Joint disctrete universality of Dirichlet L-functions. Ii Текст научной статьи по специальности «Математика»

HEEbimEBCKHft CEOPHHK TOM 16 BbinycK 1 (2015)

Y^K 519.14

JOINT DISCTRETE UNIVERSALITY OF DIRICHLET L-FUNCTIONS. II

A. LaurinCikas (Vilnius, Lithuania), D. Korsakiene, D. SiauCiunas (Siauliai, Lithuania)

To the memory of Professor A.A. Karatsuba Abstract

In 1975, S. M. Voronin obtained the universality of Dirichlet L-functions L(s,x), s = a + it. This means that, for every compact K of the strip {s € C : 2 < a < 1}, every continuous non-vanishing function on K which is analytic in the interior of K can be approximated uniformly on K by shifts L(s+ir, x), t € R. Also, S. M. Voronin investigating the functional independence of Dirichlet L-functions obtained the joint universality. In this case, a collection of analytic functions is approximated simultaneously by shifts L(s + iT,xi), ■ ■ ■ ,L(s + iT, Xr), where xi, ■ ■ ■ ,Xr are pairwise non-equivalent Dirichlet characters.

The above universality is of continuous type. Also, a joint discrete universality for Dirichlet L-functions is known. In this case, a collection of analytic functions is approximated by discrete shifts L(s + ikh, xi), ■ ■ ■, L(s + ikh, xr), where h > 0 is a fixed number and k € No = N U{0}, and was proposed by B. Bagchi in 1981. For joint discrete universality of Dirichlet L-functions, a more general setting is possible. In [3], the approximation by shifts L(s + ikh1 ,Xi), ■■■, L(s + ikhr ,xr) with different h1 > 0, ■ ■ ■ ,hr > 0 was considered. This paper is devoted to approximation by shifts L(s + ikh1,x1), ■ ■ ■ ,L(s + ikhri,xri),L(s + ikh,xri+1), ■ ■ ■ ,L(s + ikh,xr), with different h1, ■ ■ ■ ,hri,h. For this, the linear independence over Q of the set

L^^^^h^ ,h; n) = {(h1 log p : p €P ),■■■, (hri log p : p €P),

(hlogp : p € P);

where P denotes the set of all prime numbers, is applied.

Keywords: analytic function, Dirichlet L-function, linear independence, universality.

Bibliography: 10 titles.

СОВМЕСТНАЯ ДИСКРЕТНАЯ УНИВЕРСАЛЬНОСТЬ

L-ФУНКЦИЙ ДИРИХЛЕ. II

А. Лауринчикас (г. Вильнюс) Д. Корсакене, Д. Шяучюнас (г. Шяуляй, Литва)

Посвящается памяти профессора А.А. Карацубы Аннотация

В 1975 г. С. М. Воронин доказал универсальность L-функций Дирихле L(s,x), s = а + it. Это означает, что для всякого компакта K полосы {s € C : I < а < 1} любая непрерывная и неимеющая нулей в K, и аналитическая внутри K функция может быть приближена равномерно на K сдвигами L(s + ir, x), т € R. Изучая функциональную независимость L-функций Дирихле, С. М. Воронин также установил их совместную универсальность. В этом случае набор аналитических функций одновременно приближается сдвигами L(s + iT, xi), • • •, L(s + iT, Xr), где xi, • • • ,xr попарно не эквивалентные характеры Дирихле.

Такая универсальность называется непрерывной универсальностью. Также известна дискретная универсальность L-функций Дирихле. В этом случае набор аналитических функций приближается дискретными сдвигами L(s + ikh, xi), • • •, L(s + ikh, xr), где h некоторое фиксированное положительное число, а k € No = N U{0}. Такая постановка задачи была дана Б. Багчи в 1981 г., однако может рассматриваться более общий случай. В [3] было изучено приближение аналитических функций сдвигами L(s + ikhi,x1), • • •, L(s + ikhr ,xr) с различными hi > 0,^^,hr > 0. Настоящая статья посвящена приближению сдвигами L(s + ikhi,xi), • ••,L(s + ikhri,xri),L(s + ikh,xri+i), •••,L(s + ikh,xr), с различными hi, • • •, hri, h. При этом требуется линейная независимость над полем рациональных чисел для множества

L^^^^h^ ,h; п) = {(hi log p : p €P ),•••, (hri log p : p €P),

(hlogp : p € P);п},

где P - множество всех простых чисел.

