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ЧЕБЫШЕВСКИЙ СБОРНИК Том 12 Выпуск 2 (2011)
Труды VIII Международной конференции Алгебра и теория чисел: современные проблемы и приложения, посвященной 190-летию Пафнутия Львовича Чебышева и 120-летию Ивана Матвеевича Виноградова
JOINT UNIVERSALITY FOR ZETA-FUNCTIONS OF DIFFERENT TYPES
1 Introduction
The Voronin universality theorem [29] on the approximation of analytic functions by shifts of the Riemann zeta-funetion Z(s), s = a + it, opened a new direction in analytic number theory. It turned out that other zeta and L-funetions also have the universality property, and this was confirmed by many authors. We state universality theorems of different type for the function Z (s) and the Hurwitz zeta-funetion Z (s, a),
0 < a < 1. The functions Z(s) and Z(s, a) are defined, for a > 1, by
ANTANAS LAURINCIKAS*
Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania Faculty of Mathematics and Informatics, Siauliai University, P, Visinskio 19, LT-77156, Siauliai, Lithuania e-mail: [email protected]
RENATA MACAITIENE
Faculty of Mathematics and Informatics, Siauliai University, P, Visinskio 19, LT-77156, Siauliai, Lithuania e-mail: renata,macaitiene@mi,su.lt
DARIUS SIAUCIUNAS
Faculty of Mathematics and Informatics, Siauliai University, P. Visinskio 19, LT-77156, Siauliai, Lithuania e-mail: [email protected]
and
* Partially supported by grant No. MP-94 from the Research Council of Lithuania
and have analytic continuations to the whole complex plane, except for a simple poles at the point s = 1 with residue 1, Fo r D = {s G C : 2 < a < 1}, denote by K the set of compact subsets of strip D with connected complements. Moreover, for K G K, let H(K) be the set of continuous functions on K which are analytic in the interior of K, and let H0(K) be the subset of H(K) non-vanishing functions on K, As usual, meas{A} stands for the Lebesgue measure of a measurable set A C R. The Voronin universality theorem can be stated in the following form.
Theorem A. Suppose that K G K and f (s) G H0(K). Then, for every £ > 0, liminf ^meas { r G [0,T] : sup |Z(s + ir) — f (s)| < £,> > 0.
[ seK J
Theorem A is a general form of the Voronin theorem whose initial version [29]
D
Now we recall the universality theorem for the Hurwitz zeta-funetion Z(s,a).
Theorem B. Suppose that a is a transcendental or rational number not equal to 1 or 2. Let K G K and f (s) G H(K). Then, for every £ > 0,
liminf —meas < r G [0, T] : sup |Z(s + ir, a) — f (s)| < ^ > 0.
T^~ T I s€K )
First Theorem B was obtained independently by S, M, Gonek and B, Bagchi in their theses [5] and [1], respectively. Its proof is also given in [16],
We see that Theorems A and B differ one from another by the set of approximated functions: H0(K) C H(K), The universality of Theorem B often is called a strong universality. So, Theorems A and B present two different types of universality.
The cases Z(s, 1) = Z(s) and
Z (s. 1) = (2s — 1)Z(s)
are not included in Theorem B because these functions are universal but not strongly universal,
a
Z(s, a)
The most interesting form of universality is the joint universality. In this case, we deal with simultaneous approximation of more than one analytic functions. The first result in this direction is also proved by Voronin a joint universality theorem [30] for Diriehlet L-funetions, Let x be a Dirichlet character modulo q. Then the corresponding L-function L(s, x) is defined, for a > 1, by
L( ) ^ X(m)
L(s.x) = 2. ^
m=1
and by analytic continuation elsewhere. If % is a non-principal character, then the function is entire, while in the case of the principal character, the function has a simple pole at the point s = 1 with residue
We remind that two Dirichlet characters are called equivalent if they are generated by the same primitive character.
Now we state a modern version of the Voronin theorem.
Theorem C. Suppose that xi. — .Xr- are pairwise non-equivalent Dirichlet characters, and L(s, x1),..., L(s, Xr) are corresponding Dirichlet L-functions. For j = 1,..., r, let Kj G K and fj(s) G H0(Kj). Then, for every £ > 0,
Theorem C by different methods also was obtained by S, M, Gonek [5] and B, Bagchi [1], [2].
