joint universality for l-functions from the selberg class

Ukrainian Mathematical Journal, Vol. 70, No. 5, October, 2018 (Ukrainian Original Vol. 70, No. 5, May, 2018)

JOINT UNIVERSALITY FOR L-FUNCTIONS FROM THE SELBERG CLASS AND PERIODIC HURWITZ ZETA-FUNCTIONS R. Macaitien˙e

UDC 517.5

We obtain a joint universality theorem on the approximation of a collection of analytic functions by a collection of shifts formed by L -functions from the Selberg class and periodic Hurwitz zeta-functions.

1. Introduction The universality of zeta and L-functions is one of the most interesting phenomena in the analytic number theory. Roughly speaking, this means that every analytic function can be approximated with a given accuracy by shifts of the considered zeta or L-functions uniformly on compact subsets of a certain region. The first result in this field belongs to S. Voronin who discovered [20] the universality property of the Riemann zeta-function ⇣(s), s = σ + it. For a modern form of the Voronin theorem, we use the following notation: Let D=

⇢

s 2 C:

� 1 0. T !1 T s2K This theorem shows that a given function f (s) 2 H0 (K) can be approximated by shifts ⇣(s + i⌧ ), ⌧ 2 R, from a wide set with positive lower density. Later, it was discovered that the majority of other classical zeta and L-functions are also universal in the indicated sense. A full survey on the universality of zeta and L-functions can be found in the excellent paper [12]. We focus our attention on the Selberg class [17], which is one of the most extensively studied objects in the analytic number theory. The Selberg class S contains Dirichlet series L(s) = ˇ ˇ ˇ Siauliai University and Siauliai State College, Siauliai, Lithuania.

1 X a(m) , ms

m=1

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 5, pp. 655–671, May, 2018. Original article submitted September 15, 2014; revision submitted July 21, 2017. 754

0041-5995/18/7005–0754

c 2018

Springer Science+Business Media, LLC

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satisfying the following axioms: (1◦ ) analytic continuation: there exists an integer k ≥ 0 such that (s − 1)k L(s) is an entire function of finite order; (2◦ ) Ramanujan hypothesis: a(m) ⌧ m✏ for any ✏ > 0, where the implied constant may depend on ✏; (3◦ ) functional equation: L(s) satisfies a functional equation ⇤L (s) = w⇤L (1 − s), where s

⇤L (s) = L(s)Q

f Y

Γ(λj s + µj )

j=1

with positive real numbers Q and λj and complex numbers µj , w, 1, ◆ ✓ q−1 1 X m+↵ ⇣(s, ↵; a) = s . am ⇣ s, q q m=0

This equality specifies meromorphic continuation to the entire complex plane for ⇣(s, ↵; a) with a simple pole at the point s = 1. If q−1 X

am = 0,

m=0

then the function ⇣(s, ↵; a) is entire one. The problem of joint universality of zeta-functions is more complicated. In this case, a collection of analytic functions is simultaneously approximated by a collection of shifts of zeta or L-functions. The first joint universality theorem for Dirichlet L-functions was obtained by Voronin [21, 22]. Moreover, the case of joint universality in which analytic functions are approximated by shifts of zeta-functions with and without Euler product is also possible. The first result in this direction belongs to Mishou [14]. In [6], the theorem of joint universality was proved for ⇣(s, ↵; a) and a periodic zeta-function ⇣(s; b) defined, for σ > 1, by 1 X bm , ⇣(s; b) = ms m=1

where b = {bm : m 2 N} is another periodic sequence of complex numbers with minimal period k 2 N. The function ⇣(s; b), just as ⇣(s, ↵; a), admits a meromorphic continuation to the entire complex plane. In [8], Laurinˇcikas proved a generalization of the mentioned result. To state his theorem, we need additional notation.

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Let bj = {bjm : m 2 N} be a periodic sequence of complex numbers with minimal period kj 2 N and let ⇣(s; bj ) be the corresponding periodic zeta-function, j = 1, . . . , r1 , r1 > 1. By k = [k1 , . . . , kr1 ] we denote the least common multiple of the periods k1 , . . . , kr1 . Moreover, by ⌘1 , . . . , ⌘'(k) we denote the reduced residue system modulo k and define a matrix 0 1 b2⌘1 . . . br1 ⌘1 b1⌘1 B C B b C b . . . b 2⌘2 r 1 ⌘2 C B 1⌘2 B C, B=B C ... ... ... C B ... @ A b1⌘'(k) b2⌘'(k) . . . br1 ⌘'(k) where '(k) is the Euler function. In addition, suppose that, for all primes p, 1 X |bjpβ |

β=1

pβ/2

 cj < 1,

(1.2)

j = 1, . . . , r1 .

