CC BY
11
ЧЕБЫШЕВСКИЙ СБОРНИК
Том 20. Выпуск 1.
УДК 511.3 DOI 10.22405/2226-8383-2018-20-1-261-271
Обобщённая предельная теорема для периодической дзета-функции Гурвица
А. Римкявичене
Римкявичене Аудроне — доктор математики, доцент, Шяуляйская государственная коллегия, Литва.
e-mail: [email protected]
Аннотация
С времен Бора и Иессена (1910-1935) в теории дзета-фуекций применяются вероятностные методы. В 1930 г. они доказали первую теорему для дзета-функции Римана С(s)-, s = а + it, которая является прототипом современных предельных теорем, характеризующих поведение дзета-функции при помощи слабой сходимости вероятностных мер. Более точно, они получили, что при а > 1 существует предел
lim 1J {t е [0, Т ] : log С (а + it) е R} ,
Т^то Т
где R - прямоугольник на комплексной плоскости со сторонами, паралельными осям, а JA обозначает меру Жордана множества А с R. Два года спустя они распространили приведенный результат на полуплоскость а >
Идеи Бора и Иессена были развиты в работах Винтнера, Борщсениуса, Иессена, Сель-берга и других известных математиков. Современные версии теорем Бора-Иессена для широкого класса дзета-функций были получены в работах К. Матсумото.
В основном теория Бора-Иессена применялась для дзета-функций, имеющих эйлерово произведение по простым числам. В настоящей статье доказывается предельная теорема для дзета-функций, не имеющих эйлерова произведения и являющихся обобщением классичесской дзета-функции Гурвица. Пусть а, 0 < а < 1, фиксированный параметр, а a = {«m : то е No = N U {0}} - периодическая последовательность комплексных чисел. Тогда периодическая дзета-функция Гурвица С(s, а; а) в полуплоскости а > 1 определяется рядом Дирихле
то
С(s, а; а) => --m
z—' (то + a)s
m=0 4 '
и мероморфно продолжается на всю комплексную плоскость. Пусть В(C) - борелевское <г-поле комплексной плоскости, measA - мера Лебега измеримого множества А с R, а функция y(i) при t ^ To имеет монотонную положительную производную <p'(t), при t ^ ж удовлетворяющую оценкам (<p'(t))-1 = o(t) и <p(2t) maxt^„^2i(^'(«))-1 ^ t. Тогда в статье получено, что при а > 2
^meas {t е [0, Т] : С (а + i<p(t), а; а) е А} , А е В( C),
при Т ^ ж слабо сходится к некоторой в явном виде заданной вероятностной мере на (C, ß(C)).
Ключевые слова: дзета-функция Гурвица, мера Хаара, периодическая дзета-функция Гурвица, предельная теорема, слабая сходимость.
Библиография: 11 названий.
262
А. Римкявичене
Для цитирования:
А. Римкявичене Обобщенная предельная теорема для периодической дзета-функции Гурвица // Чебышевский сборник, 2019, т. 20, вып. 1, с. 261-271.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 1.
UDC 511.3 DOI 10.22405/2226-8383-2018-20-1-261-271
A generalized limit theorem for the periodic Hurwitz zeta-function
A. Rimkeviciené
Audroné Rimkeviciené — doctor of mathematics, associated professor, Siauliai State College, Lithuania.
e-mail: [email protected]
Abstract
Probabilistic methods are used in the theory of zeta-functions since Bohr and Jessen time (1910-1935). In 1930, they proved the first theorem for the Riemann zeta-function ((s), s = a + it, which is a prototype of modern limit theorems characterizing the behavior of ((s) by weakly convergent probability measures. More precisely, they obtained that, for a > 1, there exists the limit
lim ^J {t e [0, T] : log C(a + it) e R} ,
T —^^o 1
where R is a rectangle ra the complex plane with edges paralle 1 to the axes, and J A denotes the Jordan measure of a set A c R. Two years latter, they extended the above result to the half-plane a > 2-
Ideas of Bohr and Jessen were developed by Wintner, Borchsenius, Jessen, Selberg and other famous mathematicians. Modern versions of the Bohr-Jessen theorems, for a wide class of zeta-functions, were obtained in the works of K. Matsumoto.
