On joint universality of dirichler L-functions Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК Посвящается 65-ой годовщине со дня рождения профессора Сергея Михайловича Воронина Том 12 Выпуск 1 (2011)

UDK 519.14

ON JOINT UNIVERSALITY OF DIRICHLET ¿-FUNCTIONS

A. Laurincikas1 (Vilnius)

To the memory of Sergei Mikhailovich Voronin Abstract.

In the paper, we present a probabilistic proof of the Voronin theorem on joint universality of Dirichlet L-functions, and prove the universality for some composite functions.

1 Introduction

For the first time, I met Sergei Mikhailovich Voronin in 1974 at the conference "Problems of analytic number theory and its applications "held in Vilnius. At that moment, I was a doctoral student, and his lecture on the functional independence of zeta-functions made a great impression for me. In 1975, the remarkable paper

[13] on the universality of the Eiemann zeta-function appeared, I begin to study it and obtained some related results. In 1983, during the conference "Theory of transcendental numbers and applications "at Moscow Lomonosov University, Sergei Mikhailovich invited me for a dinner, and from this moment we became good friends.

S. M. Voronin was extremely talented mathematician, and his early death is a very big loss.

The present paper is related to Voronin’s work [14] on the joint universality and functional independence of Dirichlet L-functions.

Let x be a Dirichlet character modulo g, and let L(s,x) s = a + it, denote the corresponding Dirichlet L-function. In [13], Voronin observed that the function L(s, x) is universal in the following sense. Let 0 < r < |, and let f (s) be a continuous

1The author is partially supported by grant No. MP-94 from the Research Council of Lithuania

non-vanishing function on the disc |s| < r which is analytic for |s| < r. Then, for every e > 0, there exists a real number t = t(e) such that

max

|s|<r

L ( s + 4 + iT>^ - f (s)

< e.

The latter result has a joint generalization. First such a generalization in non-explicit form has been given in [14]. The full statement with proof is presented in [5].

Theorem 1. Let 0 < r < \, let xi,..., Xn be pairwise non-equivalent Dirichlet characters, and let fi(s),..., fn(s) be functions which are ana lytic for |s| < r and |s| < r |s| < r

e > 0, there exisis t > 0 such that for all j = 1,... ,n,

max

|s|<r

m s+4+iT,x^- fj(s)

< e.

L

We remind a Bagchi’s version [2] of Theorem 1,

Theorem 2. Let q > 1, and Iet xi,..., xn be distinct Dirichlet characters modulo q. For j = 1,... ,n, let Kj be a simply connected compact subset of the strip D = {s E C : I < a < 1}, and I et fj (s) be a non-vanishing continuous function on Kj which is analytic in the interior of Kj. Then the set of all t E R for which

sup sup |L(s + iT, xj) — fj(s)| < e

i<j<n sEKj

e > 0

An another proof of Theorem 2 based on probabilistic arguments was given in

[1]-

A modern version of Theorem 1 is of the following form [11],

xi, . . . , xn

j = 1,..., n, let Kj be a compact subset of the strip D with connected complement,

and let fj (s) be a non-vanishing continuous function on Kj which is analytic in the

Kj e > 0

liminf — meas -v t E [0,T] : sup sup |L(s + iT, xj) — fj(s)| < e } > 0.

T T I i<j<rseKj

Here meas{A} denotes the Lebesgue measure of a measurable set A c R.

As usual, denote by H(G) the space of analytic functions on a region G c C equipped with the topology of uniform convergence on compacta. The space H(G)

is metrisable. It is well known that there exists a sequence of compact subsets {K :

l E N} of the region G such that

G = U Ki, i=i

Ki c Kl+i, l E N and, f°r every compact subset K c G, there exists Kl such that K c ^ For gi,g2 E H(G), define

p(gi„g2) = £ 2-i sup»^Klgi<s) — g'2(s)l

l=i

1 + sups€Ki |gi (s) — g2(s)|

Then it is easily seen that p is a metric which induces the topology of uniform

H(G)

For gi = (gii,... , gin), g2 = g2i,... , g2n) E Hn(G), n E N, we use the metric

Pn(gi,g2) = max p(gij,g2j).

i < j< n

The aim of this paper is to describe the functions F : Hn(D) ^ H(D) such that the shifts F(L(s + iT, xi),..., L(s + iT, xn)) will be approximants of a given analytic function.