Further, let aj = {ajm : m 2 N0 } be another periodic sequence of complex numbers with minimal period qj 2 N and let ⇣(s, ↵j ; aj ) be the corresponding periodic Hurwitz zeta-function, j = 1, . . . , r2 , 0 < ↵j  1. Then the following statement was proved in [8]. Theorem 1.2. Suppose that the sequences b1 , . . . , br1 are multiplicative, inequalities (1.2) are satisfied, the numbers ↵1 , . . . , ↵r2 are algebraically independent over Q, and rank(B) = r1 . Let ˆ 1, . . . , K ˆ r 2 K, K1 , . . . , Kr1 , K 2 f1 (s) 2 H0 (K1 ), . . . , fr1 (s) 2 H0 (Kr1 )

and

ˆ 1 ), . . . , fˆr (s) 2 H(K ˆ r ). fˆ1 (s) 2 H(K 1 2

Then, for any " > 0, ( 1 lim inf meas ⌧ 2 [0, T ] : T !1 T

� � sup sup �⇣(s + i⌧ ; bj ) − f (s)� < ",

1jr1 s2Kj

sup sup |⇣(s + i⌧, ↵j ; aj ) − fˆj (s)| < "

)

1jr2 s2K ˆj

> 0.

The aim of the present paper is to prove the joint universality theorem for the functions L(s) 2 S and ⇣(s, ↵j ; ajl ), where 0 < ↵j  1 and ajl = {amjl : m 2 N0 } is a periodic sequence of complex numbers with minimal period qjl 2 N, j = 1, . . . , r, l = 1, . . . , lj . For j = 1, . . . , r, let qj be the least common multiple of qj1 , . . . , qjlj and let 0

a1j1

a1j2

...

a1jlj

1

C B C Ba B 2j1 a2j2 . . . a2jlj C C. Aj = B C B B ... ... ... ... C A @ aqj j1 aqj j2 . . . aqj jlj

R. M ACAITIEN E˙

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Moreover, as above, let KL be a class of compact subsets of the strip DL with connected complements and let H0L (K), K 2 KL , be the class of continuous nonvanishing functions on K analytic in the interior of K. Theorem 1.3. Suppose that L 2 S, hypothesis (1.1) is satisfied, the numbers ↵1 , . . . , ↵r are algebraically independent over Q, and rank(Aj ) = lj , j = 1, . . . , r. Let K 2 KL , let f (s) 2 H0L (K), and, for any j = 1, . . . , r and l = 1, . . . , lj , let Kjl 2 K and fjl (s) 2 H(Kjl ). Then, for any " > 0, ( � � 1 lim inf meas ⌧ 2 [0, T ] : sup �L(s + i⌧ ) − f (s)� < ", T !1 T s2K

) � � � � sup sup ⇣(s + i⌧, ↵j ; ajl ) − fjl (s) < " > 0.

sup

1jr 1llj s2Kjl

Clearly, Theorem 1.3 implies the results of the previous works [4, 10, 11, 16], where the function ⇣(s), zetafunctions of normalized Hecke cusp forms, zeta-functions of new forms, and zeta-functions of new forms with Dirichlet character were considered instead of L 2 S, respectively. Thus, we can take, for r = 2, ↵1 = 2− because the numbers ↵1 = 2

p 3

2

and ↵2 = 2

p 3

4

p 3

2

,

↵2 = 2−

p 3

4

,

are algebraically independent over Q [2].

2. Limit Theorem To prove Theorem 1.3, we establish the limit theorem on weakly convergent probability measures in the space of analytic functions. Let G be a region in the complex plane. By H(G) we denote the space of analytic functions on G equipped with the topology of uniform convergence on compact sets. Let u=

r X

lj ,

v = u + 1,

j=1

and H v = H v (DL , D) = H(DL ) ⇥ H u (D). As usual, by B(X) we denote the Borel σ -field of the space X. For brevity, let ↵ = (↵1 , . . . , ↵r ),

a = (a11 , . . . , a1l1 , . . . , ar1 , . . . , arlr ),

and � � Z(s1 , s, ↵; a, L) = L(s1 ), ⇣(s, ↵1 ; a11 ), . . . , ⇣(s, ↵1 ; a1l1 ), . . . , ⇣(s, ↵r ; ar1 ), . . . , ⇣(s, ↵r ; arlr ) .

In this section, we consider the weak convergence for df

PT (A) = as T ! 1.

