A discrete limit theorem for the Mellin transforms of the Riemann zeta-function Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК Том 11 Выпуск 1 (2010)

Труды VII Международной конференции Алгебра и теория чисел: современные проблемы и приложения, посвященной памяти профессора Анатолия Алексеевича Карацубы

A DISCRETE LIMIT THEOREM FOR THE MELLIN TRANSFORMS OF THE RIEMANN ZETA-FUNCTION

Аннотация

A discrete limit theorem for the Mellin transforms of the Riemann zeta-function is proven.

1 Introduction

As usual, denote by Z (s), s = a + it, the Riemann zeta-funetion. In analytic number theory, the modified Mellin transforms Zk (s) of powers of the function Z(s) are considered. For k > 0 and a > a0(k) > 1, the function Zk(s) is defined by

In view of the Mellin inversion formula, the function Zk (s) is very useful for the investigations of power moments

of the Riemann zeta-function.

This paper is devoted to the asymptotic behavior of Zi(s).The basic properties of the function Z1(s) were obtained in [7]. In view of the well-known estimate I1(T) = O(T log T), the integral defining Z1(s) converges absolutely for a > 1. Therefore, the function Z1(s) is analytic in the half-plane {s G C : a > 1}. In [7], the function Z1(s) has been meromorphieally continued to the half-plane {s 6 C : u > -1}, Moreover, it was obtained that the point s = 1 is a pole of order two with residue 2j0 — log 2n, where j0 is the Euler constant, M. Jutila in [8] continued meromorphieally the function Z1(s) to the whole complex plane with possible poles of order at most two at the points s = — m, m G N Recently, M. Lukkarinen proved in [12] that Z1(s) has simple poles only at s = —(2m — 1), m G N. She also found the formulae for the residues at the above points.

V. Balinskaite , A. Laurincikas

In the paper [7], also the first estimates and mean square estimates for the function Z1(s) have been obtained.

The idea of application of probabilistic methods in the theory of the Eiemann zeta-function belongs to H, Bohr and B, Jessen [3], [4]. Later, Bohr-Jessen's theory was developed by many authors, for history and results, see [11].

Let meas{A] denote the Lebesgue measure of a measurable set A e R, and let B(S) stand for the class of Borel sets of the space S. The first probabilistic limit theorem for the function Z1(s) has been obtained in [10].

Theorem 1. Let a > Then on (C,£>(C)), there exists a probability measure Pa such, that the probability measure

converges weakly to P^ as T — to.

Theorem 1 is of so called continuous character, because the imaginary part t in Z1(a + it) varies continuosly in [0,T],

Also, a discrete limit theorem for the function 2q(s) is known [1]. In this theorem, the imaginary part t in Z1 (a + it) in the definition of the probability measure takes values in an arithmetical progression. Let, for N e N U {0},

where in place of dots a condition satisfied by m is to be written. Let h > 0 be a fixed number.

Theorem 2. [1] Let a > Then on (C, £>(C)), there exists a probability measure Pa such, that the probability measure

/n(Zi(a + imh) e A), A e B(C), converges weakly to Pa as N — to.

The aim of this note is to obtain a discrete limit theorem in the space of analytic functions for Z1(s). Let D = {s G C : \ < a < 1}. Denote by H(D) the space of analytic functions on D equipped with topology of uniform convergence on compacta.

Theorem 3. On (H(D), B(H(D))), there exists a probability measure P such, that the probability measure

/N(Zi(s + imh) e A), A e B(H(D)),

1

meas{t e [0,T] : Z1(a + it) e A}, A e B(C)

0 <m<N

1

converges weakly to P as N — to.

2 A limit theorem on a torus

Let y = {s G C : |s| = 1}be the unit circle on the complex plane. Define, for a > 1,

Q = II Yu,

u£[1,a]

where Yu = Y f°r aH u G [1, a]. Since y is a compact, by the Tikhonov theorem the torus Qa is a compact topological Abelian group.