The theory of Bohr and Jessen is applicable, in general, for zeta-functions having Euler's product over primes. In the present paper, a limit theorem for a zeta-function without Euler's product is proved. This zeta-function is a generalization of the classical Hurwitz zeta-function. Let a, 0 < a < 1, be a fixed parameter, and a = {am : m e No = NU{0}} be a periodic sequence of complex numbers. The periodic Hurwitz zeta-function ((s, a; a) is defined, for a > 1, by the Dirichlet series
C(s,a; a)
(to + a)s
m=0 4 '
and is meromorphically continued to the whole complex plane. Let B(C) denote the Borel a-field of the set of complex numbers, measA be the Lebesgue me^ure of a measurable set A c R, and let the function <^(i) for t > To have ^^e monotone positive derivative <^'(i) such that (^'(t))-1 = o(t) mid ^(2t) maxt^„^2i(^'(w))-1 ^ t. Then it is obtained in the paper that, for a > 2, " "
^meas {t e [0, T] : C(^ + i<fi(t), a; a) € A} , A e B(C),
converges weakly to a certain explicitly given probability measure on (C, S(C)) as T ^ to.
Keywords: Haar measure, Hurwitz zeta-function, limit theorem, periodic Hurwitz zeta-function, weak convergence.
Bibliography: 11 titles.
For citation:
A. Rimkevicienè, 2019, "A generalized limit theorem for the periodic Hurwitz zeta-function" , Che-byshevskii sbornik, vol. 20, no. 1, pp. 261-271.
In honor of Professor Antanas Laurincikas on the occasion of his 70th birthday
1. Introduction
The idea of application of probabilistic methods in the theory of zeta-functions is due to Bohr and Jessen. In [2], they proved a theorem for the Riemann zeta-function
C, S = a + it,a> 1,
^ ms m=l
which is a prototype of a modern limit theorems on weakly convergent probability measures. Denote by J A the Jordan measure of a measurable set A c R, and let R be a rectangle on the complex
< > 1
lim Tj lie [0, T] : log ((a + it) e R} .
T^oo T
^ 1
( )
Two years later, Bohr and Jessen extended [3] the above result to the half-plane a > 2- In this case,
G ={ se C : a > \ |J |s = a + i t3 : 1 < a < a^ ,
where Sj runs over all zeros of ((s) in the region |s e C : 2 < a < 1}, and proved that there exists the limit
lim ¿J |i e [0, T] : a + ite G, log((a + it) e R} .
In the sixth decade of the last century the theory of weak convergence of probability measures was created. Therefore, it became possible to state Bohr-Jessen type theorems in the sense of weakly convergent probability measures, for results, see [6] and [8].
The present note is devoted to limit theorems for the periodic Hurwitz zeta-function. Let a, 0 < a ^ 1 be a fixed parameter, and let a = {am : m e No = N U |0}} be a periodic sequence of complex numbers with minimal period q e N. The periodic Hurwitz zeta-function ((s, a; a) was
a > 1
ф,a; a) =
n ( m + a)s'
m=0
If am = 1, then ((s, a; a) becomes the classical Hurwitz zeta-function
1
(m + a)s '
C(s, a) = У^ ---——, a > 1,
m=0
which has a meromorphic continuation to the whole complex plane with the unique simple pole at the point s = 1 with residue 1. The periodicity of the sequence a implies, for a > 1, the equality
1 1 i I \
((s, a; a) = -^ U ^T).
q i=o v q J
a
m
Therefore, the function ((s, a; a) also can be continued meromorphically to the whole complex plane with the unique simple pole at the point s = 1 with residue
def 1
a = - > ai. q ^
H 1=0
If a = 0, then the periodic Hurwitz zeta-function is entire.
