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References
1. Coppersmith D., Winograd S. Matrix multiplication via arithmetic progressions. J. Symbolic Comput., 1990. Vol. 9, no. 3, pp. 251-280.
2. Vassilevska Williams V. Multiplying Matrices Faster than Coppersmith-Winograd. Proceedings of the 44-th Symposium on Theory of Computing, STOC'12. 2012. URL: www.cs.berkeley.edu/ virgi/matrixmult.pdf (Accsessed 30, September, 2013).
3. Cohn H., Umans C. A group theoretic approach to fast matrix multiplication. Proceedings of the 44-th Annual IEEE Symposium on Foundations of Computer Science. 2003, pp. 438-449. DOI: 10.1109/SFCS.2003.1238217.
4. Cohn H., Kleinberg R., Szegedy B., Umans C. Group-theoretic algorithms for matrix multiplication. Proceedings of the 46-th Annual IEEE Symposium on Foundations of Computer Science. 2005, pp. 379-388. DOI: 10.1109/SFCS.2005.39.
5. Platonov V. P., Kuznetsov Iu. V., Petrunin M. M. O teoretiko-gruppovom podkhode k probleme bystrogo umnozheniia matrits. Matematicheskoe i komp'iuternoe modelirovanie sistem: teoreticheskie i prikladnye aspekty [A group-theoretical approach to the problem of fast matrix multiplication. Mathematical and computer
modeling of systems: theoretical and applied aspects]. Sbornik nauchnuh trudov NIISI RAN [Collection of scientific papers NIISI RAS]. Moscow, 2009, pp. 4-15 (in Russian).
6. Kuznetsov Yu. V. Nekotorye kombinatornye aspekty teoretiko-gruppovogo podkhoda k probleme bystrogo umnozheniia matrits [Some combinatorial aspects of the group-theoretic approach to fast matrix multiplication]. Chebyshevskii sbornik., 2012, vol. 13, no. 1, pp. 102-109 (in Russian).
7. Alon N., Shpilka A., Umans C. On sunflowers and matrix multiplication. Electronic Colloquium on Computational Complexity, 2011, Report no. 67, pp. 1-16.
8. Davis B. L., Maclagan D. The card game SET. Mathematical Intelligencer, 2003, vol. 25, no. 3, pp. 33-40.
9. Bateman M., Katz N. New bounds on caps sets. J. American Math. Soc., 2012. vol. 25, no. 2, pp. 585-613.
10. Edel Y. Extensions of generalized product caps. Designs, Codes and Cryptography, 2004, vol. 31, pp. 5-14.
11. Mebane P. Uniquely Solvable Puzzles and Fast Matrix Multiplication. HMC Senior Theses, 2012, 37 p.
УДК 511.3
ОБ УНИВЕРСАЛЬНОСТИ НЕКОТОРЫХ ДЗЕТА-ФУНКЦИЙ
А. Лауринчикас1, Р. Мацайтене2, Д. Мохов3, Д. Шяучюнас4
1Академик АН Литвы, доктор физико-математических наук, профессор, заведующий кафедрой теории вероятностей и теории чисел, Вильнюсский университет (Литва), [email protected] 2Доктор математических наук, профессор кафедры математики, Шяуляйский университет (Литва), [email protected]
3Магистрант факультета математики и информатики, Вильнюсский университет (Литва), [email protected] 4Доктор математических наук, профессор кафедры математики, Шяуляйский университет (Литва), [email protected]
Хорошо известно, что обобщение дзета функции Гурвица — периодическая дзета функция Гурвица — с трансцендентным параметром универсальна в том смысле, что её сдвигами приближается всякая аналитическая функция. В статье условие трансцендентности параметра заменяется более слабым условием о линейной независимости некоторого множества.
Ключевые слова: периодическая дзета функция Гурвица, пространство аналитических функций, слабая сходимость, универсальность.