� 1 meas ⌧ 2 [0, T ] : Z(s1 + i⌧, s + i⌧, ↵; a, L) 2 A , T

A 2 B(H v ),

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To state the limit theorem for PT , we need a certain topological structure. Let γ = {s 2 C : |s| = 1}. We define Y Y ˆ= γp and ⌦= γm , ⌦ p

m2N0

ˆ and ⌦ where γp = γ for all primes p and γm = γ for all m 2 N0 . By the Tikhonov theorem, the tori ⌦ ˆ ˆ with �the product � topologies and pointwise multiplication are compact topological groups. Thus, on (⌦, B(⌦)) and ⌦, B(⌦) , the respective probability Haar measures m ˆ H and mH exist and we have the probability spaces � � ˆ ˆ (⌦, B(⌦), m ˆ H ) and ⌦, B(⌦), mH . Moreover, for j = 1, . . . , r, let ⌦j = ⌦ and ˆ ⇥ ⌦ 1 ⇥ . . . ⇥ ⌦r . ⌦=⌦

Then ⌦ is also a compact topological Abelian group, and this gives one more probability space (⌦,B(⌦),mH ), ˆ onto γp , p 2 P ˆ (p) the projection of ! ˆ2⌦ where mH is the probability Haar measure on (⌦, B(⌦)). Denote by ! (P is the set of all prime numbers), and by !j (m) the projection of !j 2 ⌦j onto γm , m 2 N0 . By using this notation, in the probability space (⌦, B(⌦), mH ), we define an H v -valued random element Z(s1 , s, !, ↵; a, L), ! , !1 , . . . , !r ) 2 ⌦, by the formula ! = (ˆ � ˆ ), ⇣(s, ↵1 , !1 ; a11 ), . . . , ⇣(s, ↵1 , !1 ; a1l1 ), Z(s1 , s, !, ↵; a, L) = L(s1 , !

� . . . , ⇣(s, !r , ↵r ; ar1 ), . . . , ⇣(s, !r , ↵r ; arlr ) ,

where, for s1 2 DL , we get

1 X a(m)ˆ ! (m) ˆ) = L(s1 , ! ms1 m=1

with

! ˆ (m) =

Y

! ˆ l (p),

pl |m

m 2 N,

pl+1 -m

and, for s 2 D, ⇣(s, ↵j , !j ; ajl ) =

1 X amjl !j (m) , (m + ↵j )s

j = 1, . . . , r,

l = 1, . . . , lj .

m=0

ˆ L(s1 , ! Note that, for almost all ! ˆ 2 ⌦, ˆ ) has the Euler product [15], i.e., L(s1 , ! ˆ ) = exp

(

1 XX b(pk )ˆ ! k (p) p k=1

pks1

)

.

Denote by PZ the distribution of the random element Z(s1 , s, !, ↵; a, L), i.e., � � PZ (A) = mH ! 2 ⌦ : Z(s1 , s, !, ↵; a, L) 2 A ,

A 2 B(H v ).

R. M ACAITIEN E˙

760

Theorem 2.1. Suppose that L 2 S, hypothesis (1.1) is satisfied, and the numbers ↵1 , . . . , ↵r are algebraically independent over Q. Then PT weakly converges to PZ as T ! 1. The procedure used in the proof is sufficiently well known; see, e.g., similar theorems from [4, 10, 11]. Therefore, we present only the principal moments of the proof. Lemma 2.1. Suppose that the numbers ↵1 , . . . , ↵r are algebraically independent over Q. Then df

QT (A) =

n � 1 meas ⌧ 2 [0, T ] : (p−i⌧ : p 2 P), T

o � � � � (m + ↵1 )−i⌧ : m 2 N0 , . . . , (m + ↵r )−i⌧ : m 2 N0 ) 2 A ,

A 2 B(⌦),

weakly converges to the Haar measure mH as T ! 1.

The proof of the lemma is given in [8] (Theorem 3). However, for the convenience of the reader, we reproduce the proof. It is well known that the dual group of ⌦ is isomorphic to the group 0

G=@

M

p2P

1

Zp A

r M j=1

0 @

M

m2N0

1

Zjm A,

where Zp = Z for all p 2 P and Zjm = Z for all m 2 N0 and j = 1, . . . , r. An element � � (k, l1 , . . . , lr ) = (kp : p 2 P), (l1m : m 2 N0 ), . . . , (lrm : m 2 N0 )

of the group G, where only finitely many integers kp and l1m , . . . , lrm differ from zero, acts upon ⌦ by the formula (ˆ ! , !1 , . . . , !r ) ! (ˆ ! k , !1l1 , . . . , !rlr ) =

Y

! ˆ kp (p)

r Y Y

l

!jjm (m).

j=1 m2N0

p2P

Therefore, the right-hand side of the last equality specifies the characters of ⌦. Hence, the Fourier transform gT (k, l1 , . . . , lr ) of QT takes the form Z

gT (k, l1 , . . . , lr ) =

⌦

0 @

Y

! ˆ kp (p)

r Y Y

j=1 m2N0

p2P

1

l !jjm (m)AdQT .

Thus, by the definition of QT , 1 gT (k, l1 , . . . , lr ) = T

ZT 0

1 = T

ZT 0

0 @

Y

p−i⌧ kp

j=1 m2N0

p2P

exp

8 < :

r Y Y

0

−i⌧ @

X

p2P

1

(m + ↵j )−i⌧ ljm Ad⌧

kp log p +

r X X

j=1 m2N0

19 = ljm log(m + ↵j )A d⌧, ;

(2.1)

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where, as above, only finitely many integers kp and l1m , . . . , lrm differ from zero. Clearly, gT (0, 0, . . . , 0) = 1.