Let

S = {g E H(D) : g-i(s) E H(D) or g(s) = 0}.

Denote by U class of continuous functions F : Hn(D) ^ H(D) such that, for every open set G E H(D),

(F-iG) n Sn = 0.

xi, . . . , xn

F E U K D

f(s) K

interior of K. Then, for every e > 0,

liminf -meaJ t E [0, T] : sup |F(L(s + iT, xi),..., L(s + iT, xn)) —

s€K

f (s)| < ej > 0.

Theorem 4 is theoretical, it is difficult to check its hypotheses. We will give a simpler version of Theorem 4, Let V > 0 be an arbitrary number. Define a bounded strip

DV = {s E C : - < a < 1, |t| < V}.

Moreover, let

Sy = {g E H(Dy) : g-i(s) E H(Dy) or g(s) = 0}.

Denote by Uy the class of continuous funotions F : H(Dy) ^ H(Dy) satisfying, for each polynomial p = p(s), the condition

(F-i{p}) n sn = 0.

xi, . . . , xn

K f(s) V > 0 K c Dy

and that F E Uy, Then the assertion of Theorem 4 is valid.

Theorems 4 and 5 describe the classes of universal operators on the space of analvtie functions. It is easv to present an example. Let n = 2, and, for fi,f2 E H(Dy)

F (fi ,f2) = Cifi + C2f2, Ci,C2 E C, CiC2 = 0.

F p = p( s )

there exist two polynomials qi(s) and q2(s) such that (qi, q2) E F-i{p}. Varying the constant terms of qi and q2, we may obtain that the polvnomials qi(s^d q2(s) do not vanish for s E Dy, Thus, by Theorem 5, the linear combination ciL(s,xi) + c2L(s,x2) is universal in the sense of Theorem 4, Similar assertion is also true for

ciL2 (s,xi) + c2L2(s,x2)-

Proofs of Theorems 4 and 5 are based on the probabilistic approach to universali-L

published, therefore, we will present some of its parts,

2 Limit theorems

L

on limit theorems in the sense of weak convergence of probability measures with explicitly given limit measures in the space of analytic functions. Proofs of such theorems are standard, therefore, we state a joint limit theorem without proof.

As usual, denote by B(S) the class of Borel sets of the space S, Let

ft = n

p

where yp = {s E C : |s| = 1} for each prime p. With the product topology

and operation of pointwise multiplication, by the Tikhonov theorem, the infinitedimensional torus ft is a compact topological Abelian group. Therefore, on (ft, B(ft)), the probability Haar measure can be defined, and we obtain a probability space (ft,B(ft),mH). Denote by u(p) the projection of u E ft to the coordinate space Yp, and, on the probability space (ft, B(ft),mH), define the Hn(D)-valued random element L(s, u, x) x = (xi,..., xn) by the formula

L(s,u,x) = (L(s,xi),...,L(s,xn))

where

Ljn(1 -xj^)-i. j =-.■■■.n.

p \ F /

Let Pl be the distribution of the element L(s,u,x), i- e.,

Pl(A) = mfl (u E ft : L(s,u,x) E A) , A E B(Hn(D)).

For A E B(Hn(D)), define

Pt (A) = meas {t E [0,T ] : (L(s + iT,xi),... ,L(s + iT,xn)) E A} .

Lemma 1. Pt converges weakly to Pl as T ^ ro.