In [4], [9] and [11], limit theorems on weakly convergent probability measures on the complex plane for the function ((s,a; a) were proved. Denote by B(X) the Borel a-field of the space X. Then, for example, it was obtained in [10] that if the parameter a is transcendental and a > 2 is fixed, then, on (C, B(C)), there exists a probability measure Pa such that
T^meas {t e [0, T] : ((a + it, a; a) € A} , A e B(C),
converges weakly to as T ^ to. Moreover, the measure Pa is given explicitly. The aim of this note is a generalization of the above theorem for
PT,a,a;a(A) d=—-^meas {t e [To,T] : C+ i<p(t),a; a) € A} , A e B(C),
l — lo
for certain functions <p(t) and T0 > 0. For its statement, we need some notation and definitions. Let 7 be the unit circle on the complex plane, and
q=n7m,
m=0
where 7m = 7 fa all m e N0. With the product topology and pointwise multiplication, the torus Q is a compact topological Abelian group. Therefore, on (Q, B(Q)), the probability Haar measure m,H can be defined. This gives the probability space (Q, B(Q),m#). Denote by w(m) the mth component, m e N0, of an element w e Q, and, on the probability space (Q, B(Q), mu) define, for a > 2, the complex-valued random element ((a, a; a)
^ amw(m)
C(a, a; a) = > 7-—.
sv ' ' 7 ^ (m + a)a
m=0
Let be the distribution of the random element ((a, a; a), i.e.,
Pc,a,a;a(A) = mH {w e Q : ((a, a; a) e A} , A e B(C).
Now, define the class of functions. We say that <p e L(T0) if tp is a real differentiable function for t ^ T0 > 0 such that f'(t) is monotonic positive, ^jy = o(t) and p(2t) maxt^u^2t ^/j^j ^ t as t ^ to. For example, the function tp(t) = t4 + 2t3 +12 is an element of the class L(1). The main result of this note is the following theorem.
Theorem 1. Suppose that the parameter a is transcendental, a > 2 is fixed and <p e L(T0). Then Pr,a,a;a converges weakly to the measure P(,a,a;a T ^ to.
2. Lemmas
We start with a limit theorem for probability measures on (Q, B(Q)). For A e B(Q), let
1
T~To
QtAA) = Y~r meas {t e [To,T] : ((m + a)-lLp(t) : m e N0) e .
Lemma 1. Suppose that p £ L(T0). Then Qi,a converges weakly to the Haar measure mn as T ^ to.
proof. We apply the Fourier transform method. Let the sign " "' mean that only a finite number of integers km are distinct from zero. Denote by gT(k), k = (km : km £ Z,m £ No) the Fourier transform of QT,a. Then the definition of QT,a implies that
~ / \ tT ~'
9T,a(k) = I in ukrn (m) ) dQT,a = n (m + a)-k^)dt
1
rp rp , exp{—<£>(i) V log(m + a)}dt. (1)
T - 10 JTo
m=0
Clearly
* 9T,a(0) = 1. (2)
Since a is transcendental, the set {log(m + a) : m £ N0} is linearly independent over the field of rational numbers, thus the finite sum
defy
r =^2 km log(m + a) = 0
m=0
fork = 0. Obviously
r r r
/ exp{-irp(t)}dt = I cos(rp(t))dt — i / sin(rp(t))dt. (3)
J To J To J To
If the function p'(t) is decreasing, then (p(t))-1 is increasing. Thus, by the mean value theorem for integrals,
XTcos(-(0)dt = 1 X0 = ^pr) (t)cos(rp(t))di
d sin( r p(t)) = o(T), (4)
1 rT
p ( T)
as T ^ to, where T0 ^ £ ^T. Similarly, we find that
f sm(r<p(t))dt = o(T), T ^то. (5)
JTo
T To
p ( ) ( p ( )) - 1
that
XJ exp di=°bk))- (6)
Now, the estimates (4)-(6), and equalities (3) and (1) show that
r m A if k = 0,
lim qTa(k) = <
t^1^-' |o if k = 0.
The right-hand side of the latter equality is the Fourier transform of the Haar measure mn- This and a continuity theorem for probability measures on compact groups prove the lemma. □
Now, we will deal with absolutely convergent Dirichlet series. Let d > 2 be a fixed number, and
Define the functions
and
vn(m, a) = exp < —
\ im + a\01
(I n + a ) j
(n(s,a; a) = ^
m=0
amVn (m,a) (m + a)s
^ amw(m)vn(m, a) Us,a,u; a)=^ (m + a)a .
m=0
We note that the above series are absolutely convergent for a > 2 [5]. Consider the function un,a,a;a : Q ^ C given by the formula
Un,a,a;a(^) = (n(&, a), a > 1.