(2.2)

We now suppose that (k, l1 , . . . , lr ) 6= (0, 0, . . . , 0). Since the numbers ↵1 , . . . , ↵r are algebraically independent over Q, the set �

(log(m + ↵1 ) : m 2 N0 ), . . . , (log(m + ↵r ) : m 2 N0 )

is linearly independent over Q. Indeed, suppose that there exist l11 , . . . , l1k1 , . . . , lr1 , . . . , lrkr 2 Z\{0} and m11 , . . . , mrk1 , . . . mr1 , . . . , mrkr such that l11 log(m11 + ↵1 ) + . . . + l1k1 log(m1k1 + ↵1 ) + . . . + lr1 log(mr1 + ↵r ) + . . . + lrkr log(mrkr + ↵r ) = 0. Then (m11 + ↵1 )l11 . . . (m1k1 + ↵1 )l1k1 . . . (mr1 + ↵r )lr1 . . . (mrkr + ↵r )lrkr = 1. This implies that there exists a polynomial p(x1 , . . . , xr ) with integer coefficients such that p(↵1 , . . . , ↵r ) = 0. However, this contradicts the algebraic independence of the numbers ↵1 , . . . , ↵r . It is well known that the set {log p : p 2 P} is linearly independent over Q. Thus, under the assumption that the set � (log p : p 2 P), (log(m + ↵1 ) : m 2 N0 ), . . . , (log(m + ↵r ) : m 2 N0 )

(2.3)

is linearly dependent over Q, by analogy with the set

� (log(m + ↵1 ) : m 2 N0 ), . . . , (log(m + ↵r ) : m 2 N0 ) ,

we arrive at a contradiction with the algebraic independence of ↵1 , . . . , ↵r . Since the sums in (2.1) are finite, it follows from the linear independence of (2.3) that, in the case (k, l1 , . . . , lr ) 6= (0, 0, . . . , 0), we have X

p2P

kp log p +

r X X

j=1 m2N0

ljm log(m + ↵j ) 6= 0.

After integration, it follows from (2.1) and (2.2) that 8 > 1 for (k, l1 , . . . , lr ) = (0, 0, . . . , 0), > > > > > > ⌘o n ⇣X Xr X > > > kp log p + ljm log(m + ↵j ) < 1 − exp −iT p2P m2N0 j=1 ⇣X ⌘ gT (k, l1 , . . . , lr ) = Xr X > > iT kp log p + ljm log(m + ↵j ) > > p2P m2N0 j=1 > > > > > > : for (k, l1 , . . . , lr ) 6= (0, 0, . . . , 0).

R. M ACAITIEN E˙

762

Hence,

lim gT (k, l1 , . . . , lr ) =

T !1

8 max , 1 − , see [19], and the 2 dL 1 series for ⇣n (s, ↵j ; ajl ) is absolutely convergent for σ > . In addition, we define 2 ˆ) = Ln (s, !

✓

1 X a(m)ˆ ! (m)un (m) ms

(2.5)

m=1

and ⇣n (s, !j , ↵j ; ajl ) =

1 X amjl !j (m)un (m, ↵j ) , (m + ↵j )s

m=0

j = 1, . . . , r,

l = 1, . . . , lj .

(2.6)

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In the next step of the proof, we consider the weak convergence of the measures PT,n (A) = and

� 1 meas ⌧ 2 [0, T ] : Zn (s1 + i⌧, s + i⌧, ↵; a, L) 2 A , T

A 2 B(H v ),

� 1 PˆT,n (A) = meas ⌧ 2 [0, T ] : Zn (s1 + i⌧, s + i⌧, !, ↵; a, L) 2 A , T

A 2 B(H v ),

as T ! 1, where

� Zn (s1 , s, ↵; a, L) = Ln (s1 ), ⇣n (s, ↵1 ; a11 ), . . . , ⇣n (s, ↵1 ; a1l1 ),

. . . , ⇣n (s, ↵r ; ar1 ), . . . , ⇣n (s, ↵r ; arlr )

and

�

� ˆ ), ⇣n (s, !1 , ↵1 ; a11 ), . . . , ⇣n (s, !1 , ↵1 ; a1l1 ), Zn (s1 , s, !, ↵; a, L) = Ln (s1 , !

� . . . , ⇣n (s, !r , ↵r ; ar1 ), . . . , ⇣n (s, !r , ↵r ; arlr ) .