Ключевые слова: аналитическая функция, L-функция Дирихле, линейная независимость, универсальность.

Библиография: 10 названий.

1. Introduction

Let s = a + it be a complex variable, and x be a Dirichlet character. The corresponding Dirichlet L-function L(s,x) is defined, for a > 1, by the series

L(s,x) = £ Щ.

ms

m=l

and is analytically continued to an entire function if x is non-principal character. If X is the principal character modulo q, then L(s, x) has a meromorphic continuation to the whole complex plane with a simple pole at the point s =1 with residue

0(-Э-

where p denotes a prime number.

In [9], S. M. Voronin discovered the universality property of Dirichlet L-functions. Roughly speaking, this means that any function from a wide class of analytic functions can be approximated by shifts L(s + ir,x), t € r. A strong statement of the modern version of the Voronin theorem is the following.

Let K be the class of compact subsets of the strip D = {s € c : 2 < a < 1} with connected complements, and let, for K € K, H0(K) denote the class of continuous non-vanishing functions on K which are analytic in the interior of K. Moreover, let measA be the Lebesgue measure of a measurable set A C r.

Theorem 1. Suppose that K € K and f (s) € H0(K). Then, for every e > 0,

liminf —meas < т e [О,Т] : sup \L(s + i^x) — f(s)\ < el > О. I [ seK J

Dirichlet L-functions also are jointly universal, and this was obtained by S. M. Voronin in [10]. We state modern version of a joint universality theorem for Dirichlet L-functions which can be found in [5], [8].

Theorem 2. Suppose that xi, ■ ■ ■ ,Xr are pairwise non-equivalent Dirichlet characters. For j = 1,... ,r, let Kj e K and fj (s) G H0(Kj ). Then, for every £ > 0,

liminf -meas ! т e [О,Т] : sup sup \L(s + ^,Xj ) — fj (s)\ < el > О.

T^^ 1 I i<j<r seKj I

Theorems 1 and 2 are called continuous universality theorems because the real shift т in L(s + ir, x) takes arbitrary real values. Also, discrete universality theorem can be considered where r takes values from the discrete set {hk : k E N0 = NU{0}}, where h > 0 is a fixed number. B. Bagchi proved [1] a joint discrete universality theorem for Dirichlet L-functions which we state in a more general form. Denote by #A the number of elements of the set A.

Theorem 3. Suppose that xl,... ,xr are pairwise non-equivalent Dirichlet characters. For j = 1,... ,r, let Kj e K and fj(s) e H0(Kj). Then, for every £ > 0 and h > 0,

liminf —1—# < 0 < k < N : sup sup \L(s + ikh,xj) - fj(s)\ < £ > > 0.

N^^ N + 1 y l<j<r s£Kj j

In [3], a version of Theorem 3 with different h for each L-function L(s,Xj) was obtained. For its proof, a certain additional independence hypothesis is applied. Denote by P the set of all prime numbers, and define, for hl > 0,... ,hr > 0, the set

L(hi,...,hr; n) = {(hi log p : p eP),..., (hr log p : p eP); n} .

Theorem 4 ([3]). Suppose that xl,... ,Xr are pairwise non-equivalent Dirichlet characters, and that the set L(hl,... ,hr; n) is linearly independent over the field of rational numbers Q. For j = 1,...,r, let Kj e K and fj (s) e H0(Kj). Then, for every £ > 0,

liminf —1—# < 0 < k < N : sup sup \L(s + ikhj, xj) - fj(s)\ < ^ > 0.

NN + 1 I i<j<r seKj I

It is known [3] that the set L(hl,...,hr; n) is linearly independent over q for almost all (hl,..., hr) e with respect to the Lebesgue measure on rr. During the memorial conference of A. A. Karatsuba, Professor Yu. V. Nesterenko constructed special examples of hj. For example, in the case r = 2, the set L(1, v^; n) is linearly independent over q.

