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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 17. Выпуск 1.
УДК 519.14
A DISCRETE UNIVERSALITY THEOREM FOR PERIODIC HURWITZ
ZETA-FUNCTIONS
A. LaurinCikas, D. Mochov (Vilnius, Lithuania)
Dedicated to Gennadii Ivanovich Arkhipov and Sergei Mikhailovich Voronin
Abstract
In 1975, Sergei Mikhailovich Voronin discovered the universality of the Riemann zeta-function Z(s), s = < + it , on the approximation of a wide class of analytic functions by shifts Z(s + ir),т G R. Later, it turned out that also some other zeta-functions are universal in the Voronin sense. If т takes values from a certain descrete set, then the universality is called discrete.
In the present paper, the discrete universality of periodic Hurwitz zeta-functions is considered. The periodic Hurwitz zeta-function Z(s, a; a) is defined by the series with terms am(m + a)-s, where 0 < a < 1 is a fixed number, and a = {am} is a periodic sequence of complex numbers. It is proved that a wide class of analytic functions can be approximated by shifts
Z(s + ihkei loge2 k, a; a) with k = 2, 3,..., where h > 0 and 0 < в < 1, в2 > 0 are fixed numbers, and the set {log(m + a) : m = 0,1,2} is linearly independent over the field of rational numbers. It is obtained that the set of such k has a positive lower density. For the proof, properties of uniformly distributed modulo 1 sequences of real numbers are applied.
Keywords: periodic Hurwitz zeta-function, space of analytic functions, limit theorem, universality.
Bibliography: 15 titles.
ДИСКРЕТНАЯ ТЕОРЕМА УНИВЕРСАЛЬНОСТИ ДЛЯ ПЕРИОДИЧЕСКИХ ДЗЕТА ФУНКЦИЙ ГУРВИЦА
А. Лауринчикас, Д. Мохов (г. Вильнюс) Аннотация
В 1975 г. Сергей Михайлович Воронин открыл свойство универсальности дзета-функции Римана Z(s), s = <7 + it, о приближении широкого класса аналитических функций сдвигами Z(s + iT), т G R. Позже оказалось, что и некоторые другие дзета-функции обладают свойством универсальности в смысле Воронина. Если сдвиг т принимает значения из некоторого дискретного множества, то универсальность называется дискретной.
В работе изучается дискретная универсальность периодических дзета-функций Гур-вица. Периодическая дзета-функция Гурвица Z(s, a; a) определяется рядом с членами am(m + a)-s, m = 0,1, 2,..., где 0 < a < 1 - фиксированное число, а a = {am} - периодическая последовательность комплексных чисел. Доказано, что широкий класс аналитических функций с заданной точностью приближается сдвигами Z(s + ihkei loge2 k, a; a) с k = 2, 3,..., где h> 0 и 0 < 1, i®2 > 0 - фиксированные числа, а множество
{log(m + a) : m = 0,1, 2,... } линейно независимо над полем рациональных чисел. Получено, что множество таких сдвигов, приближающих данную аналитическую функцию, имеет положительную нижнюю плотность. При доказательстве используются свойства равномерно распределенных по модулю 1 последовательностей действительных чисел.
Ключевые слова: периодическая дзета-функция Гурвица, предельная теорема, пространство аналитических функций, универсальность.
Библиография: 15 названий.
1. Introduction
Let s = a + it be a complex variable, and a = {am : m e No = N U {0}} be a periodic sequence of complex numbers with minimal period q e N. The periodic Hurwitz zeta-function Z(s, a; a) with parameter a, 0 < a < 1 is defined, for a > 1 , by the Dirichlet series
C(s,a; a) = Y^
m=0
(m + a)s '
and was introduced in [7]. In virtue of the equality
1 q-1
A/ N 1 /-( m + «\
Z(s, a; a) = — -J,a > 1,
qs ^ v q
H m=0 H
where Z(s, a) is the classical Hurwitz zeta-function given, for a > 1, by
1
Z (s,a) = X)
m=0
(m + a)s
and meromorphically continued to the whole complex plane with unique simple pole at the point s = 1 with residue 1, the function Z(s, a; a) also has meromorphic continuation to the whole complex plane with possible simple pole at the point s = 1 with residue
q-1
E'
m=0
If the latter quantity is equal to zero, the function Z(s, a; a) is entire one.
Clearly, if am = 1, the function Z(s, a; a) becomes the Hurwitz zeta-function. If am = e ^, m G No, then Z(s, a; a) reduces to the Lerch zeta-function
L(A,a,s) =
œ g2niAm
> 1,
-i (m + a)-
m=0 v '
with A = 1. Thus, the periodic Hurwitz zeta-function is a generalization of classical zeta-functions.