L
usually they are of one type, i, e,, the approximated functions fj (s) G H0(Kj) or fj (s) G H(Kj), The joint universality for Hurwitz zeta-funetions was discussed in [27] and [14], for Lerch zeta-functions in [18], [21] and [16], for twists with character L
give a short survey on the joint universality for zeta-functions of different types, i, e,, connecting simultaneously the approximation of functions from sets H0(Kj) and H(Kj),
2 Joint universality of Z(s) and Z(s,a)
The paper of H, Mishou [26] is the first work on the joint universality for zeta-functions of different types. For brevity, further we will call this universality a mixed joint universality,
a Ki , K2 G
K, and f1(s) G H0(K1) and f2(s) G H(K2). Then, for every £ > 0,
liminf —meas < t G [0, T] T I
SUPs€Ki |Z(s + iT) - f1(s)| <^ SUPs€K2 |Z(s + iT,a) - f2(s)| <£
A proof of Theorem 1 is based on the linear independence over the field of rational numbers of the set
{(logp : p G P), (log(m + a), m G N0)}, (1)
where P and N0 denote the sets of all primes and non-negative integers, respectively. For a region G on the complex plane, denote by H(G) the space of analytic functions on G endowed with the topology of uniform convergence on compacta. Then, using the linear independence of the set (1), it is deduced by a standard method that the probability measure
—meas {t G [0,T]: (Z(s + ir),Z(s + ir, a)) G A} , A G B(H2(D)),
where B(S) denotes the a-field of Borel sets of the space S, converges weakly to a probability measure P on (H2(D), B(H2(D))) as T ^ ro, Moreover, the support of the measure P is the set
{g G H(D) : g(s) = 0 or g(s) = 0} x H(D).
The above facts are sufficient to prove Theorem 1,
3 Periodic zeta-funetions
The functions Z(s) and Z(s, a) have their natural generalizations. Let a = {am : m G N} and b = {bm : m G N0} be two periodic sequences of complex numbers with minimal periods k G N and l G N, respectively. Then generalizations of Z(s) and Z(s, a) are, respectively, the periodic zeta-function Z(s; a) and the periodic Hurwitz zeta-function Z(s,a; b), defined, for a > 1, by
Z(s;a) = S «"s and Z(s,a;b) = S .
ms (m + a)s
m=1 m=0 v 7
The periodicity of sequences ^d b, for a > 1, implies the equalities
1 k
Z (s;a) = k^s S a™Z ^, m
ks ^ v > k
m=1
and
z- l
ls ’ l
m=0 x
Z(s; a) Z(s, a; b)
complex plane, except, maybe, for simple poles at the point s = 1 with residues
1 k i 1—1
and j ^ bm,
m=1 m=0
respectively. If the later quantities are zeros, then the functions Z(s; fl^d Z(s, a; b) are entire.
The universality of the function Z (s; a) with a multiplicative sequence is given in
[25], [28] and is similar to Theorem A, The general case was obtained in [9], and is of the form,
a
exists a positive constant c0 = c0(a) such that, for every K e K with, height h(K) = maxIm(s) — minlm(s) < co, f (s) e H(K), and every e > 0,
s€K s€K
liminf -meas< r e [0, T] : max |Z(s + ir, a) — f (s)| < e > > 0.
T^ro T I sEK I
The universality of the function Z(s,a; b) with transcendental parameter a is proved in [6], and completely coincides with Theorem B,
The joint universality for periodic zeta-functions with multiplicative coefficients was discussed in [17], Suppose that, for j = 1,... , r1; aj = {amj : m e N} is a periodic sequence of complex numbers with minimal period kj, and Z(s; aj) is the corresponding periodic zeta-funetion. Denote by k = [k1,..., kri] the least common multiple of the periods k1,..., kri, and b yu1,..., the reduced system of residues modulo k, where ^(k) is the Euler function. Define the matrix
A
aui2
aU2 2
\ a«^(fc)1 a«^(fc)2
^uir
\
/
Theorem 3 ([17]). Suppose that the sequences a1,..., ari are multiplicative, rank(A) = r1t and that, for all primes p, the inequalities
£< cj < i, j = 1,
1=1 P2
. r1,
(2)
hold. For j = 1,..., r1; fet Kj e K, and fj(s) e H0(Kj). Then, for every e > 0, liminf — meas < r e [0,T] : sup sup |Z(s + ir; aj) — fj(s)| < ^ > 0.
T^1 1<j<ri sEKj
The joint universality for periodic Hurwitz zeta-functions was studied in a series of works [11]-[13], [7], [23] and [24], For j = 1,...,r, let 0 < aj < 1 and bj = {bmj : m e N0} be a periodic sequence of complex numbers with minimal period /j, and let Z(s, aj; aj) denote the corresponding periodic Hurwitz zeta-funetion. The simplest case /j = / and aj = a j = 1,..., r, was considered in [11], The hypothesis that /j = /, j = 1,..., r, was removed in [12], The general case with algebraically
a
independent over the field of rational numbers Q numbers a^..., ar and a certain rank condition on the coefficients bmj was studied in [7]. A rank condition was removed in [23], and the following theorem was proved. We set
L(a1,..., ar) = {log(m + aj) : m G N0 = N U {0}, j = 1,..., r} .