Lemma 2.2. Suppose that the numbers ↵1 , . . . , ↵r are algebraically independent over Q. Then, � v for any � ˆ fixed ! 2 ⌦, both measures PT,n and PT,n weakly converge to the same probability measure Pn on H , B(H v ) as T ! 1. Proof. We apply a standard method based on the preservation of weak convergence under continuous mappings and Theorem 5.1 in [1]. We define a function hn : ⌦ ! H v by the formula hn (!) = Zn (s1 , s, !, ↵; a, L). The absolute convergence of series (2.5) and (2.6) shows that the function hn is continuous. Moreover, hn

��

� � � � �� p−i⌧ : p 2 P , (m + ↵1 )−i⌧ : m 2 N0 , . . . , (m + ↵r )−i⌧ : m 2 N0

= Zn (s1 + i⌧, s + i⌧, ↵; a, L).

Hence, PT,n = QT h−1 n . This fact, the continuity of hn , Lemma 2.1, and Theorem 5.1 in [1] imply the weak convergence of the measure PT,n to mH h−1 n as T ! 1. Further, for fixed ! 0 2 ⌦, let h(!) = ! ! 0 for ! 2 ⌦. Then, clearly,

� � � � � ��� hn h (p−i⌧ : p 2 P), (m + ↵1 )−i⌧ : m 2 N0 , . . . , (m + ↵r )−i⌧ : m 2 N0

= Zn (s1 + i⌧, s + i⌧, ! 0 , ↵; a, L).

Therefore, repeating the arguments presented above and using the invariance of the Haar measure mH , we find ˆ that the measure PˆT,n also weakly converges to mH h−1 n as T ! 1. Thus, the measures PT,n and PT,n are both −1 weakly convergent to the measure Pn = mH hn as T ! 1.

R. M ACAITIEN E˙

764

To replace Zn with Z, we need certain approximation results. Denote by ⇢L , ⇢, and ⇢v the metrics on H(DL ), H(D), and H v , respectively, inducing the topology of uniform convergence on compact sets. We note that, for g = (g, g11 , . . . , g1l1 , . . . , gr1 , . . . , grlr ) and f = (f, f11 , . . . , f1l1 , . . . , fr1 , . . . , frlr ) 2 H v , ✓ ◆ ⇢v (g, f ) = max ⇢L (g, f ), max max ⇢(gjl , fjl ) . 1jr 1llj

Lemma 2.3. The following relation is true: 1 lim lim sup n!1 T !1 T

ZT 0

� � ⇢v Z(s1 + i⌧, s + i⌧, ↵; a, L), Zn (s1 + i⌧, s + i⌧, ↵; a, L) d⌧ = 0.

Suppose that the numbers ↵1 , . . . , ↵r are algebraically independent over Q. Then, for almost all ! 2 ⌦, 1 lim lim sup n!1 T !1 T

ZT 0

� � ⇢v Z(s1 + i⌧, s + i⌧, !, ↵; a, L), Zn (s1 + i⌧, s + i⌧, !, ↵; a, L) d⌧ = 0.

Proof. By the definition of the metric ⇢v , 1 T

ZT 0

� � ⇢v Z(s1 + i⌧, s + i⌧, ↵; a, L), Zn (s1 + i⌧, s + i⌧, ↵; a, L) d⌧ 1  T

ZT 0

� � ⇢L L(s1 + i⌧ ), Ln (s1 + i⌧ ) d⌧

lj ZT r X X � � 1 + ⇢ ⇣(s + i⌧, ↵j ; ajl ), ⇣n (s + i⌧, ↵j ; ajl ) d⌧. T j=1 l=1

(2.7)

0

Moreover, in view of Lemma 4.8 from [19], 1 lim lim sup n!1 T !1 T

ZT 0

� � ⇢L L(s1 + i⌧ ), Ln (s1 + i⌧ ) d⌧ = 0,

(2.8)

and, by the proof of Lemma 2.4 from [11], we get 1 lim lim sup n!1 T !1 T

ZT 0

� � ⇢ ⇣(s + i⌧, ↵j ; ajl ), ⇣n (s + i⌧, ↵j ; ajl ) d⌧ = 0

(2.9)

for j = 1, . . . , r, l = 1, . . . , lj . Therefore, the first assertion of the lemma follows from (2.7)–(2.9). Similarly,

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in view of Lemma 4.10 from [19], for almost all ! ˆ with respect to the measure m ˆ H , we find 1 lim lim sup n!1 T !1 T

ZT 0

� � ⇢L L(s1 + i⌧, ! ˆ ), Ln (s1 + i⌧, ! ˆ ) d⌧ = 0.

(2.10)

Let ⇢u (g, f ) = max max ⇢(gjl , fjl ) 1jr 1llj

� � and let m e H denote the Haar measure on ⌦1 ⇥ . . . ⇥ ⌦r , B(⌦1 ⇥ . . . ⇥ ⌦r ) . Thus, it follows from relation (2.6) e H, in [4] that, for almost all (!1 , . . . , !r ) 2 ⌦1 ⇥ . . . ⇥ ⌦r with respect to m 1 lim lim sup n!1 T !1 T

ZT

⇢u

0

⇣�

� ⇣(s + i⌧, !1 , ↵1 ; a11 ), . . . , ⇣(s + i⌧, !r , ↵r ; arlr ) ,

�

⇣n (s + i⌧, !1 , ↵1 ; a11 ), . . . , ⇣n (s + i⌧, !r , ↵r ; arlr )

�⌘

d⌧ = 0.

