The zeros of the Lerch zeta-function are uniformly distributed modulo one
UDC 511.311We prove that the ordinates of the nontrivial zeros of the Lerch zeta-function are uniformly distributed modulo one.
Saved in:
| Date: | 2021 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2021
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/893 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507228337864704 |
|---|---|
| author | Garunkštis, R. Panavas, T. Garunkštis, R. Garunkštis, R. Panavas, T. |
| author_facet | Garunkštis, R. Panavas, T. Garunkštis, R. Garunkštis, R. Panavas, T. |
| author_sort | Garunkštis, R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:46:40Z |
| description | UDC 511.311We prove that the ordinates of the nontrivial zeros of the Lerch zeta-function are uniformly distributed modulo one. |
| doi_str_mv | 10.37863/umzh.v73i9.893 |
| first_indexed | 2026-03-24T02:05:59Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i9.893
UDC 511.311
R. Garunkštis*, T. Panavas (Inst. Math., Vilnius Univ., Lithuania)
THE ZEROS OF THE LERCH ZETA-FUNCTION
ARE UNIFORMLY DISTRIBUTED MODULO ONE
НУЛI ДЗЕТА-ФУНКЦIЇ ЛЕРХА,
РIВНОМIРНО РОЗПОДIЛЕНI ЗА МОДУЛЕМ 1
We prove that the ordinates of the nontrivial zeros of the Lerch zeta-function are uniformly distributed modulo one.
Доведено, що ординати нетривiальних нулiв дзета-функцiї Лерха рiвномiрно розподiленi за модулем 1.
1. Introduction. Let s = \sigma + it denote a complex variable. Denote by \{ \lambda \} the fractional part of
a real number \lambda . In this paper T always tends to plus infinity and constants in big O notations may
depend on parameters \lambda , \alpha and x.
The Lerch zeta-function is defined by
L(\lambda , \alpha , s) =
\infty \sum
m=0
e2\pi i\lambda m
(m+ \alpha )s
, \sigma > 1,
where 0 < \lambda ,\alpha \leq 1. This function has an analytic continuation to the whole complex plane except
for a possible simple pole at s = 1 (see [16, 17]). The Lerch zeta-function satisfies the functional
equation (see, for example, [16], Chapter 2, or [10], formula (1))
L(\lambda , \alpha , 1 - s) = (2\pi ) - s\Gamma (s)
\Bigl(
e\pi i
s
2
- 2\pi i\alpha \lambda L(1 - \alpha , \lambda , s) +
+ e - \pi i s
2
+2\pi i\alpha (1 - \{ \lambda \} ) L(\alpha , 1 - \{ \lambda \} , s)
\Bigr)
. (1)
Next we indicate zero free regions. Let l be a straight line in the complex plane \BbbC , and denote
by \varrho (s, l) the distance of s from l. Define, for \delta > 0,
L\delta (l) =
\bigl\{
s \in \BbbC : \varrho (s, l) < \delta
\bigr\}
.
In [7, 12], for 0 < \lambda < 1 and \lambda \not = 1/2, it is proved that L(\lambda , \alpha , s) \not = 0 if \sigma < - 1 and
s \not \in L \mathrm{l}\mathrm{o}\mathrm{g} 4
\pi
\left( \sigma =
\pi t
\mathrm{l}\mathrm{o}\mathrm{g}
1 - \lambda
\lambda
+ 1
\right) .
For \lambda = 1/2, 1, from [7, 22] we see that L(\lambda , \alpha , s) \not = 0 if \sigma < - 1 and | t| \geq 1. Moreover, in [7] it
is shown that L(\lambda , \alpha , s) \not = 0 if \sigma \geq 1 + \alpha . We say that a zero of L(\lambda , \alpha , s) is nontrivial if it lies in
the strip - 1 \leq \sigma < 1 + \alpha . The nontrivial zero is denoted by \rho = \beta + i\gamma .
* R. Garunkštis is funded by European Social Fund according to the activity improvement of researchers qualification
by implementing world-class R\&D (projects of measure No. 09.3.3-LMT-K-712-01-0037).
c\bigcirc R. GARUNKŠTIS, T. PANAVAS, 2021
1170 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
THE ZEROS OF THE LERCH ZETA-FUNCTION ARE UNIFORMLY DISTRIBUTED MODULO ONE 1171
Let \zeta (s) and L(s, \chi ) be the Riemann zeta-function and the Dirichlet L-function accordingly. We
have
L(1, 1, s) = \zeta (s) and L(1/2, 1/2, s) = 2sL(s, \chi ), (2)
where \chi is a Dirichlet character \mathrm{m}\mathrm{o}\mathrm{d}4 with \chi (3) = - 1. For these two cases, the Riemann hypothesis
can be formulated. Similar cases are L(1, 1/2, s) = (2s - 1)\zeta (s) and L(1/2, 1, s) = (1 - 21 - s)\zeta (s).