Theorem 4. On (Qa, B(Qa)) there exists a probability measure Qa such, that the probability measure

QnAA) = VN((uimh : u G [1, a]) G A), A G B(Q), converges weakly to Qa as N ^

Proof. Let Z denote the set of all integers. The dual group of Q is

®u€[1,a] Zuj

where Zu = Z for all u G [1, a]. An element k = {ku : u G [1, a]} G ©ue[i,a]^t, where only a finite number of integers ku are distinct torn zero, acts on Qa by

ry. _V ryh. - II ryku

-' 7 ■' | |

n

u€[1,a]

where x = {xu : xu G 7, tt G [l,a]}. Therefore, the Fourier transform gN(k) of the probability measure QN is

1 N

gN(k) = / J] <udQN,a = j—j Y, II eimhkul°gU

° ue[1,a] m=0 ue[1,a]

1 N

= N + 1 ^ e^iimh Y1

m=0 u€[1,a]

where only a finite number of integers ku are non-zero. Since

exp{ih ku log u} = 1

ug[1,a]

if and only if there exists r G Z such that

2nr

y^ ku log u

u€[1,a]

h

hence we deduce that

1 if ku log u = ^ for some r G Z,

u€[1,a]

9N(k) = l-exp|i(W+l)/i £ fc„loguj

-^q-j--7-^^-x— otherwise,

1—exps ih ku log u f

I ue[1,a] J

Therefore,

/7N f 1 if „1 it = -^r" for some r G Z, lim gN{k) = { n ' l)

W—>oo [0 otherwise.

Thus, by the continuity theorem for probability measures on locally compact topological groups, see, for example, [6], Theorem 1.4.2, we obtain that the probability measure QNa converges weakly to a probabilitv measure Qa with the Fourier transform given by the right-band side of (1). The theorem is proved.

3 Limit theorem for integral over a finite interval

For fixed <Ti > and x > y > 1, define

v{x,y) = expj-(-) j.

V'

In this section, we will prove a discrete limit theorem in the space of analytic functions for the function

Zl,a,y(s) = J v(x,y)x sdx

with finite fixed a > 1.

Divide the interval [1, a] by points 1 = x0 < x1 < ... < xn = a into n subintervals of the same length In each interval [xj_i,xj], we take a point ^ and define the sum

n I 1 2

Sl,n,a,y(S) = Sp (2 + " A X*

j=1

with Axj = Xj — Xj—1. Define

Pn (A) = ^n (S1 ,n,a,y (s + imh) G A), A G B(H(D)).

Lemma 1. On (H(D), B(H(D))), there exists a probability measure Pn,a,y such, that the measure PN,n,a,y converges weakly to Pn,a,y o.s N — to.

Proof. Consider the function hn,ay : Qa — H(D) given bv the formula j=l

Then the function hnay is continuous, and

n 1 2

К,а,у{х-гтН) = £ <(- + | v(C3,y)C7s~imh A = ^.„(s + imft). j=i

Thus, PN,n,a,y = h-a,y, and the lemma follows from Theorem 4 and Theorem 5.1 of [2], by using the continuity of the function hnay. The limit measure Pn,a,y = Qh"^, where Q is the limit measure in Theorem 4.

Lemma 2. Let K be a compact subset of the strip D. Then the relation 1 N

lim limsup ——- } sup |SiBO!((s + imh) — Ziay(s + imh)\ = 0

i.—Von M . . . /\/ —I— 1 -^ _ t^ ' ' ' ' '

NN + 1 ^ «ек

m=0

holds.

Proof. Let L be a simple closed contour lying in D and enclosing the set K. Then, by the integral Cauchv formula

Si,n,a,y(s + imh) - Zhayy(s + imh) = —[ Sl'n'a'y^z + —2ho.,y(z +

2 JL z — s

Hence,

1

sup \Si>n>a,,y{s + imh)-Zi>a>y{s + imh)\ < - / \Si>n>a,,y{z + imh)-Ziiaiy(z+imh)\\dz\ «ек о JL

where 0 is the distance of L from the set K. Therefore,

1N

У^ sup \ Siiniaiy(s + imh) - Zhaty(s + imh) \

N + 1 m=0 seK

m=0

1 r 1 N

дг -I + imh) - Z\ ay(z + ¿mfo)|. (2)

0 Л N +1

L m=0

Clearlv, we have

ISm^

ISi,n,a,y(a + imh) - Zi,a,y(a + imh)|2

Si,n,a,y(a + imh)Si,n,a,y(a + imh) - Si^y(a + imh)Zi,a,y(a + imh)

Si,n,a,y (a + imh)Zia,y (a + imh) + Zi^y (a + imh)ZiA,y (a + imh).

