Joint disctrete universality of Dirichlet L-functions. Ii

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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 & lt- a & lt- 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 & gt- 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 & gt- 0, ¦ ¦ ¦, hr & gt- 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 & lt- а & lt- 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 & gt- 0,^^, hr & gt- 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 & gt- 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 & lt- a & lt- 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 & gt- 0,
liminf -meas & lt- т e [О, Т]: sup L (s + i^x) — f (s) & lt- el & gt- 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? & gt- 0,
liminf -meas ! т e [О, Т]: sup sup L (s +, Xj) — fj (s) & lt- el & gt- О.
T^^ 1 I i& lt-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 & gt- 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? & gt- 0 and h & gt- 0,
liminf -1-# & lt- 0 & lt- k & lt- N: sup sup L (s + ikh, xj) — fj (s) & lt-? & gt- & gt- 0.
N^^ N + 1 y l& lt-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 & gt- 0,…, hr & gt- 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? & gt- 0,
liminf -1-# & lt- 0 & lt- k & lt- N: sup sup L (s + ikhj, xj) — fj (s) & lt- ^ & gt- 0.
NN + 1 I i& lt-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 & lt- rl & lt- r, and, for hl & gt- 0,…, hri & gt- 0 and h& gt- 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? & gt- 0,
liminf a/i I 0 & lt- k & lt- N: suP suP |L (s + ikhjx) — fj (s)
N^^ N + 1 I l& lt-j<-ri s? Kj
sup sup L (s + ikh, xj) — fj (s) & lt- ^ & gt- 0.
ri& lt-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 & lt- k & lt- 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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Clearly,
9n (0) = 1.
Moreover, we observe that, for k = 0,
exM -i (kjphj log pkph log p
I j=l p& amp-r p& amp-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& amp-r p& amp-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& lt-{ -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& gt-., 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& amp-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 & lt-(-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& gt-., 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& gt-) = J 4& gt-(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 & lt- a & lt- 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 —)& quot-II (l —)& quot-,
^ L — Xri+1(p)un+1(pa & quot-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 & lt- k & lt- N: (L (s + ikh, x)) e A}, A e B (Hr (D)),
converges weakly to Pl as N ^ x& gt-. Proof. Let, for a fixed al & gt- |,
'- 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 & gt- 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 & lt- k & lt- N: Ln (s + ikh, X) e A} ,
and
PN, n, u (A) = n+Y# {0 & lt- k & lt- 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) = J2
™ sup |gi (s) — g2(s)|
-l s& amp-Ki
1 + sup |gi (s) — g2(s)|
s& amp-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), & quot- & quot-
p (gl, g2) = max po (glj, g2j).
-l -2 l& lt-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
jL (a + it, xi)2dt = O (T), a & gt- 1, j = 1,…, r, 0
and the Gallagher lemma [7], we obtain by a standard procedure that
N
n-& lt-x NN +1
lim limsup n, ! p {L (s + ikh, x), Ln (s + ikh, x)) = 0. ^^ & quot- + 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
J2L (a + ikh, u, Xj)2dt = O (N), a& gt- 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 --& gt- ^^^-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) & gt- 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& gt-. 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 ^ & lt-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& amp-r p '-
is the set S, we find that the support the random element
f^ L — x1(p)u1 ^ 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) & lt- -. (15)
1& lt-j<-r seKj 2
Define
G =(gi,…, gr) e Hr (D): sup sup gj (s) — ePj (s)|& lt-?|.
I i& lt-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) & gt- 0. Hence, liminf PN (G) & gt- 0, since, by
N-m
Theorem 6,
liminf Pn (G) & gt- 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
1. Bagchi, B. 1981, & quot-The Statistical Behaviour and Universality Properties of the Riemann Zeta-function and Other Allied Dirichlet Series& quot-, Ph. D. Thesis. Calcutta: Indian Statistical Institute.
2. Billingsley, P. 1968, & quot-Convergence of Probability Measures& quot-, New York: Wiley.
3. Dubickas, A. & amp- Laurincikas, A. 2015, & quot-Joint discrete universality of Dirichlet L-functions& quot-, Archiv Math. Vol. 104. P. 25−35.
4. Heyer, H. 1974, & quot-Probability Measures on Locally Compact Groups& quot-, Berlin, Heidelberg, New York: Springer-Verlag.
5. Laurincikas, A. 2011, & quot-On joint universality of Dirichlet L-functions& quot-, Cheby-shevskii Sb. Vol. 12, No. 1. P. 129−139.
6. Mergelyan, S. N. 1952, & quot-Uniform approximations to functions of a complex variable& quot-, Usp. Matem. Nauk. Vol. 7, No. 2. P. 31−122 (Russian) = Amer. Math. Trans. 1954. Vol. 101.
7. Montgomery, H. L. 1971, & quot-Topics in Multiplicative Number Theory. "-, Lecture Notes in Math. Vol. 227. Berlin: Springer.
8. Steuding, J. 2007, & quot-Value-Distribution of L-functions. "-, Lecture Notes in Math. Vol. 1877. Berlin, Heidelberg: Springer-Verlag.
9. Voronin, S. M. 1975, & quot-Theorem on the & quot-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.
10. Voronin, S. M. 1975, & quot-The functional independence of Dirichlet L-functions& quot-, Acta Arith. Vol. 27. P. 493−503 (Russian).
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Bagchi B. The Statistical Behaviour and Universality Properties of the Riemann Zeta-function and Other Allied Dirichlet Series. Ph. D. Thesis. Calcutta: Indian Statistical Institute, 1981.
2. Billingsley P. Convergence of Probability Measures. New York: Wiley, 1968.
3. Dubickas A., Laurincikas A. Joint discrete universality of Dirichlet L-functions // Archiv Math. 2015. Vol. 104. P. 25−35.
4. Heyer H. Probability Measures on Locally Compact Groups. Berlin, Heidelberg, New York: Springer-Verlag, 1974
5. Laurincikas A. On joint universality of Dirichlet L-functions // Чебышевский сборник. 2011. Т. 12, вып. 1. P. 129−139.
6. С. Н. Мергелян Равномерные приближения функций комплексного переменного // УМН 1952. Т. 7, №. 2. С. 31−122 = Amer. Math. Trans. 1954. Vol. 101.
7. Montgomery H. L. Topics in Multiplicative Number Theory. Lecture Notes in Math. Vol. 227. Berlin: Springer, 1971.
8. Steuding J. Value-Distribution of L-functions. Lecture Notes in Math. Vol. 1877. Berlin, Heidelberg: Springer-Verlag, 2007.
9. Воронин С. М. Теорема об & quot-универсальности"- дзета-функции Римана // Изв. АН СССР. Сер. матем. 1975. Т. 39. С. 475−486. = Math. USSR Izv. 1975. Vol. 9. P. 443−453.
10. Воронин С. М. Функциональная независимость L-функций Дирихле // Acta Arith. 1975. Vol. 27. P. 493−503.
Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-3 225 Vilnius, Lithuania.
Institute of Informatics, Mathematics and E-studies, Siauliai University, P. Visinskio str. 19, LT-77 156, Siauliai, Lithuania.
E-mail: antanas. laurincikas@mif. vu. lt, korsakiene@fm. su. lt, siauciunas@fm. su. lt Получено 18. 02. 2015

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