For all the other cases, it is expected that the real parts of zeros of the Lerch zeta-function form
a dense subset of the interval (1/2, 1). This is proved for any \lambda and transcendental \alpha [8] using
the universality property of the Lerch zeta-function. More about the universality of the Lerch zeta-
function see [6, 15, 18].
Denote by N(\lambda , \alpha , T ) the number of nontrivial zeros of the function L(\lambda , \alpha , s) in the region
0 < t \leq T. Then [7, 9]
N(\lambda , \alpha , T ) =
T
2\pi
\mathrm{l}\mathrm{o}\mathrm{g} T - T
2\pi
\mathrm{l}\mathrm{o}\mathrm{g}(2\pi e\alpha \lambda ) +O(\mathrm{l}\mathrm{o}\mathrm{g} T ). (3)
A sequence \{ a1, a2, a3, . . .\} of real numbers is uniformly distributed in the interval [a, b], if for
any subinterval [c, d] of [a, b] we have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
| \{ a1, a2, a3, . . . , an\} \cap [c, d]|
n
=
d - c
b - a
.
The notation | \{ a1, a2, a3, . . . , an\} \cap [c, d]| denotes the number of elements, out of the first n elements
of the sequence, that are between c and d. A sequence a1, a2, a3, . . . of real numbers is said to be
uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted by \{ an\} , is
uniformly distributed in the interval [0, 1].
Under the assumption of the truth of the Riemann hypothesis Rademacher [20] proved that the
imaginary parts of the nonreal zeros of the Riemann zeta-function are uniformly distributed modulo
one; Elliott [3] and (independently) Hlawka [14] gave unconditional proofs of this result. Further
extensions and generalizations can be found in the articles [2, 4, 5, 11, 21].
The main result of this paper is the following theorem.
Theorem 1. The imaginary parts of nontrivial zeros of the Lerch zeta-function L(\lambda , \alpha , s) are
uniformly distributed modulo one.
The proof of Theorem 1 relies on the following proposition.
Proposition 1. Let x be a fixed positive real number not equal to 1. Then\sum
0<\gamma \leq T
x\rho = (c(x) + d(x))
T
2\pi
+O(\mathrm{l}\mathrm{o}\mathrm{g} T ),
where c(x) and d(x) are complex numbers defined by formulas (9) and (16) below.
Proposition 1 and Theorem 1 are proved in Sections 2 and 3, respectively.
2. Proof of Proposition 1. Let B \geq 3 be a sufficiently large number which will be chosen
later. A strip 1 - B \leq \sigma \leq B contains all the nontrivial zeros and a finite number of trivial zeros.
Applying the residue theorem, we get\sum
0<\gamma \leq T
x\rho =
1
2\pi i
\int
\square
xs
L\prime
L
(\lambda , \alpha , s) ds+O(1), (4)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1172 R. GARUNKŠTIS, T. PANAVAS
where \square denotes the counterclockwise oriented rectangular contour with vertices B + i, B + iT,
1 - B + iT, 1 - B + i and
L\prime (\lambda , \alpha , s) =
\partial
\partial s
L(\lambda , \alpha , s).
To deal with the integral in formula (4) the following two lemmas will be useful.
Lemma 1. If f(s) is analytic and f(s0) \not = 0 with\bigm| \bigm| \bigm| \bigm| f(s)f(s0)
\bigm| \bigm| \bigm| \bigm| < eM
in \{ s : | s - s0| \leq r\} with M > 1, then\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f
\prime
f
(s) -
\sum
\rho \prime
1
s - \rho \prime
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < C
M
r
for | s - s0| \leq r
4
, where C is some constant and \rho \prime runs through the zeros of f(s) such that
| \rho \prime - s0| \leq
r
2
.
For the proof, see [23] (\S 3.9).
Lemma 1 is applied in the proof of the next lemma.