It is not difficult to see that

N

N 1 y, Si>n>a>y(a + imh)Si>n>a>y((j + imh)

m=0

n 1 4

j=1

j=1 k=1

j=k

X-

1 I

,2 i(N +1)h

A x7- A xk.

i^ A ^ j

i _ i ife 1 tj

Since

+ 4v2(£j>y)&2°Axj= [ k{^ + ix^v2{x,y)x-2(Tdx <oo j=1 J1

we have that

j=1 _ 1 - ra

n n

j=1

+ + «<!))= 0(1)

(3)

n n 1 2 1 2

j=1 k=1

j=k

1

i(N +1)h

X-

A xj A xk = o(1)

ih A j

(4)

1

as N — to. Similarly

1

N

N + 1

y 21)(l)?/((7 + imh)Zi:a:y(a + imh)

m=0

1

N + 1

N

E

m=0

,2+m

v(u,y)u—a—imhdu

~2+tX

v(x, y)x—

2

11

u=x

+

11

u= x

(1 \2 1 — 2

1 _ 1 X

i(N+1)h

-h— dudx = o(1)

as N — to.

In the same way we find that

1

N

N + 1

y Si,n,a,y(cr + imh)Zitaty(a + imh) = o(l)

and

1

m=0

N

N + 1

y + imh)Zltaty(a + imh) = o(l)

m=0

as N — to. From (3)-(7), we have that

1

N

lim limsup -77—— Y^ \Si>n>a,y(z + imh) - Zi>a>y(z + zm/i)| = 0.

N ^^ N +1

m=0

(5)

(6)

(7)

This and (2) prove the lemma.

Let [Ki : l G N} be a sequence of compact subsets of D such that

D = U Ki, i=1

Ki C Ki+1, l G N, and if K C D is a compact subset, then K C Ki for some l [5]. For f,g G H(D), define

p(f,g) = Y,2

-1 Pi(f,9) i + Pi(/,sO'

where

Pi(f,g) = sup |f(s) — g(s)|.

s&Ki

Then it is easily seen that p(f, g) is a metric on H(D) which induces its topology of uniform convergence on compacta.

Theorem 5. On (H(D), B(H(D))), there exists a probability measure Pay such, that the probability measure

PN,a,y(A) = (Z1>ay(s + imh) E A), A E B(H(D)), converges weakly to Pay as N — œ.

Proof. On a certain probability space (Q, B(Q), P), define a random variable 0Nby

P(0W = hm) = -j^j, m = 0,1,..., N.

Let

un (s) = Si (s + %9n ). Then Lemma 1 implies the relation

D

un

(s) Un

,a,y (s), (8)

N

where Un,ay (s) is a H(D) - valued random variable having the distribution P,

n,a,yj

where Pn,a,y is the limit measure in Lemma 1,

We will prove that the family of probability measures {Pn,ay : n E N} is tight. For

Mi > 0,

1N

P(sup \UNtnta,y\ > Ml) < Y^ sup \Sitntaty(s + imh)|.

seKi Ml(N + 1) m=0 s€Kt

This and Lemma 2 show that

limsupP(sup lUN,n,a,y(s)| > Mi)

N^tt seKl

1N

< sup limsup V" sup \Sitntaty(s + imh) |

neN N^tt Ml(N + 1) m=0 seKl 1N

< sup limsup V" sup \Sitntaty(s + imh) - Zltaty(s + imh) |

neN N^tt Mi(N + 1) m=0seKl

1N

+ lim SUP A JfAT^-W SUP \Zl>a>v(S + imh) I

N^tt M1(N + 1) seKl

1 1 N R

^ M, + MNTT) £SKZl>) - i (9)

with Rt < oo, since the integral Z1>y(s) converges absolutely for a > see [1], Lemma 8, therefore, uniformly on compact subsets of D. Now we take Ml = Ml£ =