Hn(Dy)

sequel, we remind a simple property of weak convergence of probability measures. Let Si and S2 be two metric spaces, and let h : Si ^ S2 be (B(Si),B(S2))-

measurable function. Then every probability measure P on (Si, B(Si)) induces the unique probability measure Ph-i on (S2, B(S2)) defined by Ph-i(A) = P(h-iA), A E B(S2).

Lemma 2. Suppose that P and Pn n E N, be probability measures on (Si, B(Si)^ h : Si ^ S2 is a continuous function, and Pn converges weakly to P as n ^ ro. Then Pnh-i also converges weakly to Ph-i as n ^ ro.

The lemma is a particular case of Theorem 5,1 from [3],

For V > 0, denote by PT,y and Pl,v the restrictions to the space (H(Dy), B(H(Dy))) of the measures Pt and Pl, respectively.

Lemma 3. For every V > 0 PT,V converges weakly to Pl,v as T ^ ro.

PROOF. The lemma is a result of Lemmas 1 and 2 because the restriction of Hn(D) to Hn(Dy) is a continuous function, □

For A E b(h(D)), define

Pt,f(A) = 1meas {t E [0, T] : F (L(s + iT, xi),..., L(s + iT, xn)) E A} .

Lemma 4. Suppose that F : Hn (D) ^ H(D) is a continuous function. Then the probability mmsure Pt,f converges weakly to the distribution of the random element F (L(s, u, x)) as T ^ ro.

PROOF. By the definitions of Pt and Pt,f, PT,F = PTF-i. This, the continuity of F, and Lemmas 1 and 2 show that Pt,f converges weaklv to PlF-i as T ^ ro. However, for A E B(H(D)),

PlF-i(A) = Pl (F-iA) = (u E ft : L(s,u,x) E F-iA)

= (u G ft: F(L(s,w, x)) G A.

Thus, PT,F converges weakly to the distribution of the random element F (L(s, u, X)) iis T —— oo. Cl

For A G B(H(Dy)) let

Pt,f,v(A) = T^meas {t G [0,T] : F (L(s + ir,xi),... ,L(s + ir,Xn)) G A} .

Denote by LV(s,u,x) the random element having the distribution Pl,v.

Lemma 5. Suppose that F : Hra(DV) — H(DV) is a continuous function. Then Pt,f,v converges weakly to the distribution of the random element F (s,u,x)) as

T — <o.

Proof of the lemma uses Lemmas 2 and 3, and completely coincides with that of Lemma 4,

3 Supports

For the proof of universality, we need to know the support of the limit measures in limit theorems in the space of analytic functions. Suppose that S is a separable metric space, and P is a probability measure on (S, B(S)), We remind that the minimal closed set SP C S, P(SP) = 1, is called a support of the measure P. The support SP consists of elements x such that, for every open neighbourhood G of x, P(G) > 0, The support of the distribution of a random element is called its support. In this section, we will prove that the support of the Hn(D)-valued random element L(s,u,x) is the set Sn, For this, the non-equivalence of the characters X1,..., Xn will be essentially used.

We start with statements of some known general results. Denote by Sg the support of a random element £.

Lemma 6. Let {Xm : m G N} be a sequence of independent Hn(D)-valued random elements such that the series

E Xm

m=1

converges almost surely. Then the support of the sum of this series is equal to the closure of the set of all g G Hn(D) that can be written as the sum of a convergent series,

The lemma is a particular case of Lemma 5 from [9].

Lemma 7. Suppose that the sequence : m e N} = {g1m,..., gnm : m G N} G Hn(D) satisfies the following conditions:

1° If ^1,..., ^n are complex-valued Borel measures on (C, B(C)) with compact supports contained in D and such, that

with |am| = 1 is dense in Hn(D).

The lemma is a particular case of Lemma 6 in [9].