(2.11)

ˆ H and m e H . Therefore, the second assertion of the lemma The measure mH is the product of the measures m follows from (2.10), (2.11), and the analog of equality (2.7).

Lemma 2.4. Suppose that L 2 S, hypothesis (1.1) is satisfied and the numbers ↵1 , . . . , ↵r are algebraically independent over Q. Then the probability measures PT and � 1 PˆT (A) = meas ⌧ 2 [0, T ] : Z(s1 + i⌧, s + i⌧, !, ↵; a, L) 2 A , T

A 2 B(H v ),

� � are both weakly convergent, for almost all ! 2 ⌦, to the same probability measure P on H v, B(H v ) as T ! 1. 1 Proof. The properties of the class S imply that, for σ > , 2 1 lim T !1 T

ZT 0

1 1 X X � � |a(m)|u2n (m) |a(m)|2 �Ln (σ + it)�2 dt =  < 1. m2σ m2σ m=1

m=1

This fact and the Cauchy integral formula lead, for a compact set K of DL , to the estimate 1 lim sup T T !1

ZT 0

� � sup �Ln (s1 + i⌧ )�d⌧  CK

s2K

with some CK and σK

◆ 1 1 . > max , 1 − 2 dL ✓

1 X |a(m)|2 m2σK

m=1

!1/2

(2.12)

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766

By (2.5) in [11], for a compact subset K of D, we get 1 lim sup T T !1

ZT 0

� � sup �⇣n (s + i⌧, ↵j ; ajl )�d⌧  BK

s2K

1 X

m=0

|amjl |2 (m + ↵)2ˆσK

!1/2

(2.13)

1 for all j = 1, . . . , r, l = 1, . . . , lj . 2 � � e A, P uniformly distributed on [0, 1]. In this Let ✓ be a random variable on a certain probability space ⌦, probability space, we define an H v -valued random element X T,n by the formula ˆK > with some BK > 0 and σ

X T,n = X T,n (s1 , s) � � = XT,n (s1 ), XT,n,1,1 (s), . . . , XT,n,1,l1 (s), . . . , XT,n,r,1 (s), . . . , XT,n,r,lr (s) = Zn (s1 + i✓T, s + i✓T, ↵; a, L) .

Thus, by Lemma 2.2, we conclude that D

(2.14)

X T,n −!X n , D

where −! denotes the convergence in distribution and � � X n = X n (s1 , s) = Xn (s1 ), Xn,1,1 (s), . . . , Xn,1,l1 (s), . . . , Xn,r,1 (s), . . . , Xn,r,lr (s)

is an H v -valued random element with the distribution Pn (Pn is the limit measure in Lemma 2.2). By using (2.12)–(2.14), we prove, in a standard way (see, e.g., [11, 19]), that the family of probability measures {Pn : n 2 N} is tight, i.e., that, for every ✏ > 0, there exists a compact set K = K(✏) ⇢ H v ˆ m and Km be compact sets from the definition of the such that Pn (K) > 1 − ✏ for all n 2 N. Indeed, let K metric ⇢v (the definition of the metric ⇢ can be found in [8]). Also let ˆm = C ˆ R Km

1 X |a(m)|2 )

m=1

k 2σKˆ m

!1/2

and Rjlm = BKm

1 X

m=0

|amjl |2 (m + ↵j )2σKm

!1/2

.

Assume that ✏ > 0 is an arbitrary number and ˆm = R ˆ m 2m+1 ✏−1 , M

Mjlm = Rjlm 2u+m+1 ✏−1 ,

m 2 N.

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Then, by virtue of (2.12) and (2.13), � � ˆm sup �XT,n (s1 )� > M

lim sup P T !1

ˆm s1 2K

 lim sup P T !1

!

✓ ◆! � � 9j, l : sup �XT,n,j,l (s)� > Mjlm

or

s2Km

� � ˆm sup �XT,n (s1 )� > M

ˆm s1 2K

✓

!

� � + lim sup P 9j, l : sup �XT,n,j,l (s)� > Mjlm T !1

s2Km

( 1  lim sup meas ⌧ 2 [0, T ] : T !1 T

◆

� � ˆm sup �L(s1 + i⌧ )� > M

ˆm s1 2K

)

⇢ � lj r X X � � 1 + lim sup meas ⌧ 2 [0, T ] : sup �⇣(s + i⌧, ↵j ; ajl )� > Mjlm T T !1 s2Km j=1 l=1

1  lim sup ˆm T !1 T M

+

lj r X X j=1 l=1

ZT 0

sup |L(s1 + i⌧ )|d⌧

ˆm s1 2K

1 lim sup T !1 T Mjlm

ZT 0

� � ✏ sup �⇣(s + i⌧, ↵j ; ajl )�d⌧  m . 2 s2Km

Hence, by (2.14), P

ˆm sup |Xn (s1 )| > M

ˆm s1 2K

!

or

✓

j = 1, . . . , r, l = 1, . . . , lj . We define a set n K✏v = f 2 H v :

� � 9j, l : sup �Xn,j,l (s)� > Mjlm s2Km

◆!



