О периодической дзета-функции Текст научной статьи по специальности «Математика»

УДК 519.14

ON THE PERIODIC ZETA-FUNCTION

M. Stoncelis, D. SiauCiunas (Siauliai, Lithuania)

Abstract

We present an universality theorem for the periodic zeta-function which is defined by a Dirichlet series with periodic coefficients satisfying a certain dependence condition. This simplifies the problem and allows to elucidate the universality of the periodic zeta-function.

Keywords: analytic function, Dirichlet series, periodic zeta-function, universality.

О ПЕРИОДИЧЕСКОЙ ДЗЕТА-ФУНКЦИИ

М. Стонцелис, Д. Шяучюнас (г. Шяуляй, Литва)

Аннотация

В статье доказана теорема универсальности для периодической дзета функции, которая опредеяется рядом Дирихле с периодическими коэффициентами, удовлетворяющими некоторому условию зависимости. Это упрощает задачу и разрешает осветить универсапьность периодической дзета функции.

Ключевые слова: аналитическая функция, периодическая дзета функция, ряд Дирихле, универсальность.

1. Introduction

Let s = a + it be a complex variable, and a = {am : m E N} be a periodic sequence of complex numbers with minimal period q E N. The periodic zeta-function Z(s; a) is defined, for a > 1, by the series

c(s; a) = £ ^

m=l

In view of the periodicity of the sequence a, we have that, for a > 1,

1 q

c (s; a) = q-sHaiC (s 0'

where C(s; a) is the classical Hurwitz zeta-function with parameter a, 0 < a ^ 1, which is defined, for a > 1, by the series

<x

c (s,a) = Y1

n(m + a)-'

m=0 v '

and is continued analytically to the whole complex plane, except for a simple pole at the point s = 1 with residue 1. Therefore, equality (1) gives a meromorphic

continuation for C(s; a) to the whole complex plane. If

q

def 1 V^ n a = - } ai = 0,

H i=i

then the function C(s; a) is entire one. Otherwise, C(s; a) has a simple pole at the point s = 1 with residue a.

Let x be a Dirichlet character modulo q, and L(s,x) denote the corresponding Dirichlet L-function. It is well known, that for (b,q) = 1,

cq) = 4 £ xm(s.x).

X=x(modq)

where summing runs over all p(q) Dirichlet characters modulo q, and p(q) is the Euler function. Therefore, denoting by (l,q) the greatest common divisor of the numbers l and q, we find from (1) that

1 JL / «

z (s; a) = - / , ^jZHL

¿ at((s, 4A

ti V J

i aM £ *((fe) l(s,x)

1 v- < t^D )

X_x(mod jf^ )

= £ k^W £ 4 и)L(sX (2)

X=x(mod (q )

In [9] J. Steuding, assuming that q > 2, am is not a multiple of Dirichlet character modulo q, and that am = 0 for (m, q) > 1, obtained the universality of the function Z(s; a). Let D = {s E C : 2 < a < 1}, K be the class of compact subsets of the strip D with connected complements, and H(K) and H0(K), K E K, be the classes of continuous functions on K and of continuous non-vanishing functions on K, respectively, which are analytic in the interior of K. Then J. Steuding proved the following statement, Theorem 11.8 of [9].

Теорема 1. Suppose that q and a are as above. Let K E K and f (s) E H(K). Then, for every £ > 0,

liminf ^meas < т E [0,T] : sup\((s + ir; a) - f (s)| < e \ > 0.

т^ж T [ seK J

Here and in the sequel, measA denotes the Lebesgue measure of a measurable set A с R.

If am = c, m E N, with c E C \ {0}, then the sequence a is periodic with q =1. In this case, we have that

Z(s; a) = cZ(s). (3)

If am is a multiple of a Dirichlet character x modulo q, i.e., am = cx(m), m E N, with a certain constant c E C \ {0}, then, clearly,

Z(s; a) = cL(s,x). (4)

Since the functions Z(s) and L(s,x) are universal in the Voronin sense [1, 5, 9, 11], we have that, in the cases (3) and (4), the function Z(s; a) is also universal.

Теорема 2. Suppose that am = c = 0 or am is a multiple of Dirichlet character modulo q. Let K E K and f (s) e H0(K). Then, for every £ > 0,

liminf ^meas < т E [0,T] : sup\((s + ir; a) - f (s)\ < e \ > 0. T [ seK J

We observe that the approximated function f (s) in Theorem 2 is different from that in Theorem 1: in Theorem 1, f (s) is not necessarily non-vanishing.