The function Z(s, a; a), as the majority of other zeta-functions, is universal in the Voronin sense, i.e., its shifts Z(s + ir, a; a),T G R, approximate a wide class of analytic functions. We recall some results on the universality of Z (s, a; a). Let D = {s G C : | <ct< 1}. Denote by K the class of compact subsets of the strip D with connected complements, and by H(K), K G K, the class of continuous functions on K which are analytic in the interior of K. Moreover, let
L(a) = {log(m + a) : m G N0} .
Then in [11], the following theorem was obtained.
Theorem 1. Suppose that the set L(a) is linearly independent over the field of rational numbers Q. Let K G K and f (s) G H(K). Then, for every e > 0,
lim inf1 meas {t G [0, T] : sup Z(s + ir, a; a) — f (s)
T^œ T L seK
< e \ > 0.
a
m
m-
Here meas A denotes the Lebesgue measure of a measurable set A c R. It is not difficult to see that the set L(a) is linearly independent over Q with transcendental a. This case was discussed in [2] and [3].
Theorem 1 is of continuous character because the shift r in Z(s + ir, a; a) can take arbitrary real values. Also, discrete versions of Theorem 1 are known when r takes values from the set {kh : k e No} with fixed h > 0. The first result in this direction has been obtained in [10].
Theorem 2. Suppose that a is a transcendental number, and exp | ^ | is a rational number. Let K e K and f (s) e H(K). Then, for every e > 0,
liminf —1—#(0 < k < N : sup Z(s + ikh, a; a) - f (s) < 4 > 0. NN + 1 I '
Here #A denotes the cardinality of the set A. In [13], a more general result was obtained. Let
L(a, h, n) = { (log(m + a) : m e No), h } .
Theorem 3. Suppose that the set L(a, h, n) is linearly independent over Q. Let K e K and f (s) e H(K). Then the assertion of Theorem 2 is true .
The aim of this paper is to replace the set {kh : k e No} in Theorems 2 and 3 by a more complicated one. Let 0 < < 1, fi2 > 0 and h > 0 be fixed numbers.
Theorem 4. Suppose that the set L(a) is linearly independent over Q. Let K e K and f (s) e H(K). Then, for every e > 0,
< e > 0.
liminf —^—#{2 < k < N : sup Z(s + ihk^1 log^2 k, a; a) - f (s) n^^ N — 1 I
For the proof of Theorem 4, we will apply good distribution properties of the sequence {hk^1 log^2 k : k = 2, 3,...}. In general, we will use the probabilistic method based on a limit theorem for probability measures in the space of analytic functions. Let B(X) denote the Borel ct— field of the space X, and let H(D) be space of analytic functions on D endowed with the topology of uniform convergence on compacta.
We note that the universality of zeta and L-functions was discovered by Sergei Mikhailovich Voronin who in [15] obtained universality of the Riemann zeta-function and Dirichlet L -functions, see also [6].
2. A limit theorem
We start with a limit theorem of discrete type on the torus
^ = n Ym,
m£N0
where Ym = {s e C : |s| = 1} for all m e N0. With the product topology and pointwise multiplication, the torus Q is a compact topological group. Therefore, on (Q, B(Q)), the probability Haar measure mH exists, and we have the probability space (Q,B(Q),mH). Denote by w(m) the projection of an element w e Q to the coordinate space Ym, m e N0. For A e B(Q), we set
Qn(A) = N3I#{2 < k < N : ((m + a)-^1 log"2 k : m e No) e a}.
Lemma 1. Suppose that the set L(a) is linearly independent over Q. Then the measure Qn converges weakly to the Haar measure mg as N ^
Proof. We remind that a sequence {xm : m G N} is uniformly distributed modulo 1 if, for every interval I = [a, b) C [0,1) of length |11,
1 n
ffim nt, «xk}) = rn.
n—TO n
k=1
where {u} is the fractional part of u G R and xi is the indicator function of I. By the Weyl criterion, see, for example, [5], the sequence {xm} is uniformly distributed modulo 1 if and only if, for every k G Z \ {0},
1 n
lim -V e2nikxm = 0. n—y^o n —' m=1
It is well known [5] that the sequence {ak^1 log^2 k : k = 2,3,...} with a = 0 is uniformly distributed modulo 1.
