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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 15 Выпуск 1 (2014)
УДК 519.14
О НУЛЯХ НЕКОТОРЫХ ФУНКЦИЙ, СВЯЗАННЫХ С ПЕРИОДИЧЕСКИМИ ДЗЕТА-ФУНКЦИЯМИ
А. Лауринчикас (г. Вильнюс, Литва), М. Стонцелис, Д. Шяучюнас (г. Шяуляй, Литва)
Аннотация
В статье полученно, что линейная комбинация периодической дзета-функции и периодической дзета-функции Гурвица и более общие комбинации этих функций имеют бесконечно много нулей, лежащих в правой стороне критической полосы.
Ключевые слова: нули аналитической функции, периодическая дзета-функция, периодическая дзета-функция Гурвица, универсальность.
ON THE ZEROS OF SOME FUNCTIONS RELATED TO PERIODIC ZETA-FUNCTIONS
A. LaurinCikas (Vilnius, Lithuania), M. Stoncelis, D. SiauCiunas (Siauliai, Lithuania)
Abstract
In the paper, we obtain that a linear combination of the periodic and periodic Hurwitz zeta-functions, and more general combinations of these functions have infinitely many zeros lying in the right-hand side of the critical strip.
Keywords: periodic zeta-function, periodic Hurwitz zeta-function, universality, zeros of analytic function.
1. Introduction
Let s = a + it be a complex variable, and let Z(s) and Z(s, a) with 0 < a ^ 1
denote the Riemann and Hurwitz zeta-functions, respectively. In this paper, we deal with generalizations of the functions Z(s) and Z(s,a). Let a = {am : m G N} and b = {bm : m G N0 = N U {0}} be two periodic sequences of complex numbers with minimal periods k G N and l G N, respectively. The periodic zeta-function Z(s; a) and periodic Hurwitz zeta-function Z(s, a; b) are defined, for a > 1, by the Dirichlet series
which are valid for a > 1, have analytic continuation to the whole complex plane, except for possible simple poles at the point s =1. Clearly, ((s; a) = Z(s) for am = 1, and Z(s, a; b) = Z(s, a) for bm = 1.
The distribution of zeros of the function Z(s; a) was considered in [18], see also [20]. Define
ca = max(|am| : 1 ^ m ^ k), ma = min{1 ^ m ^ k : am = 0},
Then in [18], it was obtained that Z(s; a) = 0 for a > 1 + A(a). Moreover, for a < —B(a), the function Z(s; a) can only have zeros close to the negative real axis if ma+ = ma-, and close to the straight line given by the equation
and
and, in view of the equalities
a± = {am : m G N}
and
B(a) = max {A(a±^ .
if ma+ = ma-.
Denote by p = /3 + iY the zeros of the function Z(s; a). The zeros with /3 < — B(a) are called trivial. The number of trivial zeros p with |p| ^ R is asymptotically equal to cR with some c = c(a) > 0. Other zeros of Z(s; a) are called non-trivial, and, by the above remarks, they lie in the strip — B(a) ^ a ^ 1 + A(a).
Let N(T; a) be the number of non-trivial zeros p of Z(s; a) with |y| ^ T. Then
[18]
T kT
N(T; a) = - log ----------- + O(logT).
n 2nem^ma- ma+
Moreover, the non-trivial zeros of ((s; a) are clustered around the critical line a =
In [15], it was obtained that the functions F(Z(s; a)) for some classes of operators F of the space of analytic functions have infinitely many zeros in the strip | < a < 1.
The paper [2] is devoted to zeros of the function Z(s,a; b). From properties of Dirichlet series, it follows that there exists ai > 0 such that Z(s, a; b) = 0 for a > ai. For simplicity, suppose that b0 = 1, and
z-i
q^im) = J>exp {±2nim^Y1}
k=0
Denote by p(s, l) the distance of s from the line l on the complex plane, and let, for £ > 0,
Le(f) = |s G C : p(s, f) < £ j .