✏ , 2m

(2.15)

o � � � � ˆ m , sup �fjl (s)�  Mjlm , j = 1, . . . , r, l = 1, . . . , lj , m 2 N . sup �f (s1 )�  M

ˆm s1 2K

s2Km

Then K✏v is a compact set in H v . Moreover, by (2.15),

� � Pn K✏v ≥ 1 − ✏

⇥ for all n 2 N, i.e., {Pn : n 2 N} is tight. Hence, by the Prokhorov theorem [1] (Theorem 6.1). If the family of probability measures is tight, then it is relatively compact), this�family is relatively compact. Therefore, there exist � a sequence X nk and a probability measure P on H v , B(H v ) such that D

X nk −! P. k!1

(2.16)

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768

e A, P), we define one more H v -valued random element X T by the formula On (⌦, � ⇤ X T = X T (s1 , s) = Z s1 + i✓T, s + i✓T, ↵; a, L) .

Thus, in view of Lemma 2.3, we conclude that, for every ✏ > 0,

� � lim lim sup P ⇢v (X T , X T,n ) ≥ ✏ = 0.

n!1 T !1

This fact, (2.14), and (2.16) show that all hypotheses of Theorem 4.2 of [1] are satisfied. Therefore, by this theorem, we get D

X T −! P, T !1

which is equivalent to the weak convergence of PT to P as T ! 1. This implies that the measure P is independent of the sequence Xnk . Thus, D

X n −! P. n!1

(2.17)

Similarly, by using the H v -valued random elements � � Zn s1 + i✓T, s + i✓T, !, ↵; a, L

and

� � Z s1 + i✓T, s + i✓T, !, ↵; a, L ,

and relation (2.17), we conclude that the measure PˆT also weakly converges to P as T ! 1. Proof of Theorem 2.1. By virtue of �Lemma 2.4, it �suffices to show that P = PZ . Let A be a continuity set of the measure P. In the probability space ⌦, B(⌦), mH , we define a random variable ⇠ by the formula ⇠(!) =

Lemma 2.4 implies the relation

8 0 is true. Let � SL = g 2 H(DL ) : g(s) 6= 0 or g(s) ⌘ 0 .

Theorem 3.1. Suppose that L 2 S, hypothesis (1.1) is satisfied, the numbers ↵1 , . . . , ↵r are algebraically independent over Q, and rank(Aj ) = lj , j = 1, . . . , r. Then the support of the measure PZ is the set SL ⇥ H u (D). Proof. We have H v = H(DL ) ⇥ H u (D).

(3.1)

Since the spaces H(DL ) and H u (D) are separable, by (3.1), we obtain � � � � B(H v ) = B H(DL ) ⇥ B H u (D) .

Therefore, it suffices to consider PZ (A) for A = A1 ⇥A2 with A1 2 H(DL ) and A2 2 H u (D). The measure mH

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770

is the product of the measures m ˆ H and m e H . Hence,

� � PZ (A) = mH ! 2 ⌦ : Z(s1 , s, !, ↵; a, L) 2 A

⇣ ⌘ � � = mH ! 2 ⌦ : L(s1 , ! ˆ ) 2 A1 , ⇣(s, ↵1 , !1 ; a11 ), . . . , ⇣(s, ↵r , !r ; arlr ) 2 A2 ⇣ � � ˆ : L(s1 , ! ˆ ) 2 A1 m =m ˆH ! ˆ2⌦ e H (!1 , . . . , !r ) 2 ⌦1 ⇥ . . . ⇥ ⌦r : �

⌘ � ⇣(s, ↵1 , !1 ; a11 ), . . . , ⇣(s, ↵r , !r ; arlr ) 2 A2 .

(3.2)

ˆ ) is the set SL . Note that, in [15], By Proposition 3 in [15], the support of the random element L(s1 , ! the authors considered theH(DL,N )-valued random element, where DL,N =

⇢

s 2 C : max

✓

1 1 ,1 − 2 dL

◆

�

< σ < 1, |t| < N ;

however, the proof remains valid for the entire strip DL . Thus, SL is a minimal closed subset of H(DL ) such that ⇣ ⌘ ˆ : L(s, ! ˆ2⌦ ˆ ) 2 SL = 1. m ˆH !