Theorem 4. Suppose that the set L(a1,... , ar) is linearly independent over Q. For j = 1,..., r, let Kj G K and /(s) G H(Kj). Then, for every e > 0,
liminf — meas < r G [0,T] : sup sup |Z(s + ir, aj; bj) — fj(s)| < e > > 0.
T T I i^jXrsGKj I
Clearly, the linear independence of the set L(a1,...,ar) holds whenever the numbers a1,..., ar are algebraically independent over Q, Therefore, the hypothesis on the numbers a1,..., ar in Theorem 4 is weaker than that in [7],
For periodic Hurwitz zeta-functions, a more general case of the joint universality can be considered. For j = 1,...,r, let j be a positive integer and 0 < aj < 1, for j = 1,...,r, 1 = 1,...,j, let bji = {bmj7 : m G N0} be a periodic
sequence of complex numbers with minimal period j G N, and 1 et Z(s,aj; bjl) denote the corresponding periodic Hurwitz zeta-funetion. It is clear that, for the joint universality, the functions Z(s,aj; bjl) must be independent in a certain sense. In the case of Theorem 4, this independence is expressed by linear independence over Q of the set L(a1,...,ar), The case of the functions Z(s,aj; bjl) is a more complicated, thus some additional conditions on the coefficients bmjl are needed. Denote by L the least common multiple of the periods 111,..., 11ll,..., 1r1,..., /rlr, and define the matrix
B
Moreover, let
( bm b112 . . b11li b122 . . b12l2 . . b1r1 b1r2 . . b1rlr ^
b211 b212 . . b21li b222 . . b22l2 . . b2r1 b2r2 . . b2rlr
V bL11 bL12 . b 1. bL22 . . &L2l2 . . &Lr1 bLr2 . . bLrlr /
K
E (j.
j=1
Then [13] contains the following result. Theorem 5. Suppose that the set
{log(m + aj) : m G N0, j = 1,.
is linearly independent over Q,and that rank(B) = k For j = 1, 1,..., j, le t Kj7 C K an d fjl(s) G H (Kj ). Then, for e very e > 0,
1
liminf —meas < r G [0,T] : sup sup sup |Z(s + ir, a,-; b,7) — /77(s)| < ^ > 0.
tT 1 -- 1
1<j<r 1<l<lj sEKji
It turns out that the rank condition in Theorem 5 can be made weaker. For = 1,..., r, let Lj be the least common multiple of the periods Zji,..., j., and
( bi,i b2j1
b
1,2
b2,2
\ bLjl bL,2
b1,1j
b2j1j
bL flj
\
/
Theorem 6. Suppose that the set (2) is linearly independent over Q and that rank(Bj) = j = 1,..., r. Let Kji and /^(s) be the same as in Theorem 5, Then
the assertion of Theorem 5 is true.
j
4 First generalizations of Theorem 1
Theorem 1 on a mixed joint universality of classical zeta-funetions Z(s) and Z(s,a) can be generalized for zeta-funetions with periodic coefficients, and this was done in [8],
Theorem 7. Suppose that the sequence a is multiplicative such that, for every prime p,
f ¥ < c< 1, (3)
1=1 P2
and that the number a is transcendental. Let K1,K2 G K, an d f1(s) G H0 (K1), /2(s) G H(K2). Then, for every £ > 0,
i- • r 1 / ^ rn ^1 suPseKx lZ(s + zt;a) - /1 (s)| <£,
iimini — mea^ t G [0, T J: e 1 . >> 0.
T { ’ ' SUPs€K2 |Z (s + iT,a; b) - f2(s)| <£
The condition (3) is technical, and, in our opinion, should be removed. However, if, for some p,
1 + V -P- = 0, s G D,
^ pls i=i 1
then a problem arises, does /1(s) belong to H0 (K1),
The ideas of [8] have been developed in [15] for collections of periodic and periodic Hurwitz zeta-funetions.
Theorem 8 ([15]). Suppose that the sequences a1,..., ari are multiplicative, rank(A) = r1 and inequalities (2) hold, and that the numbers a1,..., ar2 are algebraically independent over Q. Fo r j = 1,... , r1; fe t Kj G K an d /j (s) G H0(Kj), and, for j = 1,..., r2, fe t Kj G K an d /j (s) G H (Kj). Then, for e very £ > 0,
iimlnf —meas<j t G [0,TJ : sup sup |Z(s + zt; a,) — /,(s)| < £,
sup sup IZ(s + zt, a,; b,) — /,(s)| < £ > > 0.
