Proof.

log |f (r)| > σ0 . r

This is [1, Lemma 6.4.10].  

Lemma 4. Let f (s) be an entire function of exponential type, and let {ξm } be a sequence of complex numbers. Moreover, assume that there are positive real constants λ, η , and ω such that (a) lim supy→∞ log |fy(±iy)|  λ, (b) |ξm − ξn |  ω|m − n|, (c) limm→∞ ξmm = η , (d) λη < π . Then lim sup m→∞

Proof.

log |f (ξm )| log |f (r)| = lim sup . ξm r r→∞

This is [1, Theorem 6.4.12].  

Proposition 1. As in Theorem 1, let L ∈ S and assume that 2 1   a(p) = κ x→∞ π(x) lim

(2.6)

px

with some positive constant κ. For a prime p and a number c(p) ∈ γ , we define the series 



gp s, c(p) :=

∞  b(pk )c(p)k k=1

Lith. Math. J., 50(3):293–311, 2010.

pks

,

(2.7)

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H. Nagoshi and J. Steuding

which converges absolutely (at least) on the half-plane σ > σ1 , by (1.3). Then the set of all convergent series    gp s, c(p) p

with c(p) ∈ γ is dense in H(D1 ). Proof. Define g˜p (s) := gp (s, 1) =

∞  b(pk )

pks

k=1

.

(2.8)

For a number N > 0, which will be chosen later (see (2.35)), define g˜p (s) if p > N , gˆp (s) := if p  N . 0 We shall show the assertion that there exists a sequence {ˆ c(p): cˆ(p) ∈ γ} such that the series  cˆ(p)ˆ gp (s)

(2.9)

p

converges in H(D1 ). By Lemma 1(i) we have ∞

g˜p (s) =

a(p)  b(pk ) + . ps pks

(2.10)

k=2

In view of (2.2), we fix a large positive integer k0 such that   1 k0 σ1 − > 1. 2

(2.11)

By Lemma 1(ii) we have     b pk δ 2k0 − 1 pkδ for all primes p and all positive integers k  k0 . This, (1.3), and (2.4) imply that ∞  |b(pk )| k=2

pkσ1

=

 |b(pk )|  |b(pk )| + δ,k0 pkσ1 pkσ1

2k
 k0

1 p2σ1 −2δ

k  k0 1

1

+

pk0 (σ1 − 2 ) 1 − σ11− 1 2 p

k0 ,σ1

 2k
1 p2σ1 −2δ

+

1 pk(σ1 −δ)

+

1 p

k0 (σ1 − 12 )



1

k  k0

.

1 pk(σ1 − 2 )

(2.12)

By (2.4) and (2.11) we have ∞  |b(pk )| p k=2

pkσ1

< ∞.

(2.13)

Universality for L-functions in the Selberg class

299

Since by (2.5) and (2.4) we have  |a(p)|2 p2σ1

p





C2

< ∞,

p2σ1 −2δ

p

it follows from Lemma 6.5.3 of [1] that there exists a sequence {ˆ c(p): cˆ(p) ∈ γ} such that  a(p)ˆ c(p) pσ 1

p

converges. Thus, from this, (2.13), (2.10), and from a property of Dirichlet series (see, e.g., [1, p. 28]) we obtain the above assertion on the series (2.9). We shall show that the set of all convergent series 

c˜(p)ˆ gp (s) =

p



with c˜(p) ∈ γ

c˜(p)˜ gp (s)

(2.14)

p>N

is dense in H(D1 ). Obviously, it suffices to show that the set of all convergent series 

c˜(p)fp (s)

with c˜(p) ∈ γ

(2.15)

p

is dense, where fp (s) := cˆ(p)ˆ gp (s).