Then the function un,a,a;a is continuous. Moreover,
PT,n,a,a; a = QT,aU-,a,a; a. This observation together with Theorem 5.1 of fl] gives the following assertion. Lemma 2. Suppose that p e L(To). Then, for a > 2,
PT,n,a,a;a(A) d=f —-^meas{i e [To,T] : Qn(a + Mt),a; a) e A}, A e B(C),
l — lo
converges weakly to measure Pn,a,a;a = mHu-1aa-a as T ^ to.
Now we will approximate ((a, a; a) by (n(s, a; a). For this, we need a mean square estimate. Lemma 3. Suppose that <p e L(T0) and a > j is fixed. Then, for t e R rT
it0
proof. Suppose that T ^ T0. Then
I \C(a + iT + iip(t), a; a)|2di ^^a T (1 + \t|).
JTo
r2T r2T 1
\((a + iT + i<p(t),a; a)\2di = —-\((a + ir + i<p(t),a; a)\2d^(i) Jt JT v (t)
^ max
T4142T p'
max
^ i:T d(L
a;
■T+<p(t)
d I / \C(o + iu, a; a)\ d-u
To ,
2 T
r+v(t)
T4M2T <p' (t) \JTo
(a + iu, a; a)\2d-u
(7)
T
For a > 2, the estimate
rt
/ (g + iu,a; a)\ du T
J To
1
is true [5]. Therefore,
( f r+f(t) \ 2T
yjT |((o- + iu,a; a)j2duj <<7,a,0 jrj +p(2T). This together with hypothesis that p(2T) maxT^2T ^¿j < T and (7) gives
f 2T 1
|C(o + it + ip(t), a; a)|2di <^,«,0 jrj + p(2T) ma* "7m
JT 1 p ()
<a,a,aT + jrj max —^ <a,a,aT (1 + jrj) . T^t^2T p(t)
Taking 2-k-1T in place of T and summing over k £ N gives the estimate of the lemma. □ Lemma 4. Suppose that p £ L(T0) and o > 2>- ^/len
1 fT
lim limsup ——— j((o + ip(t),a; a) — (n(o + ip(t),a; a)jdi = 0.
n^ro T^ro T — T0 Jt0
Proof. Define the function
ln(s,a) = IГ (I) (m + a)
e \q,
where r(s ) is the Euler gamma-function, and the number 0 comes from the definition of vn(m, a). Then the function ((s, a; a) has the integral representation [5]
1 r-e+i^ in(z a)
(n(s,a; a) = — ф + z,a; a)^^— dz.
Je-ioo
id—iro
Then, using the residue theorem and properties of the gamma-function, we obtain that
1 '-T
/ |((a + ip(t),a; a) - (n(a + ip(t),a; a)|di
JTn
T -To Jt0
roo / i rT
a J |ln(<i + iT,a)\(^T - ^ jf |((a2 + it + ip(i),a; a)|di^ dr + o(1)
as T ^ to, where <1 < 0 and > 2- Hence, in view of Lemma 3,
l fT
-—— \((a + iLp(t),a; a) - C,n(< + ip(t),a; a)|di
T - T0 JTo
/X
|¿n(«i + iT, a)| (1 + |r|)di + o(1)
-X
as T ^ to. Thus, by the proper ties of ln(s, a),
1 fT
lim lim sup——— | C( < + ip(t),a; a) - (n(< + ip(t),a; a)|di = 0.
n^x T^X T - T0 JT0
□
We recall that Pn,a,a;a is the limit measure in Lemma 2.
Lemma 5. The sequence {Pn,a,a;a : n G N} is tight, i.e., for every e > 0; there exists a compact set, K = K(e) C C such that
Pn,a,a;a(K) > 1 -£
for all n G N.
Proof. Let £ be a random variable defined on a certain probability space with measure P, and uniformly distributed on [0,1]. Define the complex-valued random element XT,n,a;a = XT,n,a by
XT,n,a;a = (n(& + if(£T), a; a). Then the assertion of Lemma 2 is equivalent to the relation
XT,n,a:a „-> Xn a-a, (8)
T —to
where Xn,a;a(a) is the complex-valued random element having the distribution Pn,a,a;a- By Lemma 3 with t = 0, for a > 2,
T
/ K(a + i<p(t),a; a)|2di ^^a T.