Lemma 2. Let B, b > 2 be fixed. If T is such that L(\lambda , \alpha , \sigma + iT ) \not = 0 for 1 - b \leq \sigma \leq B,
then
B\int
1 - b
\bigm| \bigm| \bigm| \bigm| L\prime
L
(\lambda , \alpha , \sigma + iT )
\bigm| \bigm| \bigm| \bigm| d\sigma \ll \mathrm{l}\mathrm{o}\mathrm{g} T.
Proof. In Lemma 1, we choose s0 = B + iT and r = 4(B - (1 - b)). It is known (see, for
example, [13], Lemma 3) that, for | s - s0| \leq r,
L(\lambda , \alpha , s) \ll T c
with some c > 0. Therefore we can take M = 2c \mathrm{l}\mathrm{o}\mathrm{g} T. Then Lemma 1 gives
L\prime
L
(\lambda , \alpha , s) =
\sum
| \rho - s0| \leq r
2
1
s - \rho
+O(\mathrm{l}\mathrm{o}\mathrm{g} T ) (5)
for | s - s0| \leq
r
4
. Note that the points B+ iT and 1 - b+ iT are not very near to zeros of L(\lambda , \alpha , s).
Thus,
B\int
1 - b
\bigm| \bigm| \bigm| \bigm| L\prime
L
(\lambda , \alpha , \sigma + iT )
\bigm| \bigm| \bigm| \bigm| d\sigma \leq
B\int
1 - b
\sum
| \rho - s0| \leq r
2
\bigm| \bigm| \bigm| \bigm| 1
\sigma + iT - \rho
\bigm| \bigm| \bigm| \bigm| d\sigma +O(\mathrm{l}\mathrm{o}\mathrm{g} T ) =
=
\sum
| \rho - s0| \leq r
2
B\int
1 - b
1\sqrt{}
(\sigma - \beta )2 + (T - \gamma )2
d\sigma +O(\mathrm{l}\mathrm{o}\mathrm{g} T ) =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
THE ZEROS OF THE LERCH ZETA-FUNCTION ARE UNIFORMLY DISTRIBUTED MODULO ONE 1173
=
\sum
| \rho - s0| \leq r
2
\Bigl(
\mathrm{l}\mathrm{o}\mathrm{g}
\Bigl(
B - \beta +
\sqrt{}
(T - \gamma )2 + (B - \beta )2
\Bigr)
-
- \mathrm{l}\mathrm{o}\mathrm{g}
\Bigl(
1 - b - \beta +
\sqrt{}
(T - \gamma )2 + (1 - b - \beta )2)
\Bigr) \Bigr)
+O(\mathrm{l}\mathrm{o}\mathrm{g} T ) \ll
\ll \mathrm{l}\mathrm{o}\mathrm{g} T,
since the inequality | \rho - s0| \leq r
2
is satisfied with O(\mathrm{l}\mathrm{o}\mathrm{g} T ) many zeros \rho (see the asymptotic
formula (3)).
Lemma 2 is proved.
Proof of Proposition 1. We consider the contour integral in formula (4):\int
\square
xs
L\prime (\lambda , \alpha , s)
L(\lambda , \alpha , s)
ds =
=
\left\{
B+iT\int
B+i
+
1 - B+iT\int
B+iT
+
1 - B+i\int
1 - B+iT
+
B+i\int
1 - B+i
\right\} xs
L\prime (\lambda , \alpha , s)
L(\lambda , \alpha , s)
ds =
4\sum
j=1
Ij . (6)
Let xm+1 = xm+1(\alpha ) = (m+ \alpha )/\alpha , m = 0, 1, . . . , be the sequence X and define
S = \{ xk1xk2 . . . xkm : m \in \BbbN , k1 \in \BbbN , . . . , km \in \BbbN \}
as the set of all possible products of elements of the sequence X. Let
1 = y1(\alpha ) < y2(\alpha ) < . . . (7)
be an ordered sequence of all different numbers of S. By Lemma 8 in [11] there are \sigma 1 \geq 1
and complex numbers cn, n = 1, 2, . . . , such that the logarithmic derivative of L(\lambda , \alpha , s) has an
absolutely convergent Dirichlet series expansion
L\prime (\lambda , \alpha , s)
L(\lambda , \alpha , s)
=
\infty \sum
n=1
cn
ysn(\alpha )
, \sigma > \sigma 1.