To show this, we  apply Lemma 2. As in (2.9), we have already verified assumption (b) of Lemma 2, namely that the series p fp (s) converges in H(D1 ). It follows from Lemma 1(i), (2.5), (2.12), (2.4), (2.2), and (2.11) that, for any prime p and any s ∈ D1 , we have  ∞  k)      |a(p)|  |b(pk )| b(p  |b(p)|  |b(pk )| fp (s)   + = +    pσ 1 p σ1 pks  pkσ1 pkσ1 k=1

δ,k0 ,σ1

k 2

1 pσ1 −δ

+

1 p2σ1 −2δ

+

k 2

1 1

pk0 (σ1 − 2 )



1 pσ1 −δ

.

By this and (2.4), for any compact set K1 ⊂ D1 , we have  p

  2 maxfp (s) δ,k0 ,σ1

s∈K1

p

1 p2σ1 −2δ

< ∞,

and therefore assumption (c) of Lemma 2 holds. Next, we shall verify assumption (a) of Lemma 2. Let μ be a complex Borel measure on (C, B(C)) with compact support contained in D1 such that      fp (s) dμ(s) < ∞.   p

Lith. Math. J., 50(3):293–311, 2010.

C

(2.16)

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H. Nagoshi and J. Steuding

We put hp (s) :=

a(p)ˆ c(p) . ps

Then by (2.13) we have  p

∞    |a(p)|   |b(pk )| sup fp (s) − hp (s)  + < ∞. pσ 1 pkσ1 s∈D1 pN

p>N k=2

This and (2.16) give      hp (s) dμ(s) < ∞.   p

(2.17)

C

Define

 ρ(z) :=

exp(−sz) dμ(s). C

Then (2.17) implies    a(p)ρ(log p) < ∞.

(2.18)

p

From this we will deduce that ρ(z) vanishes identically. See (2.34) below. To prove this vanishing, we apply Lemmas 4 and 3. We choose a sufficiently large positive constant M such that the support of μ is contained in the region {s ∈ D1 : |t| < M − 1}. Then, by the definition of ρ(z), we have      ρ(±iy)  exp(M y) μ(s) C

for y > 0, and, hence, lim sup y→∞

log |ρ(±iy)|  M. y

(2.19)

M η < π.

(2.20)

Let us fix a positive real number η such that

Let A be the set of all positive integers n such that there exists a real number r ∈ ((n− 14 )η, (n+ 14 )η] satisfying   ρ(r)  exp(−σ2 r). For n ∈ N, set

 αn := exp

  1 n− η 4

 and βn := exp

  1 n+ η . 4

(2.21)

Universality for L-functions in the Selberg class

301

Fix a positive real number μ satisfying C 2 β12δ − μ2 > 0

and κ − μ2 > 0,

(2.22)

where C is as in (2.5). Define     Pμ := p: p is prime, a(p) > μ . The definition of A implies that, for n ∈ / A, the estimate   ρ(log p) > 1 pσ 2 holds for all primes p ∈ (αn , βn ]. For all n ∈ N, we have βn < αn+1 . Hence,       a(p)ρ(log p) μ ρ(log p)  p

p∈Pμ



n∈A /





n∈A /

p∈Pμ αn


  ρ(log p)

p∈Pμ αn
 1  p σ2

n∈A /



p∈Pμ αn
 1 1 = βnσ2 βnσ2 n∈A /



1.

p∈Pμ αn
This and (2.18) yield  1 βnσ2

n∈A /



1 < ∞.

(2.23)

p∈Pμ αn
We shall evaluate the sum in (2.23). Let πμ (x) := #{p  x: p ∈ Pμ }.

We have 



  a(p)2 =

p∈Pμ αn
αn
  a(p)2 +



  a(p)2 .

(2.24)

p∈ / Pμ αn
By (2.5) we obtain 



  a(p)2 

p∈Pμ αn
  C 2 βn2δ = C 2 βn2δ πμ (βn ) − πμ (αn ) .

p∈Pμ αn
By the definition of Pμ we obtain 

  a(p)2  μ2

p∈ / Pμ αn
Lith. Math. J., 50(3):293–311, 2010.

 p∈ / Pμ αn
    1 = μ2 π(βn ) − π(αn ) − πμ (βn ) − πμ (αn ) .

(2.25)

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This, (2.25), and (2.24) yield that 

       a(p)2  C 2 βn2δ − μ2 πμ (βn ) − πμ (αn ) + μ2 π(βn ) − π(αn ) .