1. INTRODUCTION
Let s = a+it be a complex variable, and a, 0 < a < 1, be a fixed parameter. The Hurwitz zeta-function Z(s,a) is defined, for a > 1, by the Dirichlet series
Z (s,a)= £
n(m + a)s '
m=0 v y
and continues analytically to the whole complex plane, except for a simple pole at s = 1 with residue 1.
© Лауринчикас А., Мацайтене Р., Мохов Д., Шяучюнас Д., 2013
A natural generalization of the function Z(s,a) is the periodic Hurwitz zeta-function. Let a = {am : m e N0 = N U {0}} be a periodic sequence of complex numbers with minimal period k e N. The periodic Hurwitz zeta-function Z(s,a; a) is defined, in the half-plane a > 1, by the Dirichlet series
Z(s,a; a) = У^ 7-m , .
sv ' ^ (m + a)s
m=0 4 '
The periodicity of the sequence a shows that, for a > 1,
1 k-1
Z(s,a; a) = aiZ (s, .
1=0
Thus, the properties of the Hurwitz zeta-function imply the analytic continuation for Z(s, a; a) to the
def k-1
whole complex plane, except for a simple pole at s = 1 with residue a = k ai -If a = 0, then the
1=0
function Z(s, a; a) is entire.
Properties of the functions Z(s,a) and Z(s,a; a) depend on the parameter a. It is known [6] that the function Z(s,a; a) with transcendental parameter a is universal in the sense that the shifts Z(s + ir, a; a), т e R, uniformly on compact subsets of the strip D = {s e C : 2 <a< 1}, approximate every analytic function. For a precise statement of the universality for Z(s, a; a), we need some notation. Denote by K the class of compact subsets of D with connected complements. For K e K, let H(K) denote the class of continuous functions on K which are analytic in the interior of K. Moreover, let measA stand for the Lebesgue measure of a measurable set A с R. Then the main result of [1] is the following theorem.
Theorem 1. Suppose that a is a transcendental number, K e K and f (s) e H(K). Then, for every £ > 0,
lim inf ^meas { т e [0, T] : sup |Z(s + iT, a; a) — f (s) | < £ 1 > 0.
т^^ T [ seK J
The aim of the present paper is to replace a hypothesis of Theorem 1 on the transcendence of the parameter a by a wider one. Define the set L(a) = {log(m + a) : m e No}.
Theorem 2. Suppose that the set L(a) is linearly independent over Q, and that K e K and f (s) e H(K). Then the same assertion as in Theorem 1 is valid.
Note that if a is a transcendental number, then the set L(a) is linearly independent over Q. On the other hand, it is known [2] that if a is an algebraic irrational number, then at least 51 percent of elements of the set L(a) are linearly independent over Q. Thus, it is possible that the set L(a) is linearly independent over Q even a is an algebraic irrational number. Unfortunately, we do not know any such a.
For the proof of Theorem 2, a probabilistic method based on limit theorems on the weak convergence of probability measures in the space of analytic functions will be applied.
2. LIMIT THEOREMS
Denote by H(D) the space of analytic functions on D equipped with the topology of uniform convergence on compacta. Let B(X) stand for the Borel field of the space X. In this section, we consider the weak convergence of the probability measure
PT(A) d=f ;1meas {t e [0, T] : Z(s + ir, a; a) e A} , A e B(H(D)),
as T ^ ro.
Let y be the unit circle on the complex plane, i. e., 7 = {s e C : |s| = 1}. We start with a limit theorem on the torus O = Yl 7m, where 7m = 7 for all m e N0. Since O with the product topology
m=0
and pointwise multiplication is a compact topological Abelian group, on (O, B(O)) the probability Haar measure mH can be defined, and we have a probability space (O, B(O),mH). Let, for A e B(O),
QT(A) =f ;1meas {r e [0, T] : ((m + a)-iT : m e No) e A} .
Lemma 1. Suppose that the set L(a) is linearly independent over Q. Then QT converges weakly to the Haar measure mH as T ^ ro.
