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VALUE DISTRIBUTION OF GENERAL DIRICHLET
SERIES
© 2007 J.Genys
1
In the paper a survey of some probabilistic results and their application to universality for general Dirichlet series is given.
Introduction
Let N, Z, R and C denote the sets of all positive integers, integers, real and complex numbers, respectively. Let {am} be a sequence of complex numbers, and let {km} be an increasing sequence of positive numbers such that lim \m = +m. Denote by s = o + it a complex variable. Then the series
is called a general Dirichlet series with coefficients am and exponents ~km. If \m = logm, then the series (1) becomes the ordinary Dirichlet series
Dirichlet series play an important role in number theory, however, their value distribution is very complicated. In many cases, difficult problems of individual values can be reduced to various average results. Among them the probabilistic methods in the investigations of Dirichlet series belongs to a famous mathematician H. Bohr. In the second decade of the last century he proposed a statistical approach in the theory of the Riemann zeta-function. H. Bohr implemented his idea in joint works with B. Jessen, [3], [4]. Later A. Wintner, V. Borchsenius, A. Selberg, P.D. T.A. Elliott, A. Ghosh, E. Stan-kus, B. Bagchi, D. Joyner, A. Laurincikas, E.M. Nikishin, K. Matsumoto, R. Garunkstis, J. Steuding and others continued the investigations of H. Bohr and B. Jessen and obtained modern probabilistic results in this field. However, the majority of their investigations were related to ordinary Dirichlet series. On the other hand, the known limit theorems for general Dirichlet series were
TO
(1)
oo
m=1
1 Genys Jonas (genys@splius.lt), Dept. of Mathematics and Informatics, Siauliai University, P. Visinskio St. 19, LT-77156 Siauliai, Lithuania.
weaker than similar theorems for ordinary Dirichlet series. All these circumstances suggested an idea of the necessity to improve the results for general Dirichlet series.
It is well known that Dirichlet series are related to almost periodic functions: almost periodic functions are presented by Dirichlet series. Therefore, we recall some results for almost periodic functions. Denote by measjAj the Lebesgue measure of a measurable set A c R.
A number t is called the e-almost period of the function g(t), —m < t < +m,
if
|g(t + t) - g(t)| < e
for all t e R. A set E c R is called relatively dense if there exists a number l > 0 such that every interval of length l contains at least one number of the set E.
A continuous on R function g(t) is called almost periodic in the Bohr sense if for every e > 0 there exists a relatively dense set of e-almost periods of the function g(t).
A. Wintner proved [29] that the distribution functions
^meas{te[-T,T\: %g(t)<x} (2)
and
ineas{te[-T,T] : 3g(t)<x}, (3)
where g(t) is an almost periodic in the Bohr sense function, converge to distribution functions at their continuity points as T ^m.
Also, almost periodic functions in the Besicovitch sense have similar statistical properties. Let p ^ 1 and, for gi(t),g2(t) e Lp,
Qbp(gl, g2) = lim sup
T —
—T
T
p
¿7 f |gl(0-g2(0lPdt
A function g(t) is called 5p-almost periodic, shortly g e Bp, if there exists a trigonometric polynomial
J^mt m^
Pn(t) = Yj b m=1
such that
lim QBp(g, Pn) = 0.
E.M. Nikishin proved [25] that if g(t) e Bi, then the distribution functions (2) and (3) also converge to some distribution functions at their continuity points as T — to. Since the class Bi is the widest one, therefore the functions g e Bp, p > 1, also have the above property.
1. Limit theorems in the space of meromorphic functions
It is well known, see for example, [24], that the Dirichlet series (1) absolutely converges in half-plane Da = {s e C : o > oa}. Denote its sum by f(s). Then f(s) is a holomorphic functions in Da. Therefore, it is reasonable to consider how often the function f(s + it) belongs to some given set of analytic functions. Denote by H(G) the space of analytic on G functions equipped with the topology of uniform convergence on compacta. Let
vK...) = ^meas{xe[0,r]: ...},
where in place of dots a conditions satisfied by t is to be written. Denote by B(S) the class of Borel sets of the space S. Then in [12] the following statement has been proved.
Theorem 1.1. On (H(Da), B(H(Da))) there exists a probability measure P such that the measure
VT(f(s + iT) e A), A eB(H(Da)),
converges weakly to P as T
When the system of exponents {km} is linearly independent over the field of rational numbers, the limit measure P in Theorem 1 can be explicitly defined. This needs one topological structure.