(3.3)

In addition, under the hypotheses of �the theorem, it was proved (in Theorem that the support of � 3.1 from [9]) u u the H (D)-valued random element ⇣(s, ↵1 , !1 ; a11 ), . . . , ⇣(s, ↵r , !r ; arlr ) is the set H (D), i.e., H u (D) is a minimal closed subset of H u (D) such that ⌘ ⇣ � � m e H (!1 , . . . , !r ) 2 ⌦1 ⇥ . . . ⇥ ⌦r : ⇣(s, ↵1 , !1 ; a11 ), . . . , ⇣(s, ↵r , !r ; arlr ) 2 H u (D) = 1.

Combining this with (3.3) and (3.2), we arrive at the assertion of the theorem. 4. Proof of Theorem 1.3

Theorem 1.3 is a consequence of Theorems 2.1 and 3.1, and Mergelyan’s theorem on the approximation of analytic functions by polynomials [13]; see also [23]. Proof of Theorem 1.3. By the Mergelyan theorem, there exist polynomials p(s) and pjl (s) such that � � ✏ sup �f (s) − p(s)� < 4 s2K

and

� ✏ � sup sup �fjl (s) − pjl (s)� < . 2 1jr 1llj s2Kjl sup

(4.1)

(4.2)

Since f (s) 6= 0 on K, we conclude that p(s) 6= 0 also on K if ✏ is sufficiently small. Therefore, there exists a continuous branch of log p(s) on K, which is analytic in the interior of K. This and the Mergelyan theorem

J OINT U NIVERSALITY FOR L-F UNCTIONS FROM THE S ELBERG C LASS AND P ERIODIC H URWITZ Z ETA -F UNCTIONS

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show that there is a polynomial q(s) such that

Therefore, inequality (4.1) implies that

We define a set G=

(

� � ✏ sup �p(s) − eq(s) � < . 4 s2K � � ✏ sup �f (s) − eq(s) � < . 2 s2K

(4.3)

) � � � � ✏ ✏ (g, g11 , . . . , grlr ) 2 H v : sup �g(s) − eq(s) � < , sup sup sup �gjl (s) − pjl (s)� < . 2 1jr 1llj s2Kjl 2 s2K

� � Thus, in view of Theorem 3.1, G is an open neighborhood of the element eq(s) , p11 (s), . . . , prlr (s) of the support of the measure PZ . Consequently, PZ (G) > 0. Therefore, by Theorem 2.1, lim inf PT (G) ≥ PZ (G) > 0 T !1

and, by the definition of G, ( � � ✏ 1 lim inf meas ⌧ 2 [0, T ] : sup �L(s + i⌧ ) − eq(s) � < , T !1 T 2 s2K

� � ✏ sup sup sup �⇣(s + i⌧, ↵j ; ajl ) − pjl (s)� < 2 1jr 1llj s2Kjl

)

> 0.

Combining this inequality with inequalities (4.3) and (4.2), we arrive at the assertion of Theorem 1.3. REFERENCES 1. P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968). 2. A. O. Gel’fond, “On the algebraic independence of transcendental numbers of certain classes,” Uspekhi Mat. Nauk, 4, No. 5, 14–48 (1949). 3. H. Heyer, Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin (1977). ˇ ci¯unas, “A mixed joint universality theorem for zeta-functions,” Math. Model. 4. J. Genys, R. Macaitien˙e, S. Raˇckauskien˙e, and D. Siauˇ Anal., 15, No. 4, 431–446 (2010). 5. J. Kaczorowski and A. Perelli, “The Selberg class: a survey,” in: Number Theory in Progress. Proc. of the Internat. Conf. in Honor of the 60th Birthday of Andrej Schinzel (Zakopane, 1997). Vol. 2: Elementary and Analytic Number Theory), Walter de Gruyter, Berlin–New York (1999), pp. 953–992. 6. R. Kaˇcinskait˙e and A. Laurinˇcikas, “The joint distribution of periodic zeta-functions,” Studia Sci. Math. Hungar., 48, Issue 2, 257–279 (2011). 7. A. Laurinˇcikas, Limit Theorems for the Riemann Zeta-Function, Kluwer Academic Publishers, Dordrecht, etc. (1996). 8. A. Laurinˇcikas, “Joint universality of zeta-functions with periodic coefficients,” Izv. Ross. Akad. Nauk, Ser. Mat., 74, No. 3, 79–102 (2010). 9. A. Laurinˇcikas and S. Skerstonait˙e, “Joint universality for periodic Hurwitz zeta-functions. II,” in: New Directions in ValueDistribution Theory of Zeta and L-functions (W¨urzburg, 2008), Shaker-Verlag, Aachen (2009), pp. 161–169. ˇ ci¯unas, “A mixed joint universality theorem for zeta-functions. III,” in: Anal. Prob. Methods Number 10. A. Laurinˇcikas and D. Siauˇ Theory. J. Kubilius Memorial Volume, TEV, Vilnius (2012), pp. 185–195.

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