(2.26)

αn
In view of (2.22), let us fix a small positive real number  satisfying 1

(κ − μ2 )(1 − e− 2 η ) − 4 > 0. 2

(2.27)

It follows from condition (2.6) that, for all large n ∈ N, 

       a(p)2 = a(p)2 − a(p)2  (κ − )π(βn ) − (κ + )π(αn )

αn
p  αn

p  βn



   = κ π(βn ) − π(αn ) −  π(βn ) + π(αn ) .

x The prime number theorem π(x) = 2 (2.21) yield that, for all large n ∈ N, βn π(βn ) − π(αn ) = αn

du log u

√ + O(x exp(−c log x)) with some constant c > 0 and definition

√ √   βn − αn   du + O βn e−c log αn + O βn e−c log αn  log u log βn 1

=

(2.28)

1

√   1 βn (1 − e− 2 η ) βn (1 − e− 2 η ) + O βn e−c log αn  . log βn 2 log βn

The prime number theorem π(x) ∼

x log x

(2.29)

gives

π(βn ) + π(αn )  2π(βn ) 

4βn . log βn

(2.30)

Combining (2.26), (2.28), (2.29), and (2.30) and recalling (2.27), we deduce that, for all large n ∈ N, 

  C 2 βn2δ − μ2 πμ (βn ) − πμ (αn )       κ − μ2 π(βn ) − π(αn ) −  π(βn ) + π(αn ) 1

(κ − μ2 )(1 − e− 2 η ) βn βn βn  − 4

; 2 log βn log βn log βn

here and in what follows, the constants implied by the symbols and  may depend on κ, η , μ, , δ , and σ2 . Thus, if n is sufficiently large, then πμ (βn ) − πμ (αn )

1 (C 2 βn2δ

−

μ2 )

βn βn

2δ . log βn βn log βn

Universality for L-functions in the Selberg class

303

From this and from (2.21) we conclude that  1 βnσ2

n∈A /



1=

p∈Pμ αn
 1   1     π (β ) − π (α ) μ n μ n σ2 σ2 πμ (βn ) − πμ (αn ) βn βn

n∈A /



n∈A / n n0

 β 1−σ2 −2δ  enη(1−σ2 −2δ) n

, log βn n

n∈A / n n0

(2.31)

n∈A / n n0

where n0 is some large positive constant. nη(1−σ2 −2δ)

1 for all n ∈ N, it follows from (2.31) and (2.23) that Since (2.4) yields e n 

1

n∈A / n n0

 enη(1−σ2 −2δ) < ∞. n

n∈A / nn0

Therefore, all large positive integers belong to A, that is, there exists a positive constant n1 such that {n ∈ N: n  n1 } ⊂ A.

By this and the definition of A, there exists a sequence {ξn : n ∈ N} such that, for all integers n  n1 , we have 

   1 1 n− η < ξn  n + η 4 4

  and ρ(ξn )  exp(−σ2 ξn ),

which implies lim

n→∞

ξn =η n

and

lim sup n→∞

log |ρ(ξn )|  −σ2 . ξn

(2.32)

If m > n  n1 , then  ξm − ξn 

   1 1 η η m− η− n+ η = (m − n)η −  (m − n). 4 4 2 2

(2.33)

Lemma 4, together with (2.19), (2.20), (2.32), and (2.33), shows that lim sup r→∞

log |ρ(r)|  −σ2 . r

However, according to Lemma 3, if ρ(z) does not vanish identically, then lim sup r→∞

log |ρ(r)| > −σ2 . r

Therefore, ρ(z) ≡ 0. Lith. Math. J., 50(3):293–311, 2010.