For the proof of the lemma, we apply the method of the Fourier transforms. Let k = {km : m G N0} with integers km. Then the Fourier transform gN(k) of the measure QN is of the form
gN(k) = j n wkm (m)dQw,
q m€No
where only a finite number of integers km are distinct from zero. Hence, we have that
N
N - 1
gN (k) = EH (m + a)-^1 log32 k
k=2 mGNo N
1 E exp j — ihk^1 log^2 k E km log(m + a)j. (1)
N-1
k=2 m£N0
The linear independence over Q of the set L(a) implies that
E km log(m + a) = 0
m£N0
if and only if k = 0. Therefore, if k = 0, then
h E km log(m + a) = 0.
m£N0
By the above remark, the sequence
hk^1 logft k
f hk^1 log^ k v-^ , , . . , n 0 I -2n- E km log(m + a) : k = 2, 3,...^
^ m€N0 '
is uniformly distributed modulo 1. Thus, in view of (1) and the Weyl criterion,
lim gN(k) = 0 (2)
N—TO
for k = 0. Obviously, by (1),
gN (0) = 1.
152
a. laurinCikas, d. mochov
This and (2) show that
Alimo gN (k) = { 0 if k = 0
Clearly,
1 if k = 0,
g(k)=i 0 if k = 0,
is the Fourier transform of the Haar measure mg. Therefore, the lemma follows by a general
continuity theorem for probability measures on compact groups, see, for example, [4]. □
Furthermore, we will deal with a limit theorem for absolutely convergent Dirichlet series. For a fixed CT > 1 and m e N0, n e N, let
v„(m,a)=exp{ — .
Define two functions
and
oo
> , x amV„(m, a)
Zn(s, a; a) = > —-■——
^ (m + a)s
m=0 oo
Zra(s,a,W; a) = ^ (m + a)s .
m=0
Then the latter series are absolutely convergent for ct > 1 [2]. From this, it follows that the function un : Q ^ H (D) given by the formula
PN,n(A) = n~[#{2 ^ k < N : Z(s + ihk^1 logft k, a; a) e a}.
un(w) = Zn(s, a, w; a), w e Q, is continuous one. For A e B(H(D)), let
1
1
Moreover, we put Pn = mHu-1, where the measure mHu-1 is defined by
mgu-1(A) = mg(u-1A), A e B(H(D)).
Lemma 2. Suppose that the set L(a) is linearly independent over Q. Then PN,n converges weakly to Pn as N ^ to.
Proof. By the definition of the function un, we have
un((m + a)"^1 log^2 k : m e N0) = Z(s + ihk^1 logft k, a; a). Therefore, for A e B(H(D)),
PN,n(A) = N"31^2 < k < N : ((m + a)"^1 log"2 k : m e N0) e a} = Qn (u"1A) = Qn u"1(A).
This, the continuity of un, Lemma 1 and Theorem 5.1 of [1] show that PN,n converges weakly to
Pn as N ^ to. □
Now we will approximate Z(s + ihk^1 log^2 k, a; a) by Zn(s + ihk^1 log^2 k, a; a) in the mean. Let p be the metric on H(D) which induces the topology of uniform convergence on compacta, see [10], or [8, 9].
Lemma 3. The equality 1 N
lim lim sup—-V p(Z (s + ihkßl logß2 k.a; a).ZU(s + ihkßl logß2 k.a; an =0
u^TO N^™ N — 1 V )
n^TO N^TO N - 1 fc=2
holds.
Proof. It was obtained in [2] that, for a > 2,
T
/|C(, + «,„. a)|2,S = 0(r). (3,
To obtain a discrete version of the latter estimate, we will use the Gallagher lemma, see Lemma 1.4 in [14]. For 2 ^ k ^ N, with sufficiently large N we have
(k + 1)ßl logß2 (k + 1) - kßl logß2 k = kßl (1 + 1) ßl (log k + log (l + 1)) - kßl logß2 k
= kßl (1 + f + + -)(log k + 1 - + •••)* - kßl logß2 k
= (kßl + ßl(ßl - 1) + ) logß2 ___)ß2
= r + ki-ßl + 2k2-ßl + "J log H1 + k log k 2k2 log k + -J
^ c logfe N
with suitable constant c > 0 not depending on N. Therefore, taking 5 = ^N"-^ in Lemma 1.4 of [14], we find that
N
E |C(a + ihk^1 logA k + it, a; a)|2 fc=2
hNlog^2 N
|2
< N1-ßl log-ß2 N j |Z(^ + iT + it. a; a)|2dr+
+
/ hNlog^2 N hN01 log^2 N \ 2
j |z(a + ir + it, a; a)|2dr j |Z'(a + ¿t + it, a; a)|2dr 11 < N + |t| < N(1 + |t|)
for a > 1 because of (3) and the estimate
T
J |Z'(o- + it. a. a)|dt = O(T)
implied by (3). Let K be a compact subset of the strip D. Then, repeating the proof of Theorem 4.1 from [6], we obtain that
1 N
lim lim sup ——- V^ sup Z(s + ihkßl logß2 k .a; a) - Zn (s + ihkßl logß2 k. a; a)
U^TO AT . _ . N - 1 ^-' TS
N^TO N - 1 k=2 s^K
l
l
l
154
A. laurinCikas, d. mochov
This and the definition of the metric p prove the lemma. □
Now we state the main limit theorem. On the probability space (Q, B(Q), mH), define the H(D)-valued random element Z(s, a, w; a) by the formula
oo
^ amw(m)
Z(s, a, w; a) = > -—.