Then in [2], it is obtained that there exist constants a0 < 0 and £0 > 0 such that Z(s, a; b) = 0 for a < a0 and
s L£o ((a - 1) log - nt = log )
where r1 = min{m G N : q+(m) = 0} and r2 = min{m G N : q- = 0}. Using the above result, non-trivial zeros of Z(s, a; b) are defined. Namely, the zero p = /3 + iY of Z(s, a; b) is called non-trivial if a0 ^ /3 ^ a1. The zero p is called trivial if
It is known that the function Z(s, a; b) has infinitely many trivial zeros.
Denote by N(T, a; b) the number of non-trivial zeros p of the function Z(s, a; b) with |y| ^ T according multiplicities. Then in [2], it was proved that
T Tk
N(T,a-,b) = — log ----------+ O(logT).
n 2nea
Moreover,
(P- 2) = -^lod + h (log |g+(ri)| + log |g-(r2)|) + O(logT).
IyI<t
The latter formula shows that the non-trivial zeros of the function Z(s, a; b) are clustered around the line a =
The aim of this paper is to show that the function Z(s, a; b) with some, for example,transcendental parameter a, and some combinations of the functions Z(s; a) and ((s,a; b) have infinitely many zeros in the strip D = {s G C : | < <r < l}. Denote by (i2, c) the assertion that, for any <7i,<t2, \ < o\ < <r2 < 1, there
exists a constant c = c(a1, a2, f) > 0 such that, for sufficiently large T, the function f (s) has more than cT zeros in the rectangle
a1 < a < a2, 0 < t < T.
Let
L(a) = {log(m + a) : m G N0} .
Theorem 1. Suppose that the set L(a) is linearly independent over the field of rational numbers Q. Then, for the function Z(s,a; b), the assertion AT(a1,a2,c) is true.
Define the function
((s, a; a, b) = Ci((s; a) + c2((s, a; b), cu c2 G C \ {0}.
Theorem 2. Suppose that the number a is transcendental, the sequence a is multiplicative, and, for each prime p, the inequality
OO
(i)
V2
m=1 1
is satisfied. Then, for the function ((s, a; a, b), the assertion At((ti, <t2, c) is true.
The next theorem is devoted to zeros of more general composite functions of ((s; a) and ((s, a; b). We recall that D = {s € C : | < cr < l}. Denote by H(D) the space of analytic on D functions equipped with the topology of uniform convergence on compacta, and H2(D) = H(D) x H(D). Let ^1 > 0 and ^2 > 0. We say that the operator F : H2(D) ^ H(D) belongs to the class Lip(^1,^2) if it satisfies the following hypotheses:
1° For each polynomial p = p(s), and any compact subset K C D with connected complement, there exists an element (g1,g2) G F-1{p} C H2(D) such that g1(s) = 0 on K;
2° For any compact subset K C D with connected complement, there exist a positive constant c, and compact subsets K1, K2 of D with connected complements such that
sup |F(g11(s),g12(s)) — F(g21(s),g22(s))| ^ c sup sup |g1j(s) — g2j(s)|^j s€K 1^i^2 seKj
for all (gr1,gr2) G H2(D), r = 1, 2.
Theorem 3. Suppose that the number a is transcendental, the sequence a is multiplicative, inequality (1) is satisfied and F G Lip(^1,^2). Then, for the function F(Z(s; a),Z(s,a; b)), the assertion AT(a1,a2,c) is true.
We note that the class Lip(^1,^2) is not empty. For example, in [6] it is proved that the operator F : H2(D) ^ H(D),
F (g1,g2) = c1#ifcl) + c2g(k2),
where c1, c2 G C \ {0}, k1, k2 G N and g(k) denotes the kth derivative of g, belongs to the class Lip(1,1). To prove this, it suffices to apply the integral Cauchy formula.