(2.34)

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H. Nagoshi and J. Steuding

Differentiating this identity r-times with respect to z and then putting z = 0, we obtain  sr dμ(s) = 0 for every r = 0, 1, 2, . . . . C

Thus, assumption (a) of Lemma 2 is also satisfied, and Lemma 2 implies the denseness of the set of all convergent series (2.15) and, hence, that of (2.14). Let f ∈ H(D1 ), K1 be a compact subset of D1 , and 1 > 0. In view of (2.12), (2.4), and (2.11), we choose the number N so large that ∞  |b(pk )| p>N k=2

pkσ1

<

1 . 4

(2.35)

From the denseness of the set of all convergent series (2.14) in H(D1 ) it follows that there exists a sequence {˜ c(p): c˜(p) ∈ γ} such that       1  maxf (s) − g˜p (s) − c˜(p)˜ gp (s) < . (2.36) s∈K1 2 pN

p>N

Set c(p) :=

1 c˜(p)

if p  N , if p > N .

Then by (2.7), (2.8), (2.36), and (2.35) we have        maxf (s) − gp s, c(p)  s∈K1

p

        = maxf (s) − g˜p (s) − gp s, c(p)  s∈K1 pN p>N                 maxf (s) − g˜p (s) − c˜(p)˜ gp (s) + max c˜(p)˜ gp (s) − gp s, c(p)  s∈K1

<

 1 +2 2

pN ∞ 

p>N k=2

s∈K1

p>N

|b(pk )| pkσ1

p>N

p>N

< 1 .

This completes the proof of the proposition.   3

A LIMIT PROBABILITY MEASURE AND ITS SUPPORT

As before, we denote by γ the compact unit circle {s ∈ C: |s| = 1} and put  Ω := γp , p

where γp = γ for each prime p. With product topology and pointwise multiplicative, Ω is a compact topological Abelian group. Thus, the probability Haar measure m on (Ω, B(Ω)) exists. For each prime p, let

Universality for L-functions in the Selberg class

305

ω(p) denote the projection of ω ∈ Ω to the coordinate space γp . The measure m on Ω is the product of the probability Haar measures mp on the coordinate spaces γp . Thus, {ω(p): p prime} is a sequence of independent complex-valued random variables defined on the probability space (Ω, B(Ω), m). For ω ∈ Ω , we set ω(1) := 1 and ω(n) :=



ω(p)ν(n;p) ,

(3.1)

p

where ν(n; p) is the exponent of the prime p in the prime factorization of n. Let σ1 be as in (2.2), and D1 be as in (2.3). For ω ∈ Ω , we define L(s, ω) :=

∞  a(n)ω(n)

ns

n=1

.

(3.2)

By (1.1), for any ω ∈ Ω , this series converges absolutely on the half-plane σ > 1. Note that, by the Euler product hypothesis, the numbers a(n) are multiplicative. Thus, this and (3.1) imply that, for any ω ∈ Ω , we have   ∞   a(pk )ω(p)k (3.3) L(s, ω) = 1+ pks p k=1

on the half-plane σ > 1. Lemma 5. As in (2.7), for a prime p and ω ∈ Ω , let ∞    b(pk )ω(p)k gp s, ω(p) := . pks k=1

Then, for almost all ω ∈ Ω , the series plane σ > σ1 . Proof.



p gp (s, ω(p))

converges and is holomorphic (at least) on the half-

As in (2.13), for any ω ∈ Ω , the sum  ∞  ∞   b(pk )ω(p)k    |b(pk )|  =   pkσ1 pkσ1 p k=2

p k=2

converges. Hence, by properties of Dirichlet series, for any ω ∈ Ω , the series ∞  b(pk )ω(p)k p k=2

pks

(3.4)

converges absolutely and is holomorphic on the half-plane σ > σ1 .  Next, we consider the series p b(p)ω(p) . We note that { b(p)ω(p) ps pσ1 : p prime} is a sequence of independent random variables on Ω . For a random variable X on Ω , let E[X] denote the expected value of X . We have   b(p)ω(p) = 0. E pσ 1 Lith. Math. J., 50(3):293–311, 2010.