sv ' ' ' 7 ^ (m + a)s
m=0
The latter series, for almost all w e Q, converges uniformly on compact subsets of the strip D, and therefore, defines a H(D)-valued random element. Denote by P^ the distribution of the random element Z(s,a, w; a), i.e.,
Pc(A) = mg{w e Q : Z(s, a, w; a) e a}, A e B(H(D)).
For A e B(H(D)), let
Pn (A) = n~T1#{2 ^ k < N : Z (s + ihkA logft k, a; a) e a}.
Theorem 5. Suppose that the set L(a) is linearly independent over Q. Then pn converges weakly to P^ as N ^ to. Moreover, the support of P^ is the whole of H(D).
Proof. Let 0N be a random variable defined on a certain probability space (Q, F, P) and having the distribution
P(0n = hk^ logft k) = ^^, k = 2,..., N.
Define the H(D)-valued random element XN,n by the formula
XN,n = XN,n(s) = Zn(s + i^N, a; a).
Moreover, let Xn be the H(D)-valued random element having the distribution Pn, where Pn is the limit measure in Lemma 2. Then the assertion of Lemma 2 can be written in the form
XN,n ———> Xn, (4)
N ^^
T>
where —> means the convergence in distribution. We will prove that the family of probability measures {Pn : n e N} is tight, i.e., for every e > 0, there exists a compact set K = K(e) c H(D) such that
Pn(K) > 1 — e
for all n e N. Since the series for Zn(s, a; a) is absolutely convergent for ct > 2, we have that
T
1 ZV/ . .. \\2j+ ^ |am|2vn(m,a)
suplimsup — \Z(ct + it, a; a)\ at = sup > —?-^—
n€N T^ Ty 1 1 n€N m=0 (m + a)2CT
^ I |2
< y < C< to.
^ (m + a)2CT
m=0
This together with the Gallagher lemma [14] implies the bound
1 N \ 2 suplimsup ——- V^ \Zn(s + ihk^1 log^2 k, a; a)
n€N N^ro N — 1 k= '
< c1 < to.
Hence,
1
N
suplimsup ——L(s + ihk^1 log^2 k, a; a)
nen n—TO N — 1 1
< C2 < TO.
(5)
fc=2
Let Kr, l G N, be compact sets from the definition of the metric p [10]. Then (6) together with the Cauchy integral formula shows that
1
N
sup lim sup ^ sup Zn(s + ihk^1 log^2 k, a; a)
neN N—TO N — 1 k=2
< R < to.
(6)
Let e > 0 be an arbitrary number, and Mr = Mr(e) = 2rRre 1. Then, taking into account (7), we find that, for l G N,
lim sup P[ sup |XN,n(s)| > M^
N —>-<TO ^ sEKi
limsup —1—#{2 ^ k ^ N : sup |(„(s + ihk^1 logft k,a; a)| > Mr)
N—TO N — 1 s€Kl J
1 N e
< limsup ———— V sup |Zn(s + ihk^1 logft k,a; a)| < -r.
n—TO (N - 1)M k=2 seKl 2
Hence, by the relation (5), we obtain that, for l G N,
e
P sup |Xn(s)| >Mr) < ^. (7)
seKi
Putting
K(e) = {g G H(D) : sup |g(s)| < Mr,l G N},
seKi
we have that K(e) is a compact subset of H(D) because it is uniformly bounded on compact subsets of the strip D. Moreover, (8) shows that, for all m G N,
P(X„(S) G K(e)) > 1 - e,
or, for all n G N,
P{K(e)) > 1 - e.
Thus, the sequence {Pn : n G N} is tight.