Proof. Denote by u the elements of O. For m e N0, let u(m) be the projection of u e O to the coordinate space Ym. Then it is well known, see, for example, [3], that the Fourier transform (k), k = (ki, k2, • • •), of the measure QT is of the form
gT(k) = T I ex^J -ir km log(m + aH dr, (1)
0 I m=o J
where only a finite number of integers km are distinct from zero. Now we essentially apply the linear independence of the set L(a). Since km log(m + a) =0 if and only if all km = 0, we deduce from (1)
m=0
that
lim gT (k) = < 1 i k 0
| o if k = 0.
This and Theorem 1.4.2 of [4] show that the measure QT converges weakly to mH as T ^ ro. □
Now we will prove a limit theorem for absolutely convergent Dirichlet series. For a fixed ai > 1/2, and m e N0, n e N, let vn(m, a) = exp j- ^m+aa) }. Define
Cn (s,a; a) = Y^
те ,4
r-^ amVn(m, a) [ s, a; a) = v
m=0
(m + a)s
Then it is known [3] that the latter series is absolutely convergent for a > 1/2. For A e B(H(D)), define PT,n(A) = Tmeas {r e [0, T] : Zn(s + ir, a; a) e A} . For u e O, define one more function
^ am^(m)vn(m, a)
Cn(s,a,w; a) =
m=0
(m + a)s
clearly, the series being absolutely convergent for a > 1. Let u0 e O be a fixed element. On (H(D), B(H(D)),mH), define one more probability measure
PT,n(A) = ;1meas {r e [0, T] : Zn(s + ir, a, U0; a) e A} .
Lemma 2. Suppose that the set L(a) is linearly independent over Q. Then the both measures PT,n and PT,n converges weakly to the same probability measure Pn on (H(D), B(H(D))) as T ^ ro.
Proof. A proof uses Lemma 1, Theorem 5.1 of [5] and the invariance of mH, and is independent on the arithmetic nature of the parameter a. Therefore, it remains the same as in the case of transcendental a [1]. □
The next step in the investigation of the weak convergence of the measure PT consists of the approximation of the function Z(s,a; a) by Zn(s,a; a) in the mean. The space H(D) is metrisable. Denote by p a metric in H(D) which induces the topology of uniform convergence on compacta.
Lemma 3. We have
T
lim lim sup— p(Z (s + ir, a; a),Zn(s + ir, a; a))dr = 0.
t^^ T J 0
Proof. A proof of the lemma in the case of transcendental a in [1] does not use the transcendence property. Therefore, it also remains the same in our case. □
Let, for ш e O,
^ / N.
am w(m)
С a) = Y^
—' (m + a)s'
m=0 v '
Then Z(s,a,w; a) is the H(D)-valued random element defined on the probability space (O, B(O),mH) [6].
The approximation of the function Z(s, a,w; a) by Zn(s,a,w; a) is more complicated, we need some elements of ergodic theory. For t e R, define aT = {((m + a)-iT : m e N0)}. Let {^T : t e R}, where ipT(w) = aTw, w e O. Then {^T : t e R} is a group of measurable measure preserving transformations of the torus O. We will prove the ergodicity of the group {^T : t e R}. We recall that the set A e B(O) is invariant with respect to the group {^T : t e R} if, for any t e R, the sets A and AT = (A) differ one from another by a set of mH-measure zero. The group {^T : t e R} is invariant if the a-field of all invariant sets consists of the sets having mH-measure 0 or 1.