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H. Nagoshi and J. Steuding

By Lemma 1(i) and (2.5) we have    2  b(p)ω(p) 2 1  = |a(p)|  E  ,  σ 2σ p 1 p 1 p2σ1 −2δ which, together with (2.4), implies that the sum    b(p)ω(p) 2    E   σ1 p p 

b(p)ω(p) conp σ1  b(p)ω(p) series p ps

converges. Thus, from [1, Theorem 1.2.11] it follows that, for almost all ω ∈ Ω , the sum

p

verges. Therefore, by the well-known properties of Dirichlet series, for these ω ∈ Ω , the converges and is holomorphic on the half-plane σ > σ1 . This and the above fact about the series (3.4) complete the proof.   From (1.1) and (1.3) it follows that, for any ω ∈ Ω , the series 1+

∞  a(pk )ω(p)k

pks

k=1

k=1

converge absolutely on the half-plane σ > σ1 . Lemma 6. For any ω ∈ Ω , we have the identity 1+

∞  a(pk )ω(p)k

pks

k=1

∞  b(pk )ω(p)k

and

 = exp

∞  b(pk )ω(p)k k=1

(3.5)

pks



pks

on the half-plane σ > σ1 . Proof. In view of the Taylor expansion exp(z) = 1 + z + z2! + · · · , for each prime p, we define the numbers dp (k) = dp (k, b, ω), k ∈ N, by  ∞  ∞  b(pk )ω(p)k  dp (k) exp = 1 + = Lp (s, ω), say. pks pks 2

k=1

k=1

To obtain the lemma, it suffices to show that, for every k ∈ N, the equality   dp (k) = a pk ω(p)k

(3.6)

holds. We have  −Lp (s, ω)

=− 1+

∞  dp (k) k=1



pks

= log p

∞  kdp (k) k=1

pks

.

(3.7)

Also, we have  −Lp (s, ω) = − exp



∞  b(pk )ω(p)k k=1

pks



 = − exp

∞  b(pk )ω(p)k k=1

pks



∞  b(pk )ω(p)k k=1

pks



Universality for L-functions in the Selberg class

 =− 1+

∞  dp (k) k=1

= log p

 − log p

pks

∞  kb(pk )ω(p)k k=1

pks

∞  kb(pk )ω(p)k k=1 ∞ 

+ log p

k=2

307



pks 1 pks



  dp (m)b p ω(p) .

(3.8)

k=m+ m1, 1

Comparing the coefficients of the series (3.7) and (3.8), we obtain dp (1) = b(p)ω(p),  2 1 dp (2) = b p ω(p)2 + dp (1)b(p)ω(p), 2

(3.9) (3.10)

and, generally, for k  2, k−1  k   1 k dp (k) = b p ω(p) + dp (k − )b p ω(p) . k

(3.11)

=1

It is well known (see [4, p. 112]) that a(p) = b(p),  2   1 b p = a p2 − a(p)b(p), 2

(3.12) (3.13)

and, generally, for k  2, k−1  k  k 1      b p =a p − a pk− b p . k

(3.14)

=1

We shall now deduce (3.6) by induction. By (3.9), (3.10), (3.12), and (3.13), we have dp (1) = a(p)ω(p)

  and dp (2) = a p2 ω(p)2 .

Let k  3 and assume that, for all r = 1, 2, . . . , k − 1, we have   dp (r) = a pr ω(p)r .

(3.15)

Then (3.11), (3.14), and assumption (3.15) yield that k−1 k−1     1   k−    1 a p b p ω(p)k + dp (k − )b p ω(p) dp (k) = a pk ω(p)k − k k

  1 = a pk ω(p)k − k  k = a p ω(p)k .

=1 k−1  =1

    1 a pk− b p ω(p)k + k

Thus, (3.6) is obtained, and the lemma is proved.   Lith. Math. J., 50(3):293–311, 2010.