Lemma 8. Let ^ be a complex-valued Borel measure on (C, B(C)) with compact support contained in the half-plane {s e C : a > ao}, and

m=1 j=1

then

/

(s) = 0

C

for all j = 1,..., n and l e N0 = N U {0};

2° For every compact subset K C D,

^ n

3° The series

is convergent in Hn(D).

Then the set of all convergent series

C

The lemma is Lemma 6,4,10 from [7],

We remind that an analytic function g(s) in an angular region | arg s| < B0,

0 < B0 < n, is called a function of exponential type if

log |fl(rei6>)| lim sup--------------< to

r

uniformly in B, |B| < B0,

Lemma 9. Suppose that g(s) is a function of exponential type, and

y log |g(x)| ^ 1

limsup-------------> -1.

Then, for all coprime I and k,

|g(log p)| = +to

p=1(mod k)

The lemma is Lemma 4,1 of [8],

We also recall the Hurwitz theorem.

Lemma 10. Suppose that {gU(s) : n e N} is a sequence of analytic functions in a region G bounded by a simple closed contour, and that

lim gra(s) = g(s)

uniformly on G, where g(s) ^ 0. Then an interior point s0 of G is a zero of g(s) if and only if there exists a sequence {sU} C G such that sU ^ s0 as n ^ to and gu(su) = 0 for n > no = no(so).

The lemma is the Hurwitz theorem, for the proof, see [12].

Lemma 11. Suppose that <^i,..., <^U are distinct homomorphisms from a group G to the multiplicative group of non-zero complex numbers. Then ^1,...,^U are linearly independent over C.

Proof of the lemma is given in [6].

Now we are ready to consider the support of the random element L.

Lemma 12. Suppose that the characters x1,..., XU are pairwise non-equivalent. Then the support of the random element L (or of the measure is the set Sn.

Proof. We have that

L(s,^,x)= in (1 + gip(s,w)) 1 ’...’ n (1 + guP(s,^)) 1 ), (1)

V p p

-i

where

For |z| < 1, define

w) — —

Xj (P)w(p

n.

z2 z3

log(1 + z) — z - — + y -

Then the functions log(1 + gjp(s, w)) are well defined for s G D j — 1,... ,n, and we can consider the support of the random element

log(1+g1P(s w)),..., - log(1+£™p(s w)). (2)

The proof that

n (1+gjp(s,w)) 1

is an H(D)-valued random element, j — 1,...,r, contains a statement that the product converges uniformly on compact subsets of the strip D for almost all w G ft. Denoting

gP(s,w) (g1p(s, w), ..., gnp(s, w)) ,

hence we have that there exists a sequence b — {bP : |bP| — 1} such that the series

Eg(s,b) (3)

converges in Hn(D), Moreover, for every compact subset K C D,

EE sup |gjp(s,

<.

p j=1

This, and the convergence of the series (3) show that the hypotheses 2° and 3° of Lemma 7 are satisfied by the sequence {g (s,b)}. It remains to verify the hypothesis 1°. _P

Let p0 be a fixed positive number. Suppose that ^1,... ,^n are complex-valued Borel measures on (C, B(C)) with compact supports contained in D such that

E

P>P0

<.

(4)

Let d be the product of the moduli of the characters x1,..., X«- Using the definition of gjP(s,b), the periodicity of the characters x1,..., X« with period d, and (4), we find that

E

P>P0 p=1(mod d)

1

j=1

1

j

2

for all / — 1,..., d with (/, d) — L For A G B(C), define

n

vi(A) — EXj(/)^j(A), 1 — l,...,d, (/,d) —1. j=1

Then v1,..., vn are again complex-valued measures with compact supports contained D

E Ipi(logp)| < ^, / — 1,...,d, (/,d) — 1, (6)

P>P0 P=i(mod d)

where

pz(z) — J e-s' dv (s), / — 1,...,d, (/, d) — 1.