Lemma 4. Suppose that the set L(a) is linearly independent over Q. Then the group {^T : t e R} is ergodic.
x
Proof. It is well known that the characters x of the group O are of the form x(w) = Yl wkm (m),
m=0
where only a finite number of integers km are distinct from zero. First let x be a non-trivial character, i. e., x(w) ^ 1. Then we have that
x ( x
x(aT) = (m + a)-iTkm = exp < -¿t ^ km log(m + «H . (2)
m=0 ^ m=0 J
x
Since the set L(a) is linearly independent over Q, ^ km log(m + a) = 0 for every finite number of
m=0
non-zero integers km. Therefore, (2) implies that there exists t0 e R \ {0} such that
x(aTo ) = 1. (3)
Let A e B(O) be an invariant set with respect to the group {^T : t e R}, and IA be the indicator function. Then we have that, for every t e R and for almost all w e O,
I A (aT w)= I a (w). (4)
Denote by g(x) the Fourier transform of the function g, i. e., g(x) = /x(w)g(w)mH(dw). Taking into
n
account (4), we find that
IA (X) = X(aTo )IA(X).
This together with (3) shows that IA (x) = 0 for every non-trivial character x.
Now let x0 be the trivial character of the torus O, i. e., x0(w) = 1, and let, for brevity, IA(x0) = b. Using the orthogonality property of characters and the equality IA(x) = 0, we find that, for every character x of the torus O,
1(x) = b J x(w)mH(dw) = bIA(x) = b(x).
n
From this the lemma easily follows. □
Lemma 5. Suppose that the set L(a) is linearly independent over Q. Then, for almost all w e O,
T
lim limsup— p (Z(s + ¿t, a, w; a),Zn (s + ¿t, a,w; a))dT = 0.
n^x t^x T J 0
Proof. In [6], the assertion of the lemma is proved in the case of transcendental a, however, the transcendence is used only for the proof of the ergodicity of the group : т e R}. Since, by Lemma 4, the group : т e R} is ergodic, the proof of the lemma runs in the same way as in [6]. □
Now we are able to obtain the weak convergence of the measure PT. However, having in mind the identification of the limit measure, we also consider the measure
PT(A) = ;1meas {т e [0,T] : Z(s + ¿T,a,w; a) e A} , A e B(H(D)).
Lemma 6. Suppose that the set L(a) is linearly independent over Q. Then the both measures PT and PT converge weakly to the same probability measure P on (H(D), B(H(D))) as T ^ ro.
Proof. The used method is the same as in the case of transcendental a [1, 6], and uses Lemmas 2, 3 and 5. □
Now we state the main limit theorem of this section.
Theorem 3. Suppose that the set L(a) is linearly independent over Q. Then the measure PT converges weakly to the distribution Pz of the random element Z(s, a,w; a).
Proof. We apply Lemmas 4, 6 and the Birkhoff-Khinchine theorem. □
3. PROOF OF THE UNIVERSALITY THEOREM
For the proof of Theorem 2, together with Theorem 3 we need the explicit form of the support of the measure Pz.
The support of Pz is independent of the arithmetic nature of the parameter a, therefore we may use the following result of [1].
Theorem 4. The support of the measure Pz is the whole of H(D).
Proof of Theorem 2. By the Mergelyan theorem [7], there exists a polynomial p(s) such that
sup |f (s) - p(s)| < §. (5)
seK
In view of Theorem 4, the polynomial p(s) is an element of the support of the measure Pz. Thus, for every open neighbourhood G of the polynomial p(s), the inequality Pz(G) > 0 is true. Let G = {g e H(D) : supseK |g(s) — p(s)| < e/2}. Using Theorem 3, an equivalent of the weak convergence of probability measures in terms of open sets and the definition of G, we obtain that
liminf ^meas! т e [0,T] : sup |Z(s + iT, a; a) — p(s)| <£/2! > 0. (6)
т^ж T [ seK J
It remains to replace in this inequality p(s) by f (s). We note that in view of (5), for such т,
sup |Z(s + iT, a; a) — f (s)| < e.
seK
Thus, we deduce from (6) that
liminf ^meas {t e [0, T] : sup |Z(s + iT, a; a) — f (s)| < el > 0. т^ж T [ seK J
The theorem is proved. □
Библиографический список
1. Javtokas A., Laurincikas A. The universality of the Heilbronn // J. London Math. Soc. 1961. Vol. 36. P. 171-periodic Hurwitz zeta-function // Integral Transforms 184.