=1 k−1  =1

    a pk− ω(p)k− b p ω(p)

308

H. Nagoshi and J. Steuding

As is shown similarly to the proof of [4, Lemma 4.1], for almost all ω ∈ Ω , the series L(s, ω) in (3.2) converges and is holomorphic on the half-plane σ > σ1 . We have  ∞    ∞   b(pk )ω(p)k  b(pk )ω(p)k exp = exp . pks pks px

px k=1

k=1

Hence, by Lemma 5, for almost all ω ∈ Ω , we have  ∞    ∞    b(pk )ω(p)k b(pk )ω(p)k exp = exp pks pks p p k=1

(3.16)

k=1

on the half-plane σ > σ1 , and this product is holomorphic on this half-plane. Therefore, recalling that identity (3.3) holds for σ > 1, we deduce from Lemma 6 and analytic continuation that, for almost all ω , the identity   ∞  b(pk )ω(p)k (3.17) L(s, ω) = exp pks p k=1

holds on the half-plane σ > σ1 . Proposition 2. For σ > σ0 + 12 , let the function f (s) be given by an absolutely convergent Dirichlet series ∞  an n=1

ns

 such that nx |an |2  x2σ0 . Suppose that f (s) is a meromorphic function in the half-plane σ > σ0 , all poles of f (s) in this region are included in a compact set, and f (σ + it)  |t|δ

for all σ  σ0 with some δ > 0. Moreover, suppose that T

  f (σ + it)2 dt  T

0

for all σ > σ0 . Then the probability measure   1 meas τ ∈ [0, T ]: f (s + iτ ) ∈ A , T

  A ∈ B H(D0 ) ,

where D0 := {s ∈ C: σ > σ0 }, converges weakly to the distribution of the random element ∞  an ω(n) n=1

as T → ∞.

ns

,

ω ∈ Ω, s ∈ D0 ,

Universality for L-functions in the Selberg class

309

Proof. This statement is rather similar to a limit theorem of Laurinˇcikas [2] and can be proved along the lines  of the proof of this result.  Let N > 0. Define the strip   D1,N := s ∈ D1 : |t| < N .

(3.18)

Let Q denote the distribution of the H(D1,N )-valued random element L(s, ω) on Ω , i.e.,   Q(A) := m ω ∈ Ω: L(s, ω) ∈ A for A ∈ B(H(D1,N )). Corollary 1. The probability measure QT defined by QT (A) :=

  1 meas τ ∈ [0, T ]: L(s + iτ ) ∈ A , T

  A ∈ B H(D1,N ) ,

converges weakly to Q as T → ∞. Proof. We apply Proposition 2 with f (s) = L(s) and σ0 = σ1 . Recall that 12  σL < σ1 . According to (1.1), Theorem 6.8 of [4], and Corollary 6.11 of [4], all assumptions of Proposition 2 are satisfied. Therefore, Proposition 2 yields that the probability measure   1 meas τ ∈ [0, T ]: L(s + iτ ) ∈ A , T

  A ∈ B H(D0 ) ,

where D0 := {s ∈ C: σ > σ1 }, converges weakly to the distribution of the random element ∞  a(n)ω(n) n=1

ns

,

ω ∈ Ω, s ∈ D0 ,

as T → ∞. Since D1,N ⊂ D0 , this gives the assertion.   Lemma 7. Let G be a simply connected ∞ region in C. Let {Xm } be a sequence of independent H(G)-valued random elements and suppose that m=1 Xm converges almost surely. Let SXm be the support of Xm . Then ∞ the support of X is the closure of the set of all f ∈ H(G) which may be written as a convergent series m m=1  f= ∞ f with f ∈ S . m m X m m=1 Proof. This is Theorem 1.7.10 of [1].   Proposition 3. The support of the measure Q is the set   SN := f ∈ H(D1,N ): f (s) = 0 for any s ∈ D1,N , or f (s) ≡ 0 . Proof. Since {ω(p)} is a sequence of independent random variables on Ω , it follows that {gp (s, ω(p))} is a sequence of independent H(D1,N )-valued random elements. The support of each ω(p) is the unit circle γ , and hence the support of each gp (s, ω(p)) is the set 

Lith. Math. J., 50(3):293–311, 2010.

 f ∈ H(D1,N ): f (s) = gp (s, c) with c ∈ γ .