Let y = {s e C : |s| = 1} be the unit circle on the complex plane, and
^ = Y\ Ym,
m=1
where Ym = Y for all m e N. With the product topology and pointwise multiplication the infinite-dimensional torus O is a compact topological Abelian group. Therefore, on (O, B(O)) the probability Haar measure mH exists, and this gives a probability space (O, B(O), mH). Denote by w(m) the projection of № e O to the coordinate space Ym, m e N and define on the probability space (O, B(O), mH) an H(Da)-valued random element f (s, №) by
f(5, œ) = ^ amœ(m)e ÀmS.
m=1
Let Pf be the distribution of the random element f(s, №), i.e., Pf (A) = mH(№ e O : f(s, w) e A), A e B(H(Da)).
Then in [12] the following theorem was obtained.
Theorem 1.2. If the .system of exponent {km} is linearly independent over the field of rational numbers then the probability measure
VT(f(s + /x) 6 A), A 6S(H(Da)),
converges weakly to the measure Pf as T ^m.
Now suppose that the function f(s) is meromorphically continuable to the half-plane o > Oi, 01 < oa, and that all poles in this region are in a compact set. Moreover, let, for o > Oi, the estimates
f(o + it) = O(|t|a), \t\ ^ to, a > 0, (4)
and
T
J |f(o + it)|2dt = O(T), T (5)
-T
be valid. Also,
\m ^ c(log m)6 (6)
with some positive constants c and 6. Let Di = {s 6 C : o > Oi}. All these conditions implies that f (s, m) is an H(Di)-valued random element defined on the probability space (O, B(O), Mh). To prove this, the pairwise orthogonality of random variables m(m), m 6 N, and the Radamacher theorem [23] are used.
Denote by M(Di) the space of meromorhpic on Di functions equipped with the topology of uniform convergence on compacta.
Theorem 1.3. [13] Suppose that the function f(s) satisfies conditions (4) and (5). Then on (M(Di),B(M(Di))) there exists a probability measure P such that the measure
VT(f(s + iT) 6 A), A e B(M(Di)),
converges weakly to the P as T
The first attempt to identify the limit measure P in Theorem C was made in [20], and the following result has been obtained.
Theorem 1.4. Suppose that the set {log2}U IJ {km} is linearly independent
m=i
over the field of rational numbers, and that conditions (4) - (6) are satisfied. Then the limit measure P in Theorem 1.3 coincides with the distribution of H(Di)-valued random element f (s, m).
Of course, the presence of the number log 2 in the statement of Theorem 1.4 is not natural. A new method of the proof gave the following result [5].
Theorem 1.5. Suppose that the system of exponents {km} is linearly independent over the field of rational numbers, and that conditions (4)-(6) are satisfied. Then the probability measure
vt(f(s + /t) 6 A), A 6 B(M(Di)),
converges weakly to the distribution of the H(Di)-valued random element f (s, m).
2. Joint limit theorems
Let, for o > oaj,
TO
fj(s) = ^ amj-e"^8
m=1
with amj e C and ~kmj e R, Xij < X2j < ..., lim Xmj = +to, j = 1,...,n. We
m—*to
assume that the functions fi(s),..., fn(s) are meromorphically continuable to the half-planes o > oii, oii < oai,...,o > oin, oin < oan, respectively, and all poles in these regions are included in a compact sets. Also, we suppose that, for o > oi j, the estimates
fj( s) = O(|t|j |t| ^ to, a j > 0, (7)
and
T
\fj(o + it)|2dt = O(T), T eTO, (8)
-T
hold, j = i,...,n. Moreover, we assume that
Xmj ^ Cj(log m)6j (9)
with some positive constants Cj and 6j, j = i,...,n. Let Dj = {s e C : o > oij}, j = i,...,n, and
Hn = H(Di) x ... x H(Dn),
Mn = M(Di) x ... x M(Dn).
Denote
O = Oi x ... x On,
where Oj = O for j = i,...,n, and let w = (wi,..., wn) for co = O and Wj e Oj, j = i,...,n. Then O is also a compact topological group, and on (O,B(O)) the probability Haar measure on (O,B(O)), j = i,...,n.
The first joint limit theorem in the space of meromorphical functions for general Dirichlet series was obtained in [22]. Define on the probability space (O,B(O),mH) an Hn-valued random element F(si,...,sn;w) by the formula
F( si,..., sn; w) = (fi( si, wi),...,fn( sn, Wn)),
where
TO
fj(sj, w) = 2 amj-e~Xmjsj, sj e Dj, j = i,..., n.
m=i
The main result of [22] is contained in the following statement.
TO
Theorem 2.1. For j = i,...,n7 suppose that the sets {log2}U U {Xmj} are
m=i
linearly independent over the field of rational numbers, and that for fj(s) conditions (7)-(9) are satisfied. Then the probability measure
Pt(A) d=fVT((fi(s + iT),..., fn(s + iT) e A), A e B(Mn),
converges weakly to the distribution of random element F( si,..., sn; m) as T ^m. However, the proof of Theorem 2.1 has a gap. For its validity the hypoth-
M
esis on the linear independence of the {log2} J IJ {Xmj}, j = i.....n, must be
m=i
nM
replaced by that on the independence of the set {log2} J J IJ {Xmj}. However,
j=i m=i
the main shortcoming of Theorem 2.1 is the presence of the number log 2 in this hypothesis.
First the consider the case Xmj = Xm, j = i,...,n. Let, for o > oaj,
M
fj( s) = am,e~XmS, j = i,...,n,
m=i
and let in the definition of the random element F(si,...,sn;m)
M
f (s j, m) = ^ am}e~Xmsj, sj 6 Dj, j = i,...,n.
m=i
Theorem 2.2 [6]. Suppose that the system of exponents {Xm} is linearly independent over the field of rational numbers, and that for f(s), j = i,...,n, conditions (7)-(9) are satisfied. Then the assertion of Theorem 2.1 is valid.
The main result of this Section is a joint limit theorem in the space Mn with different systems of exponents. To state it we need some additional notation. Define on the probability space (O, B(M), Mh) an Hn-valued random element F( si,..., sn; M) by the formula
F(si,..., sn;M) = (fi(si, Mi),..., fn(si, Mn)),
where
M
fj(sj, Mj) = 2 amjMjmer^, sj 6 Dj,
m=i
and Mj(m) is the projection of Mj 6 Oj to the coordinate space ym, j = i,...,n. We have that the distribution Pf of the random element F( si,..., sn; M) is defined by
PF(A) = mH((m1 ,...,Mn) 6 M : F(si,...,sn;M) 6 A), A 6B(Hn).
nM
Theorem 2.3 [7]. Suppose that the set U IJ {kmj} is linearly independent
j=i m=i
over the field of rational numbers, and that for fj(s), j = i,...,n, conditions (7)-(9) are satisfied. Then the probability measure Pt converges weakly to Pf as T * m .
We note that Theorems 1.5, 2.2 and 2.3 are not empty. For example, the conditions of these theorems are satisfied by Lerch zeta-function L(s, X, a) [16] for o > i, given by
L(s, X, a) =
g2n/Xm
(m + a)s'
m=0
and by analytic continuation elsewhere, with transcendental a, 0 < a ^ 1, and real X. In this case am — e and Xm = log(m + a). When X e Z, the function L(s, X, a) reduces to the Hurwitz zeta-function and has a simple pole at s = 1 with residue 1. If X £ Z, then L(s, X, a) is an entire function.
3. The joint universality
Note that the problem of universality for zeta-functions, and, in general, for Dirichlet series comes back to S.M. Voronin. In 1975 he proved [27] that, roughly speaking, every analytic function can be approximated by translations Z(s + iT) of the Riemann zeta-function Z(s). More precisely, the last version of the Voronin theorem is the following statement [10].
Theorem 3.1. Let K be a compact subset of the strip {s e C : \ < a < 1} with connected complement, and let g(s) be a non-vanishing continuous function on K which is analytic in the interior of K. Then, for every e > 0,
liminf vt(sup |Z(s + iT) - g(s)| < e) > 0.
T—TO seK
Theorem 3.1 was generalized by many authors for other zeta-functions. For references see a survey [15].
The universality of general Dirichlet series was obtained in [21]. To state a theorem from [21] we need some additional conditions. We suppose that the system of exponents {Xm} or series (1) is linearly independent over the field of rational numbers, the function f(s) is meromorphically continuable to the half-plane o > oi with some oi < o and it is analytic in the strip
D = {s e C : o1 < o < oa}.
We also require that the estimates (4) and (5) should be satisfied. Denote, for x > 0,
r(x) =2 1,
Xm
and let cm = ame-Xm°a. Suppose that, for some 6 > 0,
2 |cm| = 6r(x)(1 + o(1))
Xm
as x — to, |cm| ^ d with some d > 0, and
r(x) = C1 xK + B, (10)
where k ^ 1, Ci > 0 and |B| ^ C2. Finally, we assume that f(s) cannot be represented in the region o > oa by an Euler product over primes. Then in [21] the following statement has been proved.
Theorem 3.2. Suppose that the function f(5) satisfies all the conditions stated above. Let K be a compact subset of the strip D with connected complement, and let g(s) be a continuous function on K which is analytic in the interior of K. Then, for every e > 0,
liminf vT(sup | f (5 + /t) - g(5)| < e) > 0.
The first results on the joint universality were obtained by S.M. Voronin [28], S.M. Gonek [9] and B. Bagchi [1, 2]. Independently they proved the joint universality for Dirichlet L-functions. We state a joint Voronin's theorem [28].
Theorem 3.3. Let 0 < r < and let Xi>—>X» be pairwise non-equivalent Dirichlet characters, g1(5),..., gn(5) be continuous and non-vanishing on the disc 15| ^ r and analytic in the interior of this disc. Then, for every e > 0, there exists a real number t such that
L^s + ^ + ix,x^-gj{s)
holds for 1 ^ j ^ n.
The joint universality for Matsumoto and Lerch zeta-functions was obtained in [11] and [17], respectively. In [18] the joint universality for zeta-functions attached to certain cusp forms was discussed, and in [26] and [19] joint universality theorems were proved for twisted Dirichlet series with multiplicative coefficient and automorphic L-functions, respectively.The joint universality for a collection general Dirichlet series with the same system of exponents was proved in [14]. The number log 2 is involved in the hypothesis of [14]. As above, suppose that, for o > oaj, the series
max
M<r
<e
a a.
mj
fj(5) = £ ■
m=1
converges absolutely, f(5) is meromorphically continuable to the half-plane o > > 01j with some 01j < oaj, all poles being included in a compact set, it is analytic in the strip Dj = {5 e C : 01 j < o < oaj}, and that f(5) cannot be represented by an Euler product over primes in the region o > oaj, j = 1,...,n. Moreover, let, for o > o1 j, the estimates (7) and (8) be satisfied. Let cmj = = amj-e~^m°aj, j = 1,...,n. Additionally, we suppose that there exists r ^ n sets
r
Nfe, Nk1 n Nk2 = 0, for k1 * k2, N = U , such that cmj = bkj for m e Nk,
k=1
k = 1,...,r, j = 1,...,n. Let
bn ... b1n
L = .........
^ br1 . . . brn )
and we assume that the sequence {Xmj satisfies (10), and that
2 1 = Kkr(x)(1 + o(1)), x (11)
Xm^x, meNk
with positive Kk, k = 1,...,r. Then in [14] the following assertion has been obtained.
Theorem 3.4. Suppose that conditions (7), (8), (10), (11) are satisfied, the
TO
set {log2}U U i^m] is linearly independent over the field of rational numbers,
m=1
and that rank(L) = n. Let Kj be a compact subset of strip Dj with connected complement, and let gj(5) be continuous function on Kj which is analytic in the interior of Kj, j = 1,...,n. Then, for every e > 0,
liminf vT ( sup sup | fj(s + it) - gj(s)| < e) > 0.
T ^TO 1^J%nseKj
The number log 2 from the conditions of joint universality theorem has been removed in the following statement [8].
Theorem 3.5. Suppose that conditions (7), (8), (10), (11) are satisfied, the system {Xm] is linearly independent over the field of rational numbers, and that rank(L) = n. Then the assertion of Theorem 3.4 is true. The proof of Theorem 3.5 is based on Theorem 2.2.
We will give an example. Let {amj] be a periodic sequence with period r ^ n, j = 1,...,n, and let ~km = (m + a)1, where a is a transcendental number and |3 e (0,1). Then the series
TO
fj(s) = 2 amje"(m+a)|s m=1
converges absolutely for o > oaj- = 0, j = 1,... ,n, and cmj = amj is constant on the set
Nk = {m e N : m = k(modr)}.
Moreover,
r(x) = \ = J+ 0(1).
(m+a)1 ^x
Since a is transcendental, the system of exponents {Xm] is linearly independent over the field of rational number. Clearly, the numbers bkj can be chosen in the manner such that rank(L) = n. Therefore, if we suppose that the function fj(s) is meromorphically continuable to the region o > 01j, 01 j < oaj, j = 1,...,n, and satisfies the estimates (7) and (8), then the collection of the functions f1(s),...,fn(s) is universal.
References
[1] Bagchi, B. The Statistical Behaviour and universality Properties of the Riemann Zeta-Function and other allied Dirichlet Series: Ph.D. Thesis / B. Bagchi. - Calcutta: Indian Statistical Institute, 1981.
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Paper received 17/1X/2007.
Paper accepted 17/1X/2007.
РАСПРЕДЕЛЕНИЕ ЗНАЧЕНИЙ ОБОБЩЕННЫХ РЯДОВ
ДИРИХЛЕ
© 2007 Дж.Жонас2
В работе дан обзор некоторых вероятностных результатов и их приложений к универсальности обобщенных рядов Дирихле.
Поступила в редакцию 17//X/2007;
в окончательном варианте — 17/1X/2007.
2Дженис Жонас (genys@splius.lt), кафедра математики и информатики университета Шауляя, ул. П. Висинского 19, LT-77156 Шауляй, Литва.