Spec. Funct. 2006. Vol. 17, № 10. P. 711-722. 3. Laurincikas A., Garunkstis R. The Lerch Zeta-
2. Cassels J. W. S. Footnote to a note of Davenport and Function. Dordrecht : Kluwer, 2002. 189 p.
4. Heyer H. Probability Measures on Locally Compact Groups. Berlin : Springer, 1977. 531 p.
5. Billingsley P. Convergence of Probability Measures. N.Y. : Wiley, 1968. 272 p.
6. Javtokas A., Laurincikas A. On the periodic zeta-
function // Hardy-Ramanujan J. 2006. Vol. 29. P. 18-36.
7. Mergelyan S. N. Uniform approximation to functions of complex variable // Usp. Matem. Nauk. 1952. Vol. 7. P. 31-122.
On Universality of Certain Zeta-functions
A. LaurinCikas1, R. Macaitiene2, D. Mokhov3, D. SiauCiUnas4
1 Vilnius University, Naugarduko st. 24, LT-03225 Vilnius, Lithuania, [email protected] 2Siauliai University, P. Visinskio st. 19, LT-77156 Siauliai, Lithuania, [email protected] 3Vilnius University, Naugarduko st. 24, LT-03225 Vilnius, Lithuania, [email protected] 4Siauliai University, P. Visinskio st. 19, LT-77156 Siauliai, Lithuania, [email protected]
It is well known that a generalization of the Hurwitz zeta-function — the periodic Hurwitz zeta-function with transcendental parameter is universal in the sense that its shifts approximate any analytic function. In the paper, the transcendence condition is replaced by a simpler one on the linear independence of a certain set.
Key words: periodic Hurwitz zeta-function, space of analytic functions, universality, weak convergence.
References
1. Javtokas A., Laurincikas A. The universality of the periodic Hurwitz zeta-function. Integral Transforms Spec. Funct., 2006, vol. 17, no. 10, pp. 711-722.
2. Cassels J. W. S. Footnote to a note of Davenport and Heilbronn. J. London Math. Soc., 1961, vol. 36, pp. 171184.
3. LaurinCikas A., Garunkstis R. The Lerch Zeta-Function. Dordrecht, Kluwer, 2002, 189 p.
4. Heyer H. Probability Measures on Locally Compact Groups. Berlin, Springer, 1977, 531 p.
5. Billingsley P. Convergence of Probability Measures. New York, Wiley, 1968, 272 p.
6. Javtokas A., Laurincikas A. On the periodic zeta-function. Hardy-Ramanujan J., 2006, vol. 29, pp. 18-36.
7. Mergelyan S. N. Uniform approximation to functions of complex variable. Uspekhi Matem. Nauk, 1952, vol. 7, pp. 31-122.
УДК 511.3
К ОЦЕНКЕ ОДНОГО КЛАССА СУММАТОРНЫХ ФУНКЦИЙ
В. А. Матвеев
Аспирант кафедры компьютерной алгебры и теории чисел, Саратовский государственный университет им. Н. Г. Чернышевского, [email protected]
Для конечнозначных функций натурального аргумента Н(и), имеющих ограниченную сумматорную функцию, оцениваются сумматорные функции вида ^ Н(и)и%г, 1 < |£| < Т.
пКх
Ключевые слова: числовые характеры, сумматорные функции, степенные ряды.
В работе [1] было показано, что для числовых характеров Дирихле х при любом действительном £ имеет место оценка вида
5]х(п)пй = 0(1).
П<Х
В данной работе этот результат обобщается на случай конечнозначных функций натурального аргумента Л-(п), для которых выполняются условия:
1) £(х) = £ й(п) = 0(1);
п<х
2) функция д(х), заданная степенным рядом вида д(х) = ^ Л,(п)хп, имеет конечный предел в
П = 1
точке х = 1.
© Матвеев В. А., 2013