C

Since the functions p^ (z) are of exponential type, in view of Lemma 8, we have that either pz(z) = 0, or

log |Pi(x)| ^ , /, n 1

lim sup------------ > — 1, / — 1,..., d, (/, d) — 1.

l — 1, . . . , d (l, d) — 1

this /,

X] |pi(logp)| — ^

P>P0 P=1(mod d)

what contradicts (6), Thus, we have that p(z) = 0 for all / — 1,..., d, (¡,d) — 1, and, by the definitions of p^ (z^d

n „

E Xj (¡We-'d^j (s) = 0, / — (/,d) — 1. (7)

j=1 C

X1, . . . , Xn

non-equivalent. This, Lemma 11 and (7) lead to the equality

J e-s'd^j(s) = 0, j — 1,..., n.

C

Differentiating this equality, we easily find that

for all m G N^d j — 1,..., n, This means that the hvpothesis 1° of Lemma 7 also holds for the sequence {g (s,b) : p > p0}. Therefore, the set of all convergent series

(8)

p>p0

with |a(p)| — 1 is dense in Hn(D),

Let x0(s) — (x10(s),..., xn0(s)) be an arbitrary element of Hn(D), e > 0 and K C D be arbitrary compact set. Then there exists p0 such that

max se£ E <2

(9)

with a — {a(p) : |a(p)| — 1}- The denseness of the set of series (8) shows that there exists a — {a(p) : |a(p)| — 1}, such that

max sup

1<j<n seK

xj0(s) — X log(1 + gjp(s 1)) — X a(P)gjP(s,a)

P<P0

P>P0

e

< 2.

(10)

Denoting

a(P) —

a(p)bp if p > p0,

1 if p < P0,

we find from (9) and (10) that

and a — {a(p)},

max sup

1<j<n seK

xj0(s) — X log(1 + gjp(s,a))

< max sup

1<j<n seK

+ max sup

1<j<n seK

xj0(s) — X log(1 + gjP(s 1)) — X a(P)gjp(s,b)

P<P0

P>P0

gjp(s,a)| +

d(P)Xj (p)bp

ps

E

P>P0

Xj (p)ap

P>P0 fc=2 P>P0

This means that the set of all convergent series

X (log(1 + g^oO^..., log(1 + gnp(s,a)))

ps

< e.

with |a(p)| — 1 is dense in Hn(D),

{ w ( p ) }

on the probability space (ft, B(ft),mH), Hence,

{log(1 + glp(s,a)),...,log(1 + gnp(s,a))}

is a sequence of independent Hn(D)-valued random elements on the probability space (ft, B(ft),mH), Since the unit circle is the support of every random variable w(p), the set

{(g0,...,gn) G h«(d) : gj(s) — log(1 + gjp(s,a)), j — 1,...,n}

with a — {a(p) : |a(p)| — 1} is the support of the Hn(D)-valued random element

(log(1 + g1p(^ w)), . . . , log(1 + gnp(s, w))).

Thus, by Lemma 6, the support of the random element (2) is the closure of all convergent series (11), Since the latter set is dense in Hn(D), the support of the random element (2) is the whole of Hn(D),

Let u : Hn(D) ^ Hn(D) be given by the formula

u(g1,...,gn) — (eg1,...,egn), (g1,...,gn) G Hn(D).

u

E log(1+glp(s, w)),...,—E log(1+gnp(s, w))

to the vector

11

n(1+g1p(s,w)) 1,..., n(1+gnp(

V p p

s,w))-1 , (12)

and mapping Hn(D) onto (S \ {0})n, This and a result on the support of the random element (2) show that the support of the random elements (12) contains the set (S \ {0})n. However, the support of (12) is a closed set. Since, by Lemma 10, the closure of (S \ {0})n is Sn, the support of the random element (12) contains the set

Sn

The products

n(1+gjp(s,w)) 1, j — ^...^

D

w G ft Sn

of the random element (12), Combining this with the opposite inclusion, we obtain the assertion of the lemma, □

In the same way as Lemma 12, the following statement follows,

X1, . . . , Xn

Then the support of the measure Pl,v is the set Sin.