310

H. Nagoshi and J. Steuding

 Therefore, Lemmas 5 and 7 imply that the support of p gp (s, ω(p)) is the closure of the set of all convergent  series p gp (s, c(p)) with c(p) ∈ γ . According to Proposition 1, this closure is H(D1,N ). The map exp : H(D1,N ) → H(D1,N ),

f → exp(f ),

 is continuous, and this map sends H(D1,N ) onto SN \ {0} and, by (3.17), sends p gp (s, ω(p)) to L(s, ω). Therefore, the support SL of L(s, ω) contains SN \ {0}. On the other hand, the support SL is closed. By Hurwitz’s theorem (see [1, Lemma 6.5.6]) we obtain SN \ {0} = SN .

Thus, SN ⊂ SL . Recalling (see (3.5)) that, for any ω ∈ Ω , the series ∞  b(pk )ω(p)k k=1

pks

converges on D1 , we infer that the function  exp

∞  b(pk )ω(p)k k=1



pks

is nonzero on D1 . Hence, in view of (3.16) and (3.17), L(s, ω) is an almost surely convergent product of nonvanishing factors. Applying Hurwitz’s theorem again, we conclude that L(s, ω) ∈ SN almost surely. Therefore, SL ⊂ SN . Thus, the lemma is proved.   4

PROOF OF THEOREM 1

Since K is a compact subset of D1 , defined in (2.3), there exists a number N > 0 such that K ⊂ D1,N , where D1,N is defined in (3.18). First, we consider the case where g(s) has a nonvanishing analytic continuation to D1,N . By Proposition 3 the function g(s) is contained in the support SL of the random element L(s, ω). Denote by Φ the set of functions φ ∈ H(D1,N ) such that   maxφ(s) − g(s) < . s∈K

Since by Corollary 1 the measure QT converges weakly to Q as T → ∞ and the set Φ is open, it follows from the properties of the weak convergence and the support that

  1 lim inf meas τ ∈ [0, T ]: maxL(s + iτ ) − g(s) <  T →∞ T s∈K = lim inf QT (Φ)  Q(Φ) > 0. (4.1) T →∞

This proves the theorem in the present case. Now we consider the general case where g(s) is as in the statement of the theorem. Since g(s) is nonvanishing on K, by Mergelyan’s approximation theorem there exists a polynomial G(s) such that G(s) = 0 on K and    maxg(s) − G(s) < . (4.2) s∈K 4

Universality for L-functions in the Selberg class

311

Since the polynomial G(s) has only finitely many zeros, there exists a region G with connected complement such that K ⊂ G and G(s) = 0 on G . Hence, there exists a continuous branch log G(s) on G , and log G(s) is analytic in the interior of G . Thus, using Mergelyan’s approximation theorem again, we find a polynomial F (s) such that         maxG(s) − exp F (s)  = maxG(s)1 − exp F (s) − log G(s)  < . s∈K s∈K 4 This and (4.2) give     maxg(s) − exp F (s)  < . s∈K 2

(4.3)

From (4.1) we deduce

    1   lim inf meas τ ∈ [0, T ]: max L(s + iτ ) − exp F (s) < > 0. T →∞ T s∈K 2 In combination with (4.3), this proves the theorem. Acknowledgments. The authors are very grateful to Professor A. Laurinˇcikas for useful discussion. REFERENCES 1. A. Laurinˇcikas, Limit Theorems for the Riemann Zeta-Function, Kluwer, 1996. 2. A. Laurinˇcikas, A limit theorem in the theory of finite Abelian groups, Publ. Math. Debrecen, 52:517–533, 1998. 3. M.R. Murty, Problems in Analytic Number Theory, Grad. Texts Math., Vol. 206, Springer-Verlag, 2001. 4. J. Steuding, Value-Distribution of L-Functions, Lect. Notes Math., Vol. 1877, Springer-Verlag, 2007. 5. S.M. Voronin, Theorem on the ‘universality’ of the Riemann zeta-function, Izv. Akad. Nauk SSSR, Ser. Mat., 39, 1975 (in Russian). English transl.: Math. USSR, Izv., 9:443–445, 1975.

Lith. Math. J., 50(3):293–311, 2010.