Let B > \sigma 1. Interchanging summation and integration, we find
I1 =
\infty \sum
n=1
cn
B+iT\int
B+i
\biggl(
x
yn(\alpha )
\biggr) s
ds =
\infty \sum
n=2
cni
T\int
1
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
(B + it) \mathrm{l}\mathrm{o}\mathrm{g}(x/yn(\alpha ))
\bigr)
dt =
=
\infty \sum
n=1
cni \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
B \mathrm{l}\mathrm{o}\mathrm{g}(x/yn(\alpha ))
\bigr) T\int
1
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
it \mathrm{l}\mathrm{o}\mathrm{g}(x/yn(\alpha ))
\bigr)
dt.
In view of
T\int
1
\mathrm{e}\mathrm{x}\mathrm{p}(it \mathrm{l}\mathrm{o}\mathrm{g}(x/yn(\alpha ))) dt =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1174 R. GARUNKŠTIS, T. PANAVAS
=
\left\{ T - 1 if x = yn(\alpha ),\bigl(
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
iT \mathrm{l}\mathrm{o}\mathrm{g}(x/yn(\alpha ))
\bigr)
- \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
i \mathrm{l}\mathrm{o}\mathrm{g}(x/yn(\alpha ))
\bigr) \bigr)
/
\bigl(
i \mathrm{l}\mathrm{o}\mathrm{g}(x/yn(\alpha ))
\bigr)
otherwise,
we get
I1 = ic(x)T +O(1), (8)
where
c(x) =
\left\{ cn if x = yn(\alpha ),
0 otherwise.
(9)
To evaluate the integral
I3 = -
1 - B+iT\int
1 - B+i
xs
L\prime (\lambda , \alpha , s)
L(\lambda , \alpha , s)
ds (10)
we will use the functional equation (1). The logarithmic derivative of the functional equation is
L\prime
L
(\lambda , \alpha , s) = \mathrm{l}\mathrm{o}\mathrm{g} 2\pi \lambda - \Gamma \prime
\Gamma
(1 - s) - \pi i
2
- E\prime
E
(\lambda , \alpha , 1 - s), (11)
where, for \sigma < - 1,
E(\lambda , \alpha , 1 - s) := 1 +
\infty \sum
m=1
e - 2\pi i\alpha m\biggl(
\lambda +m
\lambda
\biggr) 1 - s + e - \pi i(1 - s)e2\pi i\alpha (1+\lambda - \{ \lambda \} )
\infty \sum
m=0
e2\pi i\alpha m\biggl(
1 - \{ \lambda \} +m
\lambda
\biggr) 1 - s
and
E\prime (\lambda , \alpha , s) =
\partial
\partial s
E(\lambda , \alpha , s).
We have
1 - B+iT\int
1 - B+i
xs \mathrm{l}\mathrm{o}\mathrm{g}(2\pi \lambda )ds = O(1). (12)
It is known (see formula 6.3.18 in [1]) that, for | \mathrm{a}\mathrm{r}\mathrm{g} s| < \pi ,
\Gamma \prime
\Gamma
(s) = \mathrm{l}\mathrm{o}\mathrm{g} s+O
\biggl(
1
| s|
\biggr)
, s \rightarrow \infty .
Thus,
1 - B+iT\int
1 - B+i
xs
\Gamma \prime
\Gamma
(1 - s)ds = ix1 - B
T\int
1
xit
\biggl(
\mathrm{l}\mathrm{o}\mathrm{g} t - \pi
2
+O
\biggl(
1
t
\biggr) \biggr)
dt = O(\mathrm{l}\mathrm{o}\mathrm{g} T ). (13)
Let
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
THE ZEROS OF THE LERCH ZETA-FUNCTION ARE UNIFORMLY DISTRIBUTED MODULO ONE 1175
F (\lambda , \alpha , 1 - s) = 1 +
\infty \sum
m=1
e - 2\pi i\alpha m\biggl(
\lambda +m
\lambda
\biggr) 1 - s , \sigma < - 1,
and
F \prime (\lambda , \alpha , s) =
\partial
\partial s
F (\lambda , \alpha , s).