Lemma 14. Suppose that x1,..., xn are pairwise non-equivalent Dirichlet character, and that F e U. Then the support of the random element F(L(s,w,x)) is H(D)

PROOF. Let x be an element of H(D), and G be any open neighbourhood of x. Since the function F is continuous, we have that F-1G is an open set, too. By the

U

therefore, there exists an element y which belongs to Sn and F-1G simultaneously. This means that F-1G is an open neighbourhood of the element y. Consequently, in virtue of Lemma 13,

(w e ft : F(L(s, w, x)) e G) — (w e ft : L(s, w, x) e F-1G) > 0.

The investigation of the support of the random element F(LV(s,w,x)) leans essentially on the famous Mergelvan theorem on the approximation of analytic functions by polynomials.

Lemma 15. Let K c C he compact set with connected complement. Then every function g(s) which is continuous on K and analytic inside K can he uniformly approximated on K by polynomials in s.

The lemma is the Mergelvan theorem, see [10], [15].

x1, . . . , xn

racier, and that F e UV. Then the support of the random element F(LV(s,w,x)) is the whole of H(DV).

G x F-1G

of Hn(DV). We have to prove that (F-1G) n Sin — 0. For this, we apply Lemma 15.

From the definition of the metric p in H(DV), it follows that in H(DV) it suffices to consider an approximation on compact subsets of DV, Indeed, let e > 0 be an arbitrary fixed number.Then there exists 10 e N such that

(F-1G) U = 0,

This and properties of a support prove the lemma.

□

Suppose that

P

sup |f (s) - P(s)| <-, f,g G H(DV).

Since K c K1+^ l e N, this implies the inequality

-

for all / = 1,..., /0, Therefore, in view of (13),

P(f,g) < є

This shows that if the functions f and g are near one from another on K with sufficiently large /, then they are near in the sense of H(DV), We note that the sets K can be chosen with connected complements.

So, let K C be a compact subset having a connected complement, and f Є H(DV), Then, by Lemma 20, we can find a polynomial p = p(s) which approximate the function f with a given accuracy uniformly on K. Hence, we have that if f Є G, then p Є G, too. By the definition of the class

(F-1{S}) n sn = 0.

p Є G

(F-1G) П Sn = 0.

Thus, by Lemma 13,

(ш Є ft : F(LV(s, ш, x)) Є G = Є ft : (s, ш, x) Є F-1G > 0.

This proves the lemma, □

4 Proof of Theorems 4 and 5

A proof uses limit theorems in the space of analytic functions as well as supports of the limit measures, and is standard,

PROOF. [Proof of Theorem 4] By Lemma 15, we can find a polynomial p(s) such that

e

sup |f (s) - p(s)| <-. (14)

-

Define

e

G — {g e H(D) : suP |f (s) - g(s)| < -}.

G

probability measures in terms of open sets yield

liminf —meas {t e [0, T] : F(L(s + ir, x1),..., L(s + ir, xn)) e G} > (L)(G) > 0

(15)

where (l) is the distribution of the random element F(L(s,w,x)) In view of Lemma 14, the polynomial p(s) is an element of the support of the measure (¿),

Therefore, the definition of G and properties of a support show that (l)(G) > 0.

This and (15) imply

liminf ^measj r G [0, T] : sup |F(L(s + ir, Xi),... , L(s + ir, xn)) — f (s)| < Д > 0.

J

Combining this with (14), we obtain the theorem, □

PROOF. [Proof of Theorem 5] The theorem follows in the same way as Theorem 4 by using Lemmas 5 and 16 in place of Lemmas 4 and 14, □

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Faculty of Mathematics and Informatics, Vilnius University, Lithuania. Received 21.06.2011