1<j<r2 seK,
Q
set L(a1,..., ar2) is not sufficient for the proof of Theorem 8 because we need the
Q
{(logp : p G P), (log(m + a) : m G N0, j = 1,... , r)}.
On the other hand, the linear independence of the latter set follows from the algebraic independence of the numbers a1,..., ar.
Now we present a generalization of Theorem 1 in the frame of Theorem 6,
Theorem 9 ([4]). Suppose that the numbers a1,..., ar are algebraically independent over Q, and that rank(Bj) = j j = 1,..., r. For j = 1,..., r and l = 1,..., j let Kj G K and let /jl(s) G H(Kj), and K G K and /(s) G H0(K). Then, for every
£>0
liminf -measi t G [0, T] : sup |Z(s + ir) — /(s)| < £,
sup sup sup |Z(s + zr, aj; ajl) — /j(s)| < £ > > 0.
1<j<r 1<j<lj s^Kji j
5 Zeta-functions of certain cusp forms
Let SL(2, Z), as usual, denote the full modular group, i, e,,
SL(2, Z) = | ^ b ^ : -, b, c, d G Z, —d — 6c = 11 .
Suppose that the function F()) is holomorphic in the upper half-plane Imz > 0, for
( — b \
some k G 2N and all ( c d ) G SL(2, Z), satisfies the functional equation
F t —Sd) = (cz+d>“F (z), (4)
and is analytic and vanishing at the cusps (rational points and to). Then F(z) is called a cusp form of weight k for the full modular group, and at to has the Fourier series expansion
œ
F (z) = f c(m)e2nimz
m=1
Moreover, we assume that F(z) is a normalized Hecke eigen cusp form, i, e,, F(z) is a simultaneous eigen function of all Hecke operators, and c(1) = 1,
Now let q G N, and let
r° = { ( C b ) G SL(2,Z) : c = 0(mod q)
be the Hecke subgroup of the full modular group. If F(z) satisfies the equation (4) fa b \
for all ( c d ) G r°, then F(z) is called a normalized Hecke eigen cusp form of weight k and level q. If F(z) is not a cusp form of level less then q, then it is called
Hb
fa b \
Now let x be a Dirichlet character modulo q, and let F(z), for all ( c ^ ) G r°,
satisfy the functional equation
F H ^ b = (cz + d)“x(d)F (z).
Then F(z) is called a cusp form of weight k and level q with character x- In the
F(z)
To a cusp form F(z) of weight k, we attach the zeta-funetion
z (s, f) = v cM, a>K±i,
sv ’ ; ^ ms ’ 2 ’
which has analytic continuation to an entire function.
6 Further generalizations of Theorem 1
In this section, we will present mixed joint universality theorems for the function Z (s, F). Le t = {s G C : K <a< }, Denote by the set of compact subsets
K C with connected complements. Moreover, for K G H0k(K) be the set
of continuous non-vanishing functions on K which are analytic in the interior of K, The universality of the function Z(s, F) when F is a normalized Hecke eigen cusp form was obtained in [19]. The universality for zeta-functions of newforms is given in [22|.
Theorem 10. Suppose that F(z) is a normalized Flecke eigen cusp form of weight k for the full modular group, and that the numbers a,..., ar are algebraically independent over Q, and rank(Bj) = lj. For j = l,...,r, and l = l,...,/j, let Kji G K and /j7(s) G H(Kj), and K G and f (s) G H0k(K). Then, for every £ > 0,
liminf — measj t G [0, T] : sup |Z(s + ir, F) — /(s)| < £,
sup sup sup |Z(s ± ¿t, aj; bj) — j(s)| < e > > 0.
1<j<r 1<l<lj s£Kji J
Theorem 11. Suppose that F(z) is a newform form of weight k and level q, and that a1,..., ar, bjl , Kjl; /^(s), K and /(s) are the same as Theorem 10, Then the assertion of Theorem 10 is true.
Theorem 12. Suppose that F(z) is a primitive cusp form of weight k and level q with character x o.nd that a1,..., ar, bj , Kjl; /j^(s), K and /(s) are ifte same as Theorem 10, Then the assertion of Theorem 10 is true.
In the cases of Theorems 10-12, the function Z(s, F) has a representation, for a > K+1, by the Euler product over primes (in each case this representation is different). This together with the method of positive densities [19] and Deligne’s estimates [3] allows to prove the universality for the function Z(s,F), Then Theorems 10-12 are derived by a method of [4],
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Received 12.10.2011