Since the sequence {Pn : n G N} is tight, by the Prokhorov theorem, see [1, Theorem 6.1], it is relatively compact. Therefore, there exists a subsequence {Pnr} C {Pn} such that Pnr converges weakly to a certain probability measure P on (H(D), B(H(D))) as n ^ to. From this,
Xnr (s) P. (8)
r — TO
Let the H(D)-valued random element XN be defined by the formula
Xn = Xn(s) = Z(s + ¿6N, a; a).
Then, by Lemma 3, for every e > 0,
lim limsupP( p(XN(s),XNn(s)) ^ e n—TO n—TO V '
1
n—M N—M N - 1
Zn(s + ihfc^1 logA k, a; a)) > e
lim lim sup —-#(2 < k < N : J((s + ihk^1 logft k,a; a),
n—M ^ N - 11 V
1 N
< lim lim sup —-— V p f Z(s + ihk^1 log^2 k, a; a),
n—M n—m (N - 1)e v
Zn(s + ihk^1 logft k, a; a)) = 0. This equality, (5), (9) and Theorem 4.2 from [1] imply the relation
Xn ---— P, (9)
N —>-<M
thus, PN converges weakly to P as N —> to. Moreover, (10) shows that the measure P is independent of the choice of the subsequence {Pnr}. Therefore, the relation
Xn -— P,
n—M
is true, and we have that the measure Pn converges weakly to P as n — to.
It remains to identify the measure P. In [11], under the hypothesis that the set L(a) is linearly independent over Q, it was obtained that the measure
T meas jr G [0,T] : Z(s + ir, a; a) G A},A g B(H(D)),
as T — to, also converges weakly to the measure P which is the limit measure of Pn as n — to, and that P coincides with P^. Since PN, as n — to, converges weakly to P, hence we have that PN also converges weakly to P^ as N — to. Moreover, in [11], it was obtained, that the support of P^
is the whole of H(D). The theorem is proved. □
3. Proof of universality
First we state two lemmas.
Lemma 4. Let K C C be a compact subset with connected complement, and let f (s) be a continuous function on K which is analytic in the interior of K. Then, for every e > 0, there exists a polynomial p(s) such that
f (s) - P(s)
sup
seK
< e.
The lemma is the Mergelyan theorem, see [12].
Lemma 5. Let Pn, n G N, and P be a probability measures on (X, B(X)). Then Pn, as n — to, converges weakly to P if and only if, for every open set G C X,
liminf Pn(G) > P(G).
n—M
The lemma is a part of Theorem 2.1 from [1].
Proof of Theorem 4. By Lemma 4, there exists a polynomial p(s) such that
sup
seK
/(s) - P(s)
e
< 2-
Define the set
G = {g G H(D) : sup g(s) - p(s) < 2).
L seK 2 J
Then G is an open set in H(D), therefore, in view of Theorem 5 and Lemma 5,
liminf PN (G) > Pc(G).
N—>oo
(10)
(11)
Moreover, G is an open neighbourhood of the polynomial p(s) which, again by Theorem 4, is an element of the support of the measure FZ• Thus, Fç(G) > 0. This, (12) and the definition of G imply the inequality
liminf—1—#Î2 ^ k ^ N : sup Z (s + ihkßl logß2 k,a; a) - p(s)
N —^^o N — 1 I seK
< ^ >0-
This and (11) prove the theorem. □
4. Conclusions
Let a = {am} be a periodic sequence of complex numbers, 0 < a < 1 and s = a + it. The periodic Hurwitz zeta-function Z(s, a; a) is defined, for a > 1 , by the series
Z(s,a; a) = E
ari
—1 (m + a)s'
m=0 v '
and by analytic continuation elsewhere. Moreover, let
L(a) = {log(m + a) : m = 0,1,2,...} .
In the paper, the following discrete universality theorem for the function Z(s,a; a) is obtained. Suppose that K be the class of compact subsets of the strip D with connected complement, and H(K), K £ K, be the class of continuous functions on K which are analytic in the interior of K. Moreover, we assume that the set L(a) is linearly independent over the field of rational numbers, and that 0 < ß < 1, ß2 > 0 and h > 0 are fixed numbers. Then the function Z(s, a; a) is universal in the Voronin sense, i.e., if K £ K, f (s) £ H(K), then, for every e > 0,
liminf —^—#{2 ^ k ^ N : sup Z(s + ihkßl logß2 k,a; a) - /(s)
N—TO N - 1 I seK
< e \ > 0.
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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.
E-mail: [email protected], [email protected] Получено 11.12.2015 г. Принято в печать 10.03.2016 г.