Ri2ie \ where e > 0 is an arbitrary number. Then (8), (9) and Lemma 4 yield, for nGN

P(sup| Untaty(s)\> Mi)<^ IgN. (10)

s&Kl 2

Define

K£ = [g G H(D) : sup |g(s)| < Mi, l G N}.

s&Ki

Then K£ is a compact subset of the space H(D), and, in view of (10)

ro 1

nUn,a,y(s) eK£)> l-£^- = l-£

i=1

for all n G N, or, by the definition of the random element Una,y (s),

Pn,a,y(Ke) > 1 — e

for all n G N. This shows that the family [Pn,ay:n&N} is tight. Hence, by the Prokhorov theorem [2], the family [Pn,a,y : n G N} is relatively compact. Therefore, there exists a sequence [Pnk,a,y} C [Pn,a,y} such that [Pnk,a,y} converges weakly to a certain probability measure [Pay} on (H(D), B(H(D))) as k — to. Thus,

Unkay(s) Pa,y. (11)

k^ro

Define

YN,a,y (s) = Z1,a,y (s + i0N ).

e > 0

lim limsupP(p(UN,n,a,y(s),YN,a,y(s)) > e)

n^ro N^ro

= lim limsup ^N(p(S1,n,a,y(s + imh), Zita,y(s + imh)) > e)

n^ro N^ro

1 N

< lim limsup y^ p(Si>n,a,y(s + imh), Zi^s + imh)) = 0.

n^ro n^roe(N+1)

This, (8), (11) and Theorem 4.2 of [2] give the relation

YN,a,y(s) — > Pa,y. N ^ro

Therefore, the measure PNa,y converges weaklv to Pa,y as N — to. The theorem is proved.

4 Limit theorem for absolutely convergent integral

This section is devoted to discrete limit theorem in the space of analytic functions for the function

ZltV(s) = J ( + v(x, y)x~sdx.

The above integral absolutely converges for a > \ (Lemma 8, [1]),

Theorem 6. On (H(D), B(H(D))), there exists a probability measure Py such, that the probability measure

PN,y(A) d=f ¡Ann(Zi,y(s + imh) E A), A E B(H(D)), converges weakly to Py as N — to.

Proof. Let 9n be the random variable defined in Theorem 5, Define H(D) - valued random element YN,ay(s) by the formula

YN,a,y (s) = Zi,a,y (s + i&N ).

Then, by Theorem 5 we have that

YN,a,y(s) Ya,y(s), (12)

N ^tt

where Ya,y(s) ^ an H(D)-valued random element having the distribution Pay. First, we will show that the family of probability measures {Pa,y : a > 1} is tight for fixed y. For arbitrary Ml > 0,

1N

P(sup \YN>a>y(s)\ > Ml) < Y, sup \Zl<a<y(s + imh)\- (13)

seKl l( + ) m=0 seKl

The integral defining Z1>y(s) converges absolutely for a > therefore, uniformly on D

Ri

limsupP(sup |yjv,a,»(s)| > Mi) < -f,

N^tt seKi Mi

( 1 \ 2

+ v(x,y)x adx < oo.

where

tt

Ri = sup seKi Ji

Now let Ml = Ml£ = Rl2le~\ where e is an arbitrary positive number. Then, in view of (12), Theorem 5,1 of [2] and (13), we find that, for all a > 1,

e

P( sup \Yaty(s)\ > Mn) < —.n G N, (14)

seKn 2n

The set He = {g G H(D) : supseKn |g(s)| < Mn,n G N} is compact in the space H(D). Moreover, (14) shows that

P(Ya,y(s) G He) > 1 - £

for all a > 1, or, by the definition of the random element Yay

Pa,y (He) > 1 - £

for all a > 1. So, we proved that the family of probabilitv measures {Pay : a > 1} is tight, therefore, by the Prokhorov theorem it is relatively compact. Thus, there exists a subsequence {Pak ,y} C {Pa,y} such that Pak ,y converges weakly to some probability measure Pa,y on (H(D), B(H(D))) as k — to, In other words,