Then, for \sigma < - 1,
E\prime (\lambda , \alpha , 1 - s)
E(\lambda , \alpha , 1 - s)
=
F \prime (\lambda , \alpha , 1 - s)
F (\lambda , \alpha , 1 - s)
+O(e - t), t \rightarrow \infty . (14)
Again, by Lemma 8 in [11] there are complex numbers dn, n = 1, 2, . . . , such that the logarithmic
derivative of F (\lambda , \alpha , s) has the Dirichlet series expansion
F \prime (\lambda , \alpha , s)
F (\lambda , \alpha , s)
=
\infty \sum
n=1
dn
ysn(\lambda )
,
which converges absolutely for \Re s \geq B if B is sufficiently large. The numbers yn(\lambda ), n = 1, 2, . . . ,
are defined by (7). This and formula (14) give
1 - B+iT\int
1 - B+i
xs
E\prime (\lambda , \alpha , 1 - s)
E(\lambda , \alpha , 1 - s)
ds = ix1 - B
\infty \sum
n=2
dn
yBn (\lambda )
T\int
1
\biggl(
x
yn(\lambda )
\biggr) it
dt+O(1) =
= id(x)T +O(1), (15)
where
d(x) =
\left\{ dnyn(\lambda ) if x = yn(\lambda ),
0 otherwise.
(16)
By formulae (10) – (13) and (15) we obtain
I3 = id(x)T +O(\mathrm{l}\mathrm{o}\mathrm{g} T ). (17)
We observe that I4 does not depend on T. Therefore,
I4 = O(1). (18)
Lemma 2 gives
I2 =
B+iT\int
1 - B+iT
xs
L\prime (\lambda , \alpha , s)
L(\lambda , \alpha , s)
ds =
B\int
1 - B
x\sigma +iT L
\prime (\lambda , \alpha , \sigma + iT )
L(\lambda , \alpha , \sigma + iT )
d\sigma \ll \mathrm{l}\mathrm{o}\mathrm{g} T. (19)
Formulae (4), (6), (8), (17), (18), and (19) prove Proposition 1.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1176 R. GARUNKŠTIS, T. PANAVAS
3. Proof of Theorem 1. In the proof of Theorem 1 the following Weyl’s criterion for the
uniform distribution will be important.
Lemma 3. A sequence of real numbers yn is uniformly distributed modulo one if and only if ,
for each integer m \not = 0,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n
n\sum
j=0
e2\pi imyj = 0.
For the proof, see [24, 25].
Let N (1)(T ) be the number of nontrivial zeros of L(\lambda , \alpha , s) in
\beta >
1
2
+ (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2/ \mathrm{l}\mathrm{o}\mathrm{g} T, 0 < t \leq T.
Let N (2)(T ) be the number of those in
\beta <
1
2
- (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2/ \mathrm{l}\mathrm{o}\mathrm{g} T, 0 < t \leq T,
and let N (3)(T ) be the number of those in
1
2
- (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2/ \mathrm{l}\mathrm{o}\mathrm{g} T \leq \beta \leq 1
2
+ (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2/ \mathrm{l}\mathrm{o}\mathrm{g} T, 0 < t \leq T.
The following clustering of the nontrivial zeros around the critical line \sigma = 1/2 will be useful in the
proof of Theorem 1.
Proposition 2. We have
N (1)(T ) \ll T \mathrm{l}\mathrm{o}\mathrm{g} T
\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T
, (20)
N (2)(T ) \ll T \mathrm{l}\mathrm{o}\mathrm{g} T
\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T
, (21)
N (3)(T ) =
T
2\pi
\mathrm{l}\mathrm{o}\mathrm{g}
T
2\pi e\alpha \lambda
+O
\biggl(
T \mathrm{l}\mathrm{o}\mathrm{g} T
\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T
\biggr)
. (22)
To prove the proposition we will need the following two lemmas.
Lemma 4. We have \sum
0<\gamma \leq T
\beta >1/2
\biggl(
\beta - 1
2
\biggr)
= O(T \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T ).
Proof. From the proof of Theorem 1 in [9] we have, for b > - 3,
2\pi
\sum
0<\gamma \leq T
\beta > - b
(b+ \beta ) =
T\int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \alpha - b+itL(\lambda , \alpha , - b+ it)
\bigm| \bigm| \bigm| dt+O(\mathrm{l}\mathrm{o}\mathrm{g} T ).
Theorem 1 in [10] gives the bound
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
THE ZEROS OF THE LERCH ZETA-FUNCTION ARE UNIFORMLY DISTRIBUTED MODULO ONE 1177
T\int
0
\bigm| \bigm| \bigm| \bigm| L(\lambda , \alpha , 12 + it)
\bigm| \bigm| \bigm| \bigm| 2 dt = O(T \mathrm{l}\mathrm{o}\mathrm{g} T ).