The aim of this note is to give a modification of Theorem 4 which idea belongs to Professor I. S. Rezvyakova. Let 1 < rl < r, and, for hl > 0,..., hri > 0 and h> 0,

L(hl,... ,hri ,h; n) = {(hl log p : p eP),..., (hri log p : p eP),

(hlogp : p e P); n}.

Theorem 5. Suppose that xl,... ,Xr are pairwise non-equivalent Dirichlet characters, and that the set L(hl,... ,hri ,h; n) is linearly independent over the field of rational numbers q. For j = 1,...,r, let Kj e K and fj (s) e H0(Kj). Then, for every £ > 0,

liminf a/i I 0 < k < N : suP suP |L(s + ikhjx) - fj(s)\

N^^ N + 1 I l<j<ri s£Kj

sup sup \L(s + ikh,xj) — fj(s)\ < ^ > 0.

ri<j<r s£Kj I

For example, in the case r = 4, we can take hl = 1, h2 = V^, h = v^.

2. Main lemmas

Let y = {s £ c : |s| = 1} be the unit circle on the complex plane. Define the

torus

per

where yp = Y for all p £P. With the product topology and pointwise multiplication, the torus Q, by the Tikhonov theorem, is a compact topological Abelian group. Therefore, denoting by B(X) the Borel a-field of the space X, we have that, on (Q, B(Q)), the probability Haar measure can be defined. Moreover, we put

Qri+1 = Q1 x • • • x Qri+i,

where Qj = Q for j = 1,... ,r1 + 1. Then, by the Tikhonov theorem again, Qri+1 is a compact topological group, and, on (Qri+1, B(Qri+1)), the probability Haar measure mH exists. Moreover, the measure mH is the product of the Haar measures mjH on (Qj, B(Qj)), j = 1,... ,r1 + 1. Denote by Uj(p) the projection of an element Uj £ Qj to the coordinate space yp, P £P, j = 1, ■ ■ ■ ,r1 + 1. Now, for A £ B(Qri+1), define

Qn(A) = {0 < k < N : ((p~ikhi : p £P),..., (p~ikhri : p £P),

(p-ikh : p £P)) £ A} .

Lemma 1. Suppose that the set L(h1,... ,hri ,h; n) is linearly independent over q. Then Qn converges weakly to the Haar measure mH as N ^

proof. We consider the Fourier transform gN(k) of the measure QN, where k = (kjp : p £ P, j = 1,... ,r1 + 1). We have that

n ri + 1

gN(k)= n II j(p)dQN,

Qri+i j=1 P^P

where only a finite number of integers kjP are distinct from zero. Thus, the definition of Qn gives

1 N ri

gN (k) = N— Y, n n p~ikkjphjU p-ikkph

k=0j=1per per

1 N ( / ri \

—1Y1 exM -ik (Y1Y1 kjphjlog p+Y1 kPh log p

k=0 I \j=1 per per J

where, for brevity, kri+1p = kp.

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a.laurinCikas, d. korsakienE, d. siauCiunas

Clearly,

9n (0) = 1.

Moreover, we observe that, for k = 0,

exM —i (kjphj log pkph log p

I \j=l p&r p&r y

= 1.

Indeed, if inequality (3) is not true, then the equality

ri

^T ^T kjphj log p + h^2kP log p = 2nl j=l p&r p&r

holds for some l e z and some finite number of integers kjp, kp. However, this contradicts the linear independence of the set L(hl, ... ,hri ,h; n) over q. Now, from (1) - (3) we find that

9n (k)

Therefore,

1

l-expi -i(N+l) ( £ J] kjphj logp+h J2 kp logp

I \j=i peP peP

-{j

(N+l) | l-exp<{ -i[ J2 J2 kjphj logp+h J2 kp logP

j=i peP peP

if k = 0, if k =0.

N1^gN(k) = {0 f k=0

This and a continuity theorem for probability measures on compact groups, see for

example, [4], prove the lemma. □

Now we will give a modification of Lemma 2.2 from [3] on the ergodicity of one transformation of Qri+l. Define

and

ahi,...,hri ,h = ((p-ihi : p eP),..., (p-ihri : p eP), (p-ih : p eP))

Phi,...,hr-, ,h(u) = ahi,hr. ,hu, u e nri+l.