Yaky(s) Py. (15)

Moreover, the uniform convergence on compact subsets of D of Z1a,y(s) to Z1>y (s) as a — to implies, for every £ > 0,

lim limsup pN(p(Z1 y(s + imh), Z1ay(s + imh)) > £) 1 N

< lim limsup y^ p(Zlty(s + imh), Zltaty(s + imh)) = 0,

N^^ £(N + 1) ^

v ' m=0

where p is the metric in H(D) defined by terms of the sequenee {Kn}. Hence, setting

Yn, y (s) = Zi , y (s + i9 n ),

we find that

lim limsupP(p(Yn,a,y(s),Yn,y(s)) > £) = 0. (16)

This, (12), (15) and Theorem 4.2 of [2] complete the proof of the theorem.

5 Approximation of Z^s) by Z1y(s)

To pass from the function Z1, y(s) to Z1(s), we need an approximation in the mean of Z^s) by Z1, y (s).

D

1N

lim limsup —-\ sup \Zi(s + imh) — Ziy(s + imh)\ = 0.

NN + 1 ^ s&K

m=0

Proof. Let

s Wi /

For s e K, suppose that \ + 9X < a < 1 - d2, > 0, 62 > 0. We put a2 = \ + y. Then we have that

1 n o"2-e+itt dz

zi,y(s) = / Zi(s + z)ly(z) —

2ni Ja2-a-itt z

+Zi(s) + Ry (s), (17)

where

Ry (s) = Re Sz=i-sZi(s + z)ly (z)z-1.

L D K | L|

L

sup \Z\{s + imh) — Zi,y(s + imh)\ < —— [ \Z\(z + imh) — Zi,y(z + imh)\\dz\, seK ' 2nd J

L

where d is the distance of L from the set K. Hence,

1N

sup \Z\(s + imh) — Z1>y(s + zm/i)|

N +1 seK

m=0

1 , N

<C —- / |dz| ^^ |^i(Rez + imft + zlmz) — -2q;:y(Rez + imft + zlmz)|

L m=0

|L| N

—- sup } \Zi(a + imh + it) — Ziy(a + imh + it)\. (18)

Nd seL ^

m=0

From (17) we have

Zi(a + imh + it) — Zi>y (a + imh + it) = — Ry (a + imh + it)

+0^ |Zi(a2 + imh + it + ir) |ly (a2 — a + ir )| dr^j.

This gives

1N

— ^^ |-2-i(c + + it) ~ zi,y(cr + imh + ¿¿)|

N

m=0

1N

< — |i?y(iJ + imh + it)|

m=0

1 N . \ly(<72 - <7 + ir)\ (— Y^ + imh + it + it) | J dr. (19)

"TO m=0

We always can choose L to be bounded, Then t is bounded, we obtain that

1 N

— \Zi(a2 + imh + it + it)\

m=0

1 N 1 < (— J] 1^1 (<J2 + imh + it + it)\2Y<.1 + |r|. (20)

m=0

Moreover, simple calculations show that

1 N

-J2\Ry(<J + imh + it)\ = o(l), (21)

m=0

as N —y to.

Now suppose that, for s G L,

1 39i

and 5 > Then estimates (18)-(21) yield, as N —> to,

N

sup |Zi (s + imh) — Z1y (s + imh) I

1N

N +1 ^ s^K'

m=0

< sup Ily(a2 — a + iT)|(1 + |tI)dr + o(1)

s&l j —to

/TO

|ly (a + it)I(1 + |t|)dt + o(1).

Since

lim sup Ily (a + it)I(1 + |t|)dt = 0,

y—>-<TO

a

hence the theorem follows.