J To
Hence, the Cauchv inequality implies
JT K(a + i<p(t),a; a)|2di < (^(T — Tq) ^ |((a + i<p(t),a; a)|2di^ <(J,a;a T.
Therefore, using Lemma 4, we obtain that, for a > 2,
sup lim sup
1 T
T — T( '
/ |(n(^ + ip(t),a; a)|di ^ ca,a;a < to. (9)
J To
n€N T—>-to 1 — 1q Jto
Let e > 0 be an arbitrary fixed number, and M = Ma,a;a(e) = Ca,a;ae-1. Then, by (9), sup lim supP (|XT,n,a;a| > M) = sup lim sup 1 meas {t e [Tq,T] : |{n(^ + i<p(t),a; a)| > M}
n€N T—to ' n€N T —to — Jq
1 fT
< suplimsu^———— |(n(^ + i^(t),a; a)|di < e.
n€N T—to (1 — JT0
This together with (8) shows that
P (|*n,a;a| >M) < £ (10)
for all n e N The set K = K(e) = {s e C : |s| ^ M} is compact, and, by (10),
P (Xn,a;a e K) ^ 1 — £
for all n e N, or equivalentlv,
Pn,a,a;a(K) ^ 1 — £ for all n e N. Thus, the sequence {Pn,a,a;a : n e N} is tight. □
3. Proof of Theorem 1
The existence of the limit measure for PT,a,a;a as T ^ to easily follows from Lemmas 4 and 5, relation (8) and Theorem 4.2 of [1].
proof. [Proof of Theorem 1] By the Prokhorov theorem [1, Theorem 6.1], and Lemma 5, the sequence {Pn,a,a;a : n e N} is relatively compact, i.e., every subsequence {Pnk,a,a;a} C {Pn,a,a;a} contains a weakly convergent subsequence. Thus, there exists a subsequence {Pnr,a,a;a} such that Pnr,a,aa converges weakly to a certain probability measure Pa,a;a on (C, B(C)) &s r ^ to. In other words,
Xnr,a:a(^) -> Pa,a;a. (H)
^ r—TO
Define one more complex-valued random element
XT,a;a = XT,a;a(a) = ((a + ip(£T), a; a).
Then Lemma 4 shows that, for every e > 0 and a > 2,
lim limsupp (\XT,a;a(o) - XT,n,a-,a(^)| ^ ¿) n—TO t—to
= lim lim sup ———meas {t e [To,T] : (a + ip(t),a; a) - (n(^ + i<p(t),a; a)| ^ e}
n—TO T —)-to - J0
1 fT
< lim lim sup ———— + up(t),a; a) - (n(^ + np(t),a; a)|di = 0.
n—TO T—TO (J- - +o)£ JTo
The later equality, relations (8) and (11), and Theorem 4.2 of fl] prove that
XT,a,a;a(&) P°,a;a. (12)
1 —TO
in other words, PT,a,a;a converges weakly to Pa,a;a as T ^ <x>. Moreover, relation (12) shows that the measure Pa,a;a does not depend of the subsequence PnrU,a;a- Therefore, we have the relation
Xn,a:a(&) ^ P(j,a\a.
n—TO
This relation allows to identify the measure Pa,a;a. Namely, in [5], it was proved that, for a > 2,
T^meas {t e [0, T] : ((a + it, a; a) e A} , A e B(C),
as T ^ <x, also converges weakly to the limit measure Pa,a;a of Pn,a,a;a as n ^ <x, and that Pa,a;a coincides with P(,a,a;a- Therefore, PTi(T,a;a converges weakly to P(,a,a;a as T ^ <. The theorem is proved. □
4. Conclusions
In the paper, a generalized limit theorem for the periodic Hurwitz zeta-function
TO
Ca) = E (^r^, Res = 1
m=0 v '
where 0 < a ^ 1 is a fixed transcendental parameter and a = {am : m e N0} is a periodic sequence of complex numbers, is obtained. More precisely, it is proved that, for a > 2,
.,. * meas {t e [To,T] : ((a + i<p(t),a; a) e A} , A e B(C),
1 - ±o
converges weakly to the explicitly given probability measure on (C, B(C)) as T ^ <. Here the function tp(t) for t ^ T0 has a monotone positive derivative p'(t) satisfying the estimates (f'(t))-1 = o(t) and f(2t) maxt^tjuj ^ t as t ^ <x>. The theorem obtained generalized previous author's results with tp(t) = t. Moreover, it can be extended to a collection of periodic Hurwitz zeta-functions. Also, the case of rational a can be considered.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Billingslev P. Convergence of Probability Measures. New York: John Wiley and Sons, 1968.