2. Lemmas
Proof of Theorems 1-3 are based on universality theorems for the corresponding functions, and the classical Rouche theorem. We remind that the universality of zeta-functions was discovered by S. M. Voronin who proved [21] an universality theorem for the Riemann zeta-function. For brevity, we denote by K the class of compact subsets of the strip D with connected complements, by H0(K), K G K, the class of non-vanishing continuous functions on K which are analytic in the interior of K, and by H(K), K G K, the class of continuous functions on K which are analytic in the interior of K. Let measA stand for the Lebesgue measure of a measurable set A C R. Then the latest version of the Voronin theorem is the following assertion, see, for example, [8].
Lemma 1. Suppose that K G K, and f (s) G H0(K). Then, for every e > 0, lim ^meas j r G [0, T] : sup \((s + ir) — /(s)| < e 1 >0.
T—too -L 1^ s^K J
The majority of other zeta and L-functions, among them the periodic zeta-function, [14], [5], the Hurwitz zeta-function with transcendental [10] or rational parameter [3], [1], the periodic Hurwitz zeta-function with transcendental parameter [4], zeta-functions of cusp forms [12], [13], L-functions from the Selberg class [19], [16], and others are universal in the Voronin sense. We state universality theorems for periodic and periodic Hurwitz zeta-functions.
Lemma 2. Suppose that the sequence a is multiplicative and inequality (1) is
satisfied. Let K G K, and f (s) G H0(K). Then, for every e > 0,
lim —meas | r G [0, T] : sup |£(s + ir; a) — f(s)\ < £ 1 > 0.
T —>oo T I s&K J
Proof of the lemma is given in [14].
Lemma 3. Suppose that the set L(a) is linearly independent over Q. Let K G K, and f (s) G H(K). Then, for every £ > 0,
lim —meas j r G [0, T] : sup |£(s + ir, a; b) — f(s) \ < £ 1 > 0.
T—>oo T [ s&K J
The lemma with transcendental parameter a has been obtained in [4], and, under hypotheses of the lemma, has been proved in [11].
In universality theory of zeta-functions, an important role is played by joint universality theorems when a collection of given analytic functions is approximated simultaneously by shifts of a collection of zeta-functions. The first joint universality result also was obtained by S. M. Voronin. In [22], investigating the functional independence of Dirichlet L-functions, he first of all infact obtained their joint universality. We remind a modern version of the Voronin theorem, see, for example, [9].
Lemma 4. Suppose that xi,---,Xr be pairwise non-equivalent Dirichlet characters, and L(s,x1), ■ ■ ■, L(s,xr) be the corresponding Dirichlet L-functions. For
j = 1,... ,r, let Kj G K, and fj(s) G H0(K). Then, for every £ > 0,
lim —meas |rG [0,T] : sup sup |L(s + ir,Xj) ~ fj(s)I < £ ^ > 0.
T—s-oo J- I 1 s£Kj
The joint universality of the periodic zeta-function and the periodic Hurwitz zeta-function has been considered in [6], and the following assertion has been proved.
Lemma 5. Suppose that the sequence a is multiplicative, inequality (1) is satisfied, and the number a is transcendental. Let K1,K2 G K, and f1(s) G H0(K1) and
f2(s) G H(K2). Then, for every £ > 0,
lim —meas IrG [0,T] : sup |£(s + *r; a) — fi(s)\ < £ ,
T—too 1 1^ sGK\
sup |Z(s + ir, a; b) — f2(s)| < £ > > 0.
seK2 J
Now we state a generalization of Lemma 5 from the paper [7].
Lemma 6. Suppose that the sequence a is multiplicative, inequality (1) is satisfied, the number a is transcendental, and that F G Lip(p1} @2). Let K G K and f (s) G H(K). Then, for every £ > 0,
lim —meas It G [0, T] : sup |F (((s + ir; a), ((s + ir, a; b)) — f(s)\ < £ 1 >0.
T—too 1 I s£:K J
For the proof of theorems on the number of zeros of zeta-functions and their certain combinations, the classical Rouche theorem is useful. For convenience, we state this theorem as a separate lemma.