The aim of this note is to consider an example of the function ((s; a) with prime q, where

1 9-1

= ^qy Eai- (5)

This example elucidates the situation.

We say that ((s; a) is universal if the inequality of universality

liminf -meaJ т e [0,T] : sup\((s + ir; a) - f (s)\ < el > 0 т^ж T { s&K J

with every e > 0 is satisfied for all K e K and f (s) e H0(K). If this inequality is satisfied for all K e K and f (s) e H(K), we say that ((s; a) is strongly universal. Let, for brevity,

q-1

b(q,x) = ^2 ai x(l), x = x(modq)-

1=1

We suppose that am ^ 0, m e N. Then the following statement is true.

Теорема 3. Suppose that q is a prime number, and that the periodic sequence a = {am : m e N} with minimal period q satisfies equality (5). 1° If the sequence a satisfies at least one of hypotheses

i) am = c, m e N;

ii) am is a multiple of a Dirichlet character modulo q;

iii) q = 2;

iv) only one number b(q,x) = 0, then the function ((s; a) is universal.

2° If at least two numbers b(q, x) = 0, then the function ((s; a) is strongly universal.

The proof of assertion 2° of Theorem 3 is based on the Voronin theorem on joint universality of Dirichlet L-functions.

2. Voronin theorem

We remind that two Dirichlet characters are equivalent if they are generated by the same primitive characters.

Теорема 4. Suppose that xi,---,Xr are pairwise non-equivalent Dirichlet characters and L(s,x1), ■ ■ ■, L(s,xr) are the corresponding Dirichlet L-functions. For j = 1,... ,r, let Kj e K, and fj (s) e H (Kj). Then, for every e > 0,

1

< r e [0,T] : sup sup \L(s + ir; xj) - fj(s)\ < e \

I Kj^r seKj I

liminf —mea^ t £ [0,T] : sup sup \L(s + ir; Xj) _ fj(s)| < W > 0.

Theorem 4, for circles in place of the sets Kj, was obtained by S.M. Voronin and applied for the functional independence of Dirichlet L-functions in [10]. A full proof of this case is given in [4]. The Voronin theorem in the form of Theorem 4 can be found in [9] and [6]. Modifications of Theorem 3 also were obtained by S.M. Gonek [3] and B. Bagchi [1, 2].

A very good survey on universality of zeta and L-functions is given in [7].

3. Mergelyan theorem

Approximation theory of analytic functions is one of the most important fields of mathematics, and has a long and rich history. In universality of zeta-functions, a very useful is the Mergelyan theorem on the approximation of analytic functions by polynomials. This theorem is a generalization of results of many authors, and is a final point in the field.

We state the Mergelyan theorem in a convenient for us form.

Teopema 5. Suppose that K C C is a compact subset with connected complement, and that F(s) is a function continuous on K and analytic in the interior of K. Then, for every £ > 0, there exists a polynomial p(s) such that

sup \f(s) — p(s)| < £.

s&K

Proof of Theorem 5 is given in [8], see also [12].

4. Proof of Theorem 3

The cases i) and ii) of assertion 1° are contained in Theorem 2. Since q is prime, we have that (l,q) = 1 for l = 1,... ,q — 1, and (l, q) = q for l = q. Therefore, we deduce from (2) that

1 —

Z (s; a) = qs E x(1)L(s,x) + a E x(l)L(s,x)

X=x(mod1) 1=1 x=x(modq)

1 q-i

= Of Z (s) + ai E x(l)L(s,x). (6)

l=1 x=x(modq)

It is well known that if xo is the principal character modulo q, and p denotes a prime number, then

L(s,Xo) = Z(s)n(1 — f) = Z(s) (1 —

p\q

in our case. This and (6) show that

1 ( 1 — ^ z(s) —

Z(s; a) = * ^ — -q) E ai) Z(s) + -q) E ai

1 q- 1 _

+ -qyEai E x(l)L(s,x)

l= 1 X=x(modq)

Z ( ) q-1 1 q-1

= —| ^01 + W) ^a' ^ X(DL(s,x) (7)

1=1 1=1 x=x(modq)

X=X0

in view of (5). In the set {x : X = x(m°dq)}, replace the principal character modulo q by the character x(m°d1), and preserve the notation x = x(modq). Then (7) we can rewrite in the form

l q-1 l

Z (s; a) = piqj^ a ^ X(l)L(s,X) = —q) Y1 L(s,X)b(q,X)- (8)

l=1 X=x(modq) x=x(modq)

Now we consider the case iii) of 1°, i.e., q = 2. By (8), we find that, in this case,

Z(s; a) = Z(s)b(2,x) = aiZ(s).