J - CO

6 Proof of the main theorem

Let Yy(s) be an H(D)- valued random element having the distribution Py. Define

YN,y (s) = Ziy (s + i&N ). Then by Theorem 6 we have that

YNy(s) -——— Yy(s). (22)

N

DO

For Ml > 0,

P(sup |lWs)| > Mi) < ' J] sup \Zlty(s + imh)|. (23)

s^Kl J( + ) m_O S^Kl

1 N

JK y m_0 '

The integral defining Z1>y(s) converges absolutely for a > therefore uniformly on compact subsets of D. Thus, by (23),

limsupP(sup \YN>y(s)\ > Mi)

N^x s&Ki MJ

where RJ < oo, Let, for arbitrarv e > 0,MJ = RJ2Je-1. Then, using (22) and Theorem 5,1 of [2], we deduce that

e

P(sup| Yy(s)\>Mi)<-,leN. (24)

s£Kn 2

Bv the compactness principle, the set H£ = {f G H(D) : supseKl |f (s)| < MJ : l G N} is compact in H(D), and by (24)

P(Yy(s) G H) > 1 - e

for all y > 1, or

Py(He) > 1 - e

for all y > 1, So, we proved that the family of probabilitv measures {Py} is tight. Moreover, Theorem 7 implies

lim limsup^N(p(Z1(s + imh), Z1y(s + imh)) > e)

n^x N^x

1N

< lim lim sup——-p(Zi(s + imh), Z\ y(s + imh)) = 0.

n^x e(N + 1) ^

m_0

Hence, taking YN(s) = Zi(s + i6N), we find that

lim limsupP(p(YN(s),YNy(s)) > e) = 0. (25)

N^x

The tightness of the family {Py} gives its relative compactness. Let {Pyk} C {Py} be such that Pyk converges weakly, sav, to P as k — oo. Then

Yyk P, (26)

and the theorem is a result of (22),(25), (26) and Theorem 4,2 of [2].

СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ

[1] V. Balinskaite, A. Laurincikas, Discrete limit theorems for the Mellin transform of the Riemann zeta-function // Acta Arithmetica, 131.1 (2008), 29-42.

[2] P. Billingslev, Convergence of Probability Measures, John Wiley & Sons, New York, 1968.

[3] H. Bohr, B. Jessen, Uber die Wertverteilung der Riemannschen Zetafunktion, Erste Mitteilung, Acta Math., 54(1930), 1-35.

[4] H. Bohr, В. Jessen, Uber die Wertverteilung der Riemannschen Zetafunktion, Zweite Mitteilung, Acta Math., 58(1932), 1-55.

[5] J.B. Conwav, Functions of One complex Variable, Springer, Berlin, 2001.

[6] H. Heyer, Probability Measures on Locally Compact Groups, Springer, Berlin, 1977.

[7] A. Ivic, M. Jutila and Y. Motohashi, The Mellin transform of powers of the zeta-function //Acta Arithmetica, 95 (2000), 305-342.

[8] M. Jutila, The Mellin transform of the square of Riemann's zeta-function // Period. Math. Hungar., 42 (2001), 179-190.

[9] A. Laurincikas, Limit Theorems for the Riemann Zeta-Function, Kluwer Academic Publishers, Dordrecht, Boston, London, 1996.

[10] A. Laurincikas, Limit theorems for the Mellin transform of the square of the Riemann zeta-function. I //Acta Arithmetica, 122.2(2006), 173-184.

[11] A. Laurincikas, Corrigendum to the paper "Limit theorems for the Mellin transform of the square of the Riemann zeta-function. I"// Acta Arithmetica, 122.2(2006), 173-184, Acta Arithmetica (to appear).

[12] M. Lukkarinen, The Mellin transform of the square of Riemann's zeta-function and Atkinson formula, Ann. Acad. Sei. Fenn. Math. Diss., 140, 2005.

A. Laurincikas

Department of Mathematics and Informatics

Vilnius University

Naugarduko 24

03225 Vilnius

Lithuania

V, Balinskaite

Faculty of Social Informatics Mvkolas Romeris University Ateities 20, 08303,Vilnius Lithuania

E-mail: [email protected] Получено 13.05.2010

V. Balinskaite College of Social Science Ulonu 5, 08240, Vilnius Lithuania