2. Bohr H., Jessen В. Uber die Wertverteilung der Riemanschen Zetafunktion, Erste Mitteilung // Acta Math. 1930. Vol. 54. P. 1-35.
3. Bohr H., Jessen B. 'Uber die Wertverteilung der Riemanschen Zetafunktion, Zweite Mitteilung // Acta Math. 1932. Vol. 58. P. 1-55.
4. Geniene D., Rimkeviciene A. A joint limit theorem for periodic Hurwitz zeta-functions with algebraic irrational parameters // Math. Modelling and Analysis. 2013. Vol. 18, no. 1. P. 1 19 159.
5. Javtokas A., Laurincikas A. On the periodic Hurwitz zeta-function // Hardv-Ramanujan J. 2006. Vol. 29, no. 3. P. 18-36.
6. Laurincikas A. Limit Theorems for the Riemann Zeta-Function. Dordrecht, Boston, London: Kluwer, 1996.
7. Laurincikas A. The joint universality for periodic Hurwitz zeta-functions // Analysis. 2006. Vol. 26, no. 3, P. 419-428.
8. Matsumoto K. Probabilistic value-distribution theory of zeta-functions // Sugaku Expositions. 2004. Vol. 17. P. 51-71.
9. Misevicius G., Rimkeviciene A. Joint limit theorems for periodic Hurwitz zeta-functions. II // Annales Univ. Sei. Budapest., Sect. Comp. 2013. Vol. 41. P. 173-185.
10. Rimkeviciene A. Limit theorems for the periodic Hurwitz zeta-function // Siauliai Math. Semin. 2010. Vol. 5(13). P. 55-69.
11. Rimkeviciene A. Joint limit theorems for the periodic Hurwitz zeta-functions // Siauliai Math. Semin. 2011. Vol. 6(14). P. 53-68.
REFERENCES
1. Billingslev, P. 1968, "Convergence of Probability Measures", John WTilev and Sons, New York.
2. Bohr, H. к Jessen, В. 1930, "Uber die Wertverteilung der Riemanschen Zetafunktion, Erste Mitteilung", Acta Math., vol. 54, pp. 1-35.
3. Bohr, H. к Jessen, B. 1932, "Uber die Wertverteilung der Riemanschen Zetafunktion, Zweite Mitteilung", Acta Math., vol. 58, pp. 1-55.
4. Geniene, D. к Rimkeviciene, A. 2013, "A joint limit theorem for periodic Hurwitz zeta-functions with algebraic irrational parameters", Math. Modelling and Analysis, vol. 18, no. 1, pp. 149-159.
5. Javtokas, A. к Laurincikas, A. 2006, "On the periodic Hurwitz zeta-function", Hardy-Ramanu-jan J., vol. 29, no. 3, pp. 18-36.
6. Laurincikas, A. 1996, "Limit Theorems for the Riemann Zeta-Function", Kluwer, Dordrecht, Boston, London.
7. Laurincikas, A. 2006, "The joint universality for periodic Hurwitz zeta-functions", Analysis, vol. 26, no. 3, pp. 419-428.
8. Matsumoto, K. 2004, "Probabilistic value-distribution theory of zeta-functions", Sugaku Expositions, vol. 17, pp. 51-71.
9. Misevicius, G. k, Rimkeviciene, A. 2013, "Joint limit theorems for periodic Hurwitz zeta-functions. II", Annales Univ. Sei. Budapest., Sect. Comp., vol. 41, pp. 173-185.
10. Rimkeviciene, A. 2010, "Limit theorems for the periodic Hurwitz zeta-function", Siauliai Math. Semin., vol. 5(13), pp. 55-69.
11. Rimkeviciene, A. 2011, "Joint limit theorems for the periodic Hurwitz zeta-functions", Siauliai Math. Semin. vol. 6(14), pp. 53-68.
Получено 05.12.2018 г. Принято в печать 10.04.2019 г.