Lemma 2. Suppose that the set L(hl,... ,hri ,h; n) is linearly independent over q. Then the transformation phi>..,h ,h is ergodic.

Proof. The characters ^(u), u = (ul,... ,uri+-]) e Qri+l of the group Qri are of the form

ri+l

#u)=R l[ukjp (p), j=l p&P

where, as in Lemma 1, only a finite number of integers kjp are distinct from zero. Thus, in view of (3),

j-i( ¿Y kjphj logp + h Y h logp) 1 = 1. (4)

I \j=i per per J )

,h) = exp <( -i ( ^ ^ , 7 y,„p-

v j=i per per

Let A E B(e QT1+1) be an invariant set of the transformation yhl...,h ,h, i.e., the sets A and phl>..,h ,h(A) can differ one from another at most by a set of zero mH-measure, let Ia be the indicator function, and let g denote the Fourier transform of g. Then, for almost all u E Qri+1,

lA(ahi,...,hr1 ,hu) = Ia(U). (5)

If ^ is a non-trivial character, then (5) and the invariance of the measure mH show that

I a(4>) = J 4>(u)lA(w)mu(du) = ^(ahi,..,hri,h)lA(4>)-

nri+1

Therefore, by (4),

ZaWO = 0. (6)

Now, suppose that is the trivial character of Qri+1, and let I A(ip0) = u. Then (6) together with orthogonality of characters shows that, for every character

I a(^) = u J ^(u)mH(du) = ul(^) = u(^).

Qri+i

Therefore, IA(u) = u for almost all u E Qri+1. Hence, mH(A) = 1 or mH(A) = 0,

in the other words, the transformation phit.,h ,h is ergodic. □

3. A limit theorem

Let, for brevity, h = (h1,... ,hri ,h), x = (Xb • • •, Xr) and

L(s + ikh, x) = (L(s + ikh1, x1),..., L(s + ikhri, xri), L(s + ikh, Xri+1),... L(s + ikh, Xr)).

Denote by H(D), D = { s E c : 2 < a < l}, the space of analytic functions on D endowed with the topology of uniform convergence on compacta, and, on the probability space (Qri+1, B(Qri+1),mH), define the Hr(D)-valued random element

L(s,u,x) = (n (l - )"II (l - )",

^ L - Xri+1(p)un+1(pa "1 ^ L - Xr(p)un+i(pA p\ p j p \ p J J

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

Pl(A) = mn (u e nri+l : L(s,u,x) e A) , A e B(Hr(D)).

Theorem 6. Suppose that the set L(hl,... ,hri ,h; n) is linearly independent over q. Then

Pn(A)= N+T# {0 < k < N : (L(s + ikh,x)) e A} , A e B(Hr(D)),

converges weakly to Pl as N ^ x>. Proof. Let, for a fixed al > |,

' m\ ai n

Define auxiliary functions

xj (m)Vn(m)

m=l

and

xj (m)uj (m)vn(m)

vn(—) = exp j — ^—j |, —,n E N.

r ( \ \ - Xj —)vn—) .

Ln(s, Xj) = -—-, j = h---,r,

\ - Xj (—)Uj (—)vn(—) .

—s m=1

oo

Ln(s,Ur1+i,Xj ) = -s-' J = ri +

—s m=1

the series being absolutely convergent for a > 2, where, for — E n,

Uj(—) = n ua(p), j = l,...,ri + 1.

pa\\m

Further, we put

Ln(s + ikh,X) = (Ln(s + ikhi,Xi),... ,Ln(s + ikhri ,Xri), Ln(s + ikh, Xri+i),..., Ln(s + ikh, Xr))

and

Ln(s + ikh,u,X) = (Ln(s + ikhi,ui,Xi),... ,Ln(s + ikhr1 ,Ur1 ,Xr1 ),

Ln(s + ikh,Uri+i,Xri+i),... ,Ln(s + ikh,Uri+i,Xr)).