Thus we obtain the case i).

We note that at least one number b(q,x) in (8) is non-zero. Suppose that only one of the numbers b(q,x) is non-zero. Let b(q,x) = 0. Then, by (8),

1 1 q-1 Z (s; a) = —q)Z (sS)b(q^ = —q)Z (s) £ 01-

Thus,

l q- 1

a1 = • • • = aq-1 =

—(q) 1=1

y^ai = Oq,

and we have again the case i).

Now let b(q, x) = 0 for some x = x(modq). Then (8) gives

1 q-1

Z(s;

a) = —q) L(s,x) £ ai x(l)

Hence,

q— 1

ai = X(l)Y] aiX(l) —(q) t!

for all l e N, and

_ _ 1 q—1 _

a1X(1) = • • • = aq—1X(q - 1) = Yl aix(l)-

Thus, al = a1 x(l), l £ N, and we have the case ii).

It remains to prove 2°. Denote by H(D) the space of analytic functions on the strip D equipped with the topology of uniform convergence on compacta. Preserving

the above notation, i.e., in place of x0(modq) using x(mod1), we define the operator F : Hv(q)(D) — H(D) by the formula

F (g x(s) : x(modq)) — ^q) gx(s)b(Q,x),

X=x(modq)

(gx(s) : x(modq)) E H^(q)(D).

First we will prove that, for every K e K and polynomial p = p(s), there exists (gx(s) : x(modq)) e F-1{p} such that gx(s) = 0 on K for all x(modq). Suppose that

b(q,Xj) = 0, j = 1, 2.

Since the set K is bounded, there exists a constant C E C such that

p(s) + C = 0 on K,

and

C

v(q)

Y b(q,x) = o.

X=x(modq) X=X1X2

We take

gxi (s) = 'Aq)b-i(q,xi)(p(s) + C),

(

gx2 (s) = Ф)ь (q,X2)

C

q)

Y1 b(q,x)

V

x=x(modq)

x=xi ,x2

/

and gx(s) — 1 for x — Xi, Then we have that gx(s) — 0 on K for all x — x(modq), and

F (gx(s): x — x(modq)) — p(s)■

Let, for brevity,

q-i

M — Y, \ai\,

i=i

and let t e R satisfy the inequality

sup sup\L(s + ir,x) - gx(s)\ < 2M'

X=x(m°dq) s£K 2M

where gx(s) has the above properties. Then, for such t, by (8) suP K(s + iT; a) - P(s)\

s&K

1

1

sup \F (L(s + ir,x) : X = X(modq)) — F (9x(s) : x = x(modq))\

seK

M

^ sup s&K -(q)

E \L(s + iT,x) — g%(s)\

X=x(modq)

£

^ sup sup\L(s + iT,x) — gx(s)\ < 2•

X=x(modq) s£K 2

:iq)

Since q is prime, the characters x = x(modq), where x0 is replaced by x, are pairwise non-equivalent. Therefore, by Theorem 1, the set of t E R satisfying (9) has a positive lower density. However, (9) implies (10). Thus

{

liminf — meas { t E [0, T] : sup \Z(s + iT; a) — p(s)\ < - 1 > 0 t^^ T | S^K 2 J

s&K

2

11)

for every polynomial p(s).

It remains to replace the polynomial p(s) by f (s). By the Mergelyan theorem (Theorem 5), we can find a polynomial p(s) such that

If t e R satisfies

then, in view of (12),

This shows that

!

sup \f(s) — p(s)\ < s&K 2

sup\Z(s + iT, a) — p(s)\ < ^ s&K 2

sup \Z(s + -it, a) — f (s)\ <£.

s&K

t E [0,T] : sup \Z(s + iT, a) — p(s)\ <-

s&K 2

12)

}

{

c t e [0, T] : sup\Z(s + iT, a) — f (s)\ < -

s&K 2

Then, by (11),

1

liminf ^meas < t E [0,T] : sup\Z(s + iT; a) — f (s)\ < £ [> > 0.

T^ T { s&K

. )

The theorem is proved.

REFERENCES

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3. Gonek S.M. Analytic properties of zeta and L-functions. Ph. D. Thesis. University of Michigan, 1979.

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

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Шяуляйский университет, Литва.

Siauliai University, Lithuania. Получено 12.09.2014