Choosing b = - 1
2
, by the concavity of the logarithm we get
2\pi
\sum
0<\gamma \leq T
\beta >1/2
\biggl(
\beta - 1
2
\biggr)
=
T\int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \bigm| \alpha sL(\lambda , \alpha ,
1
2
+ it)
\bigm| \bigm| \bigm| \bigm| dt+O(\mathrm{l}\mathrm{o}\mathrm{g} T ) =
=
T\int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| \bigm| L(\lambda , \alpha , 12 + it)
\bigm| \bigm| \bigm| \bigm| dt+O(T ) \leq
\leq 1
2
T \mathrm{l}\mathrm{o}\mathrm{g}
\left( 1
T
T\int
0
\bigm| \bigm| \bigm| \bigm| L(\lambda , \alpha , 12 + it)
\bigm| \bigm| \bigm| \bigm| 2 dt
\right) +O(T ) \ll
\ll T \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T.
Lemma 4 is proved.
Lemma 5. Let b \geq 3 be a constant. For 0 < \lambda ,\alpha \leq 1,\sum
0<\gamma \leq T
(b+ \beta ) =
\biggl(
b+
1
2
\biggr)
T
2\pi
\mathrm{l}\mathrm{o}\mathrm{g}
T
4\pi e\alpha \lambda
+
T
4\pi
\mathrm{l}\mathrm{o}\mathrm{g}
\alpha
\lambda
+O(\mathrm{l}\mathrm{o}\mathrm{g} T ).
For the proof, see [9] (Theorem 1) and [13] (Lemma 1).
Proof of Proposition 2. From Lemma 4 and the definition of N (1)(T ) we see that
0 \leq N (1)(T )(\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2/ \mathrm{l}\mathrm{o}\mathrm{g} T \leq O(T \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T ).
This proves the bound (20).
Next we consider the bound for N (2)(T ). Let \delta = (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2/ \mathrm{l}\mathrm{o}\mathrm{g} T. We have\sum
0<\gamma \leq T
(b+ \beta ) =
\sum
0<\gamma \leq T
\beta \geq 1/2 - \delta
(b+ \beta ) +
\sum
0<\gamma \leq T
\beta <1/2 - \delta
(b+ \beta ). (23)
Definitions of N (1)(T ), N (3)(T ), and Lemma 4 give\sum
0<\gamma \leq T
\beta \geq 1/2 - \delta
(b+ \beta ) \leq
\sum
0<\gamma \leq T
\beta >1/2
\biggl(
\beta - 1
2
\biggr)
+
\biggl(
b+
1
2
\biggr)
(N (1)(T ) +N (3)(T )) \leq
\leq O(T \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T ) +
\biggl(
b+
1
2
\biggr)
(N (1)(T ) +N (3)(T )).
For the second sum in the right-hand side of formula (23), we obtain
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1178 R. GARUNKŠTIS, T. PANAVAS
\sum
0<\gamma \leq T
\beta <1/2 - \delta
(b+ \beta ) \leq
\biggl(
b+
1
2
- (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2/ \mathrm{l}\mathrm{o}\mathrm{g} T
\biggr)
N (2)(T ).
The last three formulas yield the inequality\sum
0<\gamma \leq T
(b+ \beta ) \leq O(T \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T ) +
\biggl(
b+
1
2
\biggr)
N(\lambda , \alpha , T ) - (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2N (2)(T )/ \mathrm{l}\mathrm{o}\mathrm{g} T.
By this, Lemma 5, and formula (3) we see that
0 \leq O(T \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T ) - (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2N (2)(T )/ \mathrm{l}\mathrm{o}\mathrm{g} T.
This proves formula (21). Formulae (20) and (21) together with the asymptotic formula (3) yield the
equality (22).
Proposition 2 is proved.
Proposition 2 leads to the following lemma.
Lemma 6. We have \sum
0<\gamma \leq T
\bigm| \bigm| \bigm| \bigm| \beta - 1
2
\bigm| \bigm| \bigm| \bigm| \ll T \mathrm{l}\mathrm{o}\mathrm{g} T
\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T
.
Proof. As in the proof of Proposition 2 we denote \delta = (\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T )2/ \mathrm{l}\mathrm{o}\mathrm{g} T. Then
\sum
0<\gamma \leq T
\bigm| \bigm| \bigm| \bigm| \beta - 1
2
\bigm| \bigm| \bigm| \bigm| =
\left