Then, using Lemma 1 and Theorem 5.1 of [2], we find, in view of the invariance of the Haar measure —H, that

PN,n(A) = {0 < k < N : Ln(s + ikh, X) e A} ,

and

PN,n,u(A) = n+Y# {0 < k < N : Ln(s + ikh, u, x) E A}

where A e B(Hr (D)), both converge weakly to the same probability measure Pn on (Hr(D), B(Hr (D))) as N ^ to.

It remains to pass from PN,n to PN. For gl,g2 e H(D), let

Po(gi,g2) = J^2

™ sup |gi(s) — g2(s)|

-l s&Ki

1 + sup |gi(s) — g2(s)|

s&Ki

where {Ki : l e n} is a sequence of compact subsets of the strip D such that

oo

D = U Kl, i=1

Ki C Kl+l for all l e n, and if K C D is a compact set, then K C Kl for some l e n. Then we have that p0 is a metric on H(D) which induces its topology of uniform convergence on compacta. Now let, for gl = (gll,..., glr), g2 = (g2l,..., g2r) e Hr (D), " "

p(gl,g2) = max po(glj ,g2j).

—l —2 l<3<r

Then p is a metric on Hr (D) inducing its topology. Using the estimate

N

£|L(a + ikhj ,Xj )|2 = O(N ),

k=0

which follows from the bound T

j\L(a + it,xi)\2dt = O(T), a > 1, j = 1,...,r, 0

and the Gallagher lemma [7], we obtain by a standard procedure that

N

n—<x NN +1

lim limsup n , ! p {L(s + ikh, x),Ln(s + ikh, x)) = 0. ^^ " + k=0

Also, standard arguments imply, for almost all u E Q, the estimate

T

i |L(a + it,u,Xj)12dt = O(T),

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a.laurinCikas, d. korsakienE, d. siauCiunas

and this leads to the bound

N

J2\L(a + ikh,u,Xj)\2dt = O(N), a> 1 j = 1,...,r.

k=0

Hence, for almost all u E Qri+1, we deduce that

N

n—m N—m^ N +1

lim limsup —-V p (Lis + ikh, u, x),Ln(s + ikh, u, x)) = 0.

i—m . . - N + 1 ' J v — — '

k=0

Equality (7) allows to show the weak convergence of the measure PN. Let 6N be a discrete random variable on a probability space (Q, F, p) such that

P(^n = k) = N+1, k = 0,1,..., N,

and

X Nn(s) = Ln(s + i^N hx).

Then the weak convergence of PN,n to Pn can be rewritten in the form

XN,n —-> ^^^—n, (9)

' N—m

where Xn is the Hr (D)-valued random element with the distribution Pn. Using the latter relation, it is not difficult to prove that the family of probability measures {Pn : n E n} is tight. Thus, it is relatively compact, and there exists a sequence {nk} and a probability measure P on (Hr(D), B(Hr(D))) such that

Xnfc -- P. (10)

k k—m

Now putting

X n (s) = L(s + i0N h,x)

and using (7), we find that

lim limsupp (p(XN,XNn) > e) =0. (11)

n—m N—m '

Relations (9) - (10) show that all hypotheses of Theorem 4.2 of [2] are satisfied. Therefore,

XN P,

N N—m

and this implies the weak convergence of PN to P as N ^ x>. It remains to identify the limit measure P. For this, define

L(s + ikh, u, x) = (L(s + ikh1,u1, x1),..., L(s + ikhri ,uri, xri),

L(s + ikh, uri+1, xri+1),..., L(s + ikh, un+1,xr))

and

Pn,u,(A) = {0 ^ k ^ N : L(s + ikh, u,x) e A} , A e B(Hr(D)).

Then, using the weak convergence of PN,n,u to Pn, equality (9) and repeating the above arguments, we obtain that PN,u also converges weakly to P as N ^ <x>. Thus, if A is a continuity set of the measure P, then we have that

lim Pnu(A) = P(A). (12)

N ^^

On the space (Qri+1, B(Qri+1),mH), define the random variable

£,)fl if L(s,u,x) e A,

[0 otherwise.