Lemma 7. Let the functions g1(s) and g2(s) are analytic in the interior of a closed contour L and on L, and let on L the inequalities g1(s) = 0 and |g2(s)| < |g1(s)| be satisfied. Then the functions g1(s) and g1(s) + g2(s) have the same number of zeros in the interior of L.
Proof of the lemma can be found, for example, in [17].
3. Proofs of theorems
Proof of Theorem 1. Let
®1 + @ 2 ®2 — &1
1 r =-------------1
0 2 ’ 2 ’
and let the number £ > 0 satisfy the inequality
1 r
£< — min Is —<70| = —• (2)
10 k-aobr 1 U| 10 V 7
Suppose that t G R satisfies the inequality
sup |Z(s + iT,a; b) — (s — a0)| < £. (3)
|s-ooKr
Then, in view of (2), we have that the functions Z(s + iT, a; b) — (s — a0) and s — a0 in the disc | s — a0| ^ r satisfy the hypotheses of Lemma 7. Hence, the function Z(s,a; b) has a zero in the disc |s — a0| ^ r . Since, by Lemma 3, the set of t satisfying inequality (3) has a positive lower density, we obtain that there exists a constant c = c(a1,a2,a, b) > 0 such that for the function Z(s,a; b) the assertion a2, c) is true. □
Proof of Theorem 2. We preserve the notation for a0 and r, and take in Lemma 5
fi(s) = e, f2(s) = —(s — <70),
c2
where the positive number £ satisfies the inequality
1r (|ci| + |c2|)e<— min \s - (701 = —. (4)
10 |s-(70\=r 10
Suppose that t G R satisfies the inequalities
suP |Z(s + iT; a) — f1(s) | < £ (5)
|s-CTo|^r
and
sup |Z(s + iT,a; b) — f2(s)| <£. (6)
|s-^o|^r
Then, for these t, we have that
sup |(c1Z(s + iT; a) + C2Z(s + iT,a; b)) — (c1f1(s) + c^f2(s))|
|s-^o|^r
< 2(|c1| + |c2 |)£.
Moreover, by the definition of f1(s) and f2(s),
sup | c1 f 1 (s) + c^f2(s) — (s — ao)| = |c11£.
|s-^o|^r
Therefore,
sup |(c1Z(s + iT; a) + c2Z(s + iT, a; b)) — (s — oo)| < 3(|c11 + |c21)£.
|s-cto|=P
This and (4) show that the functions
c1Z(s + iT; a) + c2Z(s + iT, a; b)) — (s — oo)
and s — o0 on the disc |s — o0| ^ r satisfy the hypotheses of Lemma 7. Therefore, the function c1Z(s + iT; a) + c2Z(s + iT,a; b)) has a zero in the disc |s — o0| ^ r. However, by Lemma 5, the set of t satisfying inequalities (5) and (6) has a positive lower density. Hence, there exists a constant c = c(o1, o2,a, a, b) > 0 such that, for the function Ci£(s + %t\ a) + c2((s + ir, a; b)), the assertion AT(ai, a2, c) is valid. □
Proof of Theorem 3. We argue similarly as above. Suppose that t G R satisfies the inequality
sup |F(Z(s + iT; a),Z(s + iT,a; b)) — (s — oo)| < £. (7)
|s-^o|^r
and £ satisfies (2). Then the functions
F(Z(s + iT; a), Z(s + iT, a; b)) — (s — 00)
and s — o0 in the disc |s — o01 ^ r satisfy the hypotheses of Lemma 7. Therefore, the function F(Z(s + iT; a), Z(s + iT,a; b)) has a zero in the disc |s — o0| ^ r. However, in view of Lemma 6, the set of t satisfying inequality (7) has a positive lower density. Thus, there exists a constant c = c(o1, o2,a, a, b, F) > 0 such that, for the function F(((s + ir; a), ((s + ir, a; b)), the assertion AT(a1} a2, c) is valid. □
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Вильнюский университет (Литва) Шяуляйский университет (Литва) Поступило 06.02.2014
