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VALUE DISTRIBUTION THEOREMS FOR THE ESTERMANN ZETA-FUNCTION
© 2007 A. Laurincikas1
In the paper, a survey of mean-value estimates, zero distribution, universality and limit theorems in the sense of weak convergence of probability measures for the Estermann zeta-function is presented.
Introduction
This paper is a text of author's talk given at the International Conference "Algebra and Number Theory" dedicated to the 80th anniversary of Professor V.E.Voskresenskii. (Samara, Russia, May 21-28, 2007). The author thanks the organizers of this conference for hospitality and for financial support.
Denote by N, Z, R and C the sets of all positive integers, integers, real and complex numbers, respectively. For arbitrary a e C and m e N, the generalized divisor function oa(m) is defined by
Oa(m) = da.
d/m
We have that
Oo(m) < me. Since, aa(m) = maa-a(m), the estimate
oa(m) << me+max(Rea'0) (1)
is valid.
Let, as usual, s = a+it denote a complex variable, and (k, l) = 1. The Estermann zeta-function for o > max(l + Rea, 1), is defined by
E(s> 7' a)= Z ^r exp(2jT im~j\-
m=1
For analytic continuation of the function E^s; j, a) to the whole complex plane, we recall the definition of the Lerch zeta-function. Let X e R and P e R, 0 < P ^ 1. The Lerch zeta-function L(X, P, s), for a > 1, is defined by
L(l,B,s) = ) --—.
v 1 ' ZJ (m + RV
m=0
1 Laurincikas Antanas ([email protected]), Dept. of Number Theory and Probability Theory, Vilnius University, 24, 03225 Vilnius, Lithuania.
If X £ Z, then L(X, s) is analytically continuable to an entire function, while for X e Z, the function L(X, |3, s) becomes the Hurwitz zeta-function
TO 1
(m + |)s
m=0 v 1 '
The function Z(s, |) is meromophically continuable to the whole complex plane where it has a simple pole at s = 1 with residue 1. It is not difficult to see that, for o > max(Rea + 1,1),
E(s- -v a)= r~> J expf^'y }Z(1, s - o)l(j , 1,(2)
V=1
The latter equation shows that the function E^s;j,aj is analytic in the whole complex plane, except for two simple poles at s = 1 and s = 1 + a if a + 0, and a double pole 5=1 if a = 0.
Let k be defined by kk = 1(mod l). Then (2) and the functional equation for the Lerch zeta-function, see [11], imply the following functional equation for
k \ 1 / 2n \2s—1—a
/ k \ i /2n\zs-i-a \
E(s;-,a)=-(—) r(l - s)r(l + a - s))x
/ na / k \ / na \ / k \\
(cos + a ~ 7' °v cos^^5 —2~j i + a ~ 7'ajj-
def
Therefore, without loss of generality we may assume that a = Rea ^ 0. Note that the function E^s;j,aj, for a = 0, was introduced by T. Estermann in [3]. The case of a e [—1,0] was considered in [9].
In the lecture, we discuss the following value distribution problems for the Estermann zeta-function:
• Mean square estimates
• Zero distribution
• Universality
• Probabilistic limit theorems
1. Mean square of E(s; f, a)
s: y, a]
Asymptotics and estimates for mean values of zeta-functions play an important role in analytic number theory. For example, the famous Lindelof hypothesis for the Riemann zeta-function Z(s) which says that, for every e > 0,
ib")
«E tE, t ^ t0 > 0,
is equivalent to the mean value estimates
7ÎNH
2k
«k,e Te, k e N
There exists a conjecture that, for all k ^ 0 and T
^"'fi KH
2k
dt - c(k)(log T)
k2
(3)
with some constant c(k) > 0. G.H. Hardy and J.E. Littlewood proved [5] that c(l) = 1, and A.E.Ingham found [7] the value c(2) = Let u ^ 0 be bounded
1. Of course,
by a constant. Then in [10] it was obtained that c(- _
■\/2 log log 7 the conjecture (3) is very complicated.
Also, the estimates for Ik(T) are known. The first results in this direction were obtained by K. Ramachandra. For example, he proved in [17] that
h2(T) « T(log T)1!.
The further progress in the field belongs to D.R. Heath-Brown. In [6], he proved the estimate
Ik(T) »kT(log T)k2 (4)
for all rational k ^ 0, and the estimate
Ik(T) <k T(log T)k2 (5)
for k = m e N. Moreover, he obtained under the Riemann hypothesis (all non-trivial zeros of Z(s) lie on the critical line) that (4) holds for all k ^ 0 and (5) is true for 0 ^ k ^ 2. To prove this, D.R. Heath-Brown applied the Gabriel convexity theorems, see, for example, [10].
For the Estermann zeta-function, the mean square was studied in [18], see also
[19].
Theorem 1.
For o > 5,
I f EC T J,
Moreover, if a < 0, then
lim
T —
■ k x it; -j, a)
I o + it: -, a.
r t
IK
k \ [a + it,-, a)
dt
2 U2a-2a)l2(2o-a)U2o) 3(4 o - 2a)
if o>\,
if o=\,
if a + \ < a <
if a = a + j,
if a < a +
T,
T log2 T,
T 2(1-a)
T1-2a log2 T,
T 3-4a+2a
For the proof, a representation of by Dirichlet L-functions is used.
Denote by ^(m) the Euler function,
qp(m) = m ]~~[(l--),
p/m
by ^(m) the Möbius function,
1, if m = 1, ^(m) = ^ (-1)r, if m = p1 ...pr, pj is prime, j = 1,..,r, 0, otherwise.
Moreover, let
m i
t(x)= J] x(«)exp{2jT7-}
m mod q ^
be the Gauss sum associated to the Dirichlet character x mod q. Then, for all s,
d\l u V\d) b\l2 x(modfe)
(h ±_\= i primitive
V ' db}
xX(s, d, x, a)L(s, x)L(s - a, x), where L(s, x) denote a Dirichlet L-function, and X(a + it, m, x, a) ^ |oa(m)|. From this it follows that
XT k 2 1 i nT r-T ^
E(o + it, - a) dt « ^ 2(1 + ~ a + ^4dtY'
b\l x(mod b) 1 primitive
and to prove Theorem 1 it remains to apply the results for the fourth moment of Dirichlet L-functions.
Y. Kamiya in [8] obtained an average mean square estimate for E^s;j,aj. He proved that, for A > 49 and T ^ra,
l 1 k 2 V £(- + 7Y;-,0) dt «; IT log4 IT.
f=1 ^[-T,T ]\[-A,A] 2 l
(k, l) = 1
The latter estimate was improved in [20]. Theorem 2. Uniformly for l ^ T as T ^m,
so i fl4+4»)l***twt.
(k, l) = 1
If l is prime, then
Y1 CT| /1 k l5 - l4 + 7l3 - Ill2 + 5l + 1 , 4 _ , 3 N
g J, K 2 + "■ 7 2jt2(/ - 1)P(/ + 1) T l0g 7 + °(r l0g ^
2. Zero distribution of e(s; y, a)
The zero distribution of zeta-functions is one of the most interesting problems and has numerous applications. B. Riemann was the first who observed a close relation of the Riemann zeta-function to the distribution of prime numbers. In 1896 de la Vallee Poussin and Hadamard proved independently that Z(1 + it) + 0, and this allowed them to obtain the asymptotic law of prime numbers:
XX
■¡—, X —> oo,
pX
du log u
As it was noted in Section 2, the famous Riemann hypothesis (RH) asserts that all non-trivial zeros of t,(s) lie on the critical line o = j- If this hypothesis is true, then
du „ ,1
jt(x) = | """ + 0(xi logx).
J2 log u
From the latter estimate RH also follows.
The zero distribution of the function E^s;j,aj depends on the parameters | and a. It is not difficult to see that
0
for o > 3. The functional equation for E^s;j,aj shows that, for o <-2 + Rea,
E^s;j,aj= 0 near the real axis. Zeros p = P + zy of E^s;j,aj in this region are called trivial. It is easily seen that
T < tf{p is trivial : |p| ^ T} < T. The non-trivial zeros of E^s; j , a) lie in the region {seC: -2 + Rea ^ a ^ 3}. Denote by N^T; f,a) the number of non-trivial zeros of E^s; f,a) with |y| ^ T. Then in [21] the following asymptotic formula has been obtained. Theorem 3. Let T ^m. Then
1 rp irri
N(T-,a)=—log— + 0(logT). i j l 2ne
We see that the main term in the formula for N(T;j,aj does not depend on the parameters k and a.
Theorem 3 is a corollary of a general result obtained in [21]. Recall that a = = Rea. Let B > 3 - a be a constant, and T ^m. Then
T Tl
^ (B + p) = (25 + a + 1)- log — + 0(log T).
P > -B
ivK t
This and Theorem 3 imply the asymptotics for the mean value of the real parts of non-trivial zeros. Theorem 4. [21]. Let T ^m. Then
*-\T-k-va) ^ p = £11+ 0(7-1).
p non - trivial
| Y | < T
Theorem 4 suggests an idea that the non-trivial zeros of E[s; f,a) lie on the line o = However, if RH holds, this is not true in general. Really, by the definition of E^s; j,aj
E(s; 1, a) = Z(s)Z(s - a). (6)
Thus, if RH holds, then E(s; l,a) = 0 on the lines o = j and o = j + a, and E(s; l,a) * 0 on the line a =
Denote by N(a,T;j,aj the number of non-trivial zeros p = |3 + iy of the function E^s; j,aj with |3 > o and |y| ^ T. Then in [21] the following bounds for N(a,T; j, aj were obtained.
Theorem 5. Let T ^ra. Then uniformly in 6 > 0
and, for fixed o > j,
For the proof the Littlewood theorem, the Jensen formula and the Jensen inequality on convex functions are applied.
Theorem 5 shows that the set of zeros on the right of the curve
1 log log t
2 log t
where y(t) > 0 and y(t) ^ ra as t ^ ra, has zero density in the set of all non-trivial zeros. Example (6) leads to the following conjecture. Conjecture. At least a positive proportion of the non-trivial zeros of E^s;j,aj is clustered around the lines a = ^ and a = j + a.
3. Universality
In [22] S.M.Voronin obtained the universality of the Riemann zeta-function. Let 0 < r < j, and let f(s) be a continuous non-vanishing function on the disc |s| ^ r which is analytic in the interior of this disc. Then he proved that, for every e > 0, there exists a real number t = x(e) such that
max
|s|<r
Z,(s + \ + n)-f(s)
< e.
4
Later, S.M. Gonek, A.Reich, B.Bagchi, K. Matsumoto, J. Stending, R. Stending, W. Schwarz, R. Garunkstis, H. Mishou, J. Genys, V. Garbaliauskiene, H. Nagoshi, R. Macaitiene, the author and others improved and generalized the Voronin theorem. Define
vt(-) = -meas{x e [0,7] : ...},
where meas{A} denotes the Lebesgue measure of a measurable set A c R, and in place of dots a condition satisfied by t is to be written. The final version of the Voronin theorem is the following [10].
Theorem 6. Let K be a compact subset of the strip Dq = {s e C : ^ < o < 1} with connected complement, and let f (s) be a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every e > 0,
liminf vT(sup |Z(s + iT) - f (s)| < e) > 0.
T seK
The case of the function E(s; j,aj is more complicated, since the factor exp|2jt7>WjJ is not multiplicative. Let % be the Dirichlet character mod/, and, for o > 1 (we recall that a ^ 0),
Oa(m)
E(s; x, a) = V -2—x(m).
¿—I ms
ms
m=1
Thus, in the definition of E^s;j,aj the arithmetic function exp|2ra/Wj| is replaced by a multiplicative function %(m). It turns out that E^s;j,aj is a linear combination of the functions E(s; x, a). For simplicity, suppose that l is a prime number.
Theorem 7. [4]. Let l be prime. Then k
T"> <p (/)
E(s; y, a)= -J- Yj x(%)x(k)E(s; a) + A(s, a)E(s; %o, a),
X( mod l) X * X0
where, for o > 0,
2l -l1-s - ls
A(s, a) =
ls(l - 1)(1 - l-s)2'
l — l1+a-s _ l1+2a + i1+2a-s _ is + l®-+s
if a = 0, otherwise.
ls(l - 1)(1 - la)(1 - l-s)(1 - la-s) '
The statement of Theorem 7 is also valid in the opposite direction. Theorem 8. [4]. Let l be prime, and x be a character modl. Then
m
E(s^>a) =zL Yj X(m)E(s; j,a)
r(y) ¿—L ^ ' v ' /
yh> m(modl)
if X * X0, and otherwise
(1 - l-s)2E(s; 1,0), if a = 0,
ls _ ls+a _ 1 + la-s + ^ _ l2a-s
E( s; X0, a) =
ls(1 - la)
-E(s;1, a), otherwise.
In any case,
the Euler product representation is valid for o > max(1 + a, 1) while the later formula holds for all s. If x + X0, then E(s; x, a) is an entire function. E(s; X0, a) has simple poles at s = 1 and s = 1 + a.
The proofs of Theorems 7 and 8 are based on the following assertions. Let (k, l) = 1. Then
exp{2ra-}=— ^ T(x)xW
X(modl)
and
<%)%(k)= 2 X(m) exp|2jt7—J.
m(mod l)
Moreover,
expi2jT4)=^o ^ Ky) Z
m |l '/(mod m)
^mj — )= 1 primitive
Since the function £(s; x, a) has the Euler product, a joint universality for it can be proved. Note that the first joint universality theorem for Dirich-let L-functions with pairwise non-equivalent characters was obtained by S.M Voronin in [23].
Theorem 9. [4]. Suppose that a < -1, l ^ 5 is prime, and that, for p = 2,3,
(7)
Z—i pmfi v 7
m=l "
with some |3 e l). For 1 ^ j ^ qp(/)7 let %j be a Dirichlet character mod I, Kj be a compact subset of the strip Dp = {s e C : |3 < o < 1} with connected complement, and let gj(s) be a continuous non-vanishing function on Kj which is analytic in the interior of Kj. Then, for every e > 0,
lim inf vT( sup sup |£(s + iT; Xj, a) - gj(s)| < e) > 0.
TKj^tp(l) seKj
Now Theorems 7 and 9 imply the universality of the Estermann zeta-func-tion.
Theorem 10. [10]. Suppose that k + 1, l + 1, a < -1, l ^ 5 is prime, and that, for p = 2,3, (7) holds. Let K be a compact subset of the strip Dp with connected complement, and let f(s) be a continuous function on K which is analytic in the interior of K. Then, for every e > 0,
lim inf vjYsup e(s + ix; -, a)-/(s) < e)> 0.
T seK l
Note that in Theorem 10, differently from Theorem 6, the function f (s) is not necessarily non-vanishing on K. This difference is explained by the existence of the Euler product for t,(s) while, for k + 1, I + \, the function E^s; j, a) has not this product.
Theorem 10 gives some information on the zero distribution of the function E(s; f,a).
Corollary. Under the assumptions of Theorem 10, for fixed o e 1) and T ^ra,
T «7V(o, r;y,a)«r.
Moreover, the real parts of zeros of the function E^s;j,aj lie dense in the interval 1).
In Theorem 10, the number l is prime. However, we conjecture that the function E^s;j,aj is universal in the Voronin sense for all I.
4. Limit theorems
The first probabilistic result for zeta-functions was obtained by H. Bohr and B. Jessen. Let R be any closed rectangle on the complex plane with the edges parallel to the axes, and let L0(T,R) denote the Jordan measure of the set
{t e [0, T] : log Z(o + it) e R}.
Suppose that o > 1. Then in [1] they proved that there exists the limit
Lo(T, R) ,m
lim ---= WJR)
T —*to T
which depends only on o and R. In [2] an analogous result was obtained for o > Let
G = {s e C : o > [J {s = a + itj : ^ < a < a,-},
sj=oj+itj
where Sj runs through all zeros of t,(s) in the region ^ < o < 1. Denote by L1,o(T, R) the Jordan measure of the set
{t e [0, T] : o + it e G, log Z(o + it) e R}.
Then in [2] H. Bohr and B. Jessen proved that there exists the limit
L1,o(T,R) T„ /m
lim ---= W\JR)
T—TO T
which depends only on o and R. For the proof of the above results the theory of sums of convex curves was used.
K. Matsumoto estimated [15], [16] the rate of convergence in Bohr-Jessen's theorems.
Bohr-Jessen's ideas were developed by A. Wintner, V. Borchsenius, A. Sel-berg, P.D.T.A. Elliott, A.Ghosh, B.Bagchi, K. Matsumoto, J. Steuding, W. Schwarz, R. Kacinskaite, R. SleZevicien^-Steuding, J. Genys, R. Macaitiene, V. Garbaliauskiene, the author and others.
The modern version of Bohr-Jessen's results can be stated in the following form. Let BB(S) stand for the class of Borel sets of the space S, and let Pn and P, n e N, be probability measures on (S, B(S)). We recall that Pn converges weakly to P as n — to if, for every real continuous bounded function f on S,
lim
im f fdPn = f fdP. Js
Theorem 11. [10]. Suppose that o> j. Then on (C,S(C)) there exists a probability measure Po such that the probability measure
vt(Z(o + it) e A), A e B(C),
converges weakly to Po as T — to.
Note that the explicit form of the limit measure PO can be given. Now let y = {s e C : |s| = 1} denote the unit circle on the complex plane, and
O =
where *p = Y for each prime p. By the Tikhonov theorem, 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 can be defined, and this gives the probability space (O, B(O), mh). Denote by №(p) the projection of № e O to the coordinate space *p, and put, for m e N,
№(p) = Y №r(p),
p ||m
where pr\\m means that pr|m but pr+1 {m. On the probability space (O,B(Q.),mH), define, for o > the complex-valued random element i?(a; |,a, (o)
by
/ k v ^ oa(m)m(m) , . k 1 E(a; j, a, co)= ^-—-exp{2ramy},
m=1
and denote by PCo its distribution, i.e.,
P^ a(A) = mH{a) e O : £(o; y, a, co)e A), A e £(C).
Theorem 12. [12]. Suppose that a > a ^ 0 and k + 1, I + 1. T/ien i/ie probability measure
k ~V
vr(£(o + 7x; y, a)e 4), 4 e 8(C),
converges weakly to PCo as T
Theorem 12 admits a joint generalization. Let, for o > max(1,1 + Reay), (kj,lj) = 1,
l kj \ -m Oaj (m) , kj 1
%-'«;)= Zj ^^exP|2jT™-}' J =
j m = 1 j
Denote
Cr = C x ... x C.
r
Suppose that ay ^ 0, j = 1 , ...,r, and for mini^y^roy > ^ and co e il, define
/1 lr
where
1 kj v ^ Oaj(mMm) f. . kj, , £(oy;-,ay,(o)= ^-^-exp|2jT7ff2 — j, j = 1 ,..,r.
j m = 1 j
Theorem 13. [14]. Suppose that mini^y^roy > cij ^ 0 and kj ± 1, lj ± 1, j =
= 1,...,r. Then the probability measure
vt((e(o i + h; ...,E[ar + h; y, <xr))e A), A e
l1 lr
converges weakly to the distribution of the random element E(o1,...,or;m) as T — to.
Another generalization of Theorem 12 is a limit theorem in the space of meromorphic functions. Let D\ = {s e C : o > and let M(D\) denote the space of meromorphic on D1 functions equipped with the topology of uniform convergence on compacta. Moreover, H(D{) is the space of analytic on D1 functions with the same topology. H(D{) is a subspace of M(D{). On the probability space (O, b(O), mh) define the H(D1)-valued random element
E{s; -,a, co)= ^-—-exp^ramyj,
m=1
and let
PEiH(A) = mH{CO e Q.: e(s; -,a, co)e A), A e B(H(D)),
be its distribution. Then we have the following result [13]. Theorem 14. Suppose that a ^ 0 and k + 1, l + 1. Then the probability measure
k T
converges weakly to Pe,h as T — to.
A joint version of Theorem 14 also can be obtained.
vr(£(s + /t; -, <x)e A), A e S(M(Di)),
References
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Paper received 17/1X/2007. Paper accepted 17/1X/2007.
НЕКОТОРЫЕ ТЕОРЕМЫ О РАСПРЕДЕЛЕНИИ ЗНАЧЕНИЙ ДЛЯ ДЗЕТА-ФУНКЦИЙ ЭСТЕРМАНА
© 2007 А. Лауринцикас2
В работе представлен обзор следующих результатов: оценки главных значений, распределение нулей, универсальные и предельные теоремы в смысле слабой сходимости вероятностных мер дзета-функции Эстерманна.
Поступила в редакцию 17/IX/2007; в окончательном варианте — 17/IX/2007.
2Лауринцикас Антанас ([email protected]), кафедра теории чисел и теории вероятности Вильнюсского университета, 24, 03225 Вильнюс, Литва.