Then

EC = J £dmn = Pl(A). (13)

n

In view of Lemma 2, we can apply the ergodic Birkhoff-Khintchine theorem, which, for almost all u e Qri+1, gives

N

fc N+I^ ^......(14)

n^ N + 1 ^^ Vni•■■■ 'hri k=0

Therefore, relations (13) and (14) imply

lim Pn ,,w(A) = PL(A).

N

Hence, by (12), we obtain that P(A) = Pl(A) for every continuity set A of P.

Therefore, P = Pl, and the theorem is proved. □

4. Proof of Theorem 5

Theorem 5 is a consequence of Theorem 6 and the Mergelyan theorem on the approximation of analytic functions by polynomials [6]. Let S = {g e H(D) : g(s) = 0 or g(s) = 0}. It is known [5] that the support of the random element

fn A _ Xri + 1(p)uri + 1(p)\-1 n A _ XApU+Jp^ ^

Kper^ PS J pep\ PS J J

is the set Sr-ri. Moreover, the measure mH is the product of the Haar measures mjH on Qj, j = l,... ,r1 + l. Since the support of

1 _ xMuM)-1 j = 1 r

n L - Xj CpM (p) V

p&r \ p '

is the set S, we find that the support the random element

f^ L - x1(p)u_1 ^ L - xri(p)uri(p)\

Ps J peP\ Ps J J

is the set Sri. These remarks show, in view of Theorem 6, that the support of Pl is the set Sr.

By the Mergelyan theorem [6], there exist polynomials p1(s),..., pr (s) such that

£

sup sup \fj(s) - epj(s)\ < -. (15)

1<j<r seKj 2

Define

G =\(gi,...,gr) e Hr (D) : sup sup \gj (s) - ePj (s)|<£|.

I i<j<r seKj 2 I

Obviously, G is an open set in Hr(D). Moreover, (epi(s),..., ePr(s)) G Sr, i.e., is an element of the support of PL. Thus, PL(G) > 0. Hence, liminf PN(G) > 0, since, by

N—m

Theorem 6,

liminf Pn(G) > Pl(G).

N—m

This, the definition of G and inequality (15) prove the theorem.

5. Conclusions

In the paper, a discrete joint universality theorem for Dirichlet L-functions L(s, x) is obtained. In this theorem, a collection of analytic functions f1 (s),..., fr(s) is approximated by shifts L(s + ikh1, xi),..., L(s + ikhri, xri), L(s + ikh, xri+i),..., L(s + ikh, xr), where x1,... ,xr are pairwise non-equivalent Dirichlet characters, and h1,... ,hri,h are such positive numbers that the set

L(h1,... ,hri ,h; n) = {(h1 log p : p EP),..., (hri log p : p EP),

(hlogp : p E P); n}.

is linearly independent over the field of rational numbers.

REFERENCES

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3. Dubickas, A. & Laurincikas, A. 2015, "Joint discrete universality of Dirichlet L-functions" , Archiv Math. Vol. 104. P. 25-35.

4. Heyer, H. 1974, "Probability Measures on Locally Compact Groups" , Berlin, Heidelberg, New York: Springer-Verlag.

5. Laurincikas, A. 2011, "On joint universality of Dirichlet L-functions" , Cheby-shevskii Sb. Vol. 12, No. 1. P. 129-139.

6. Mergelyan, S. N. 1952, "Uniform approximations to functions of a complex variable" , Usp. Matem.. Nauk. Vol. 7, No. 2. P. 31-122 (Russian) = Amer. Math. Trans. 1954. Vol. 101.

7. Montgomery, H. L. 1971, "Topics in Multiplicative Number Theory." , Lecture Notes in Math. Vol. 227. Berlin: Springer.

8. Steuding, J. 2007, "Value-Distribution of L-functions." , Lecture Notes in Math. Vol. 1877. Berlin, Heidelberg: Springer-Verlag.

9. Voronin, S. M. 1975, "Theorem on the "universality" of the Riemann zeta-function." , Izv. Akad. Nauk SSSR. Vol. 39. P. 475-486 (in Russian) = Math. USSR Izv. 1975. Vol. 9. P. 443-453.

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СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ

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Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania.

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E-mail: [email protected], [email protected], [email protected] Получено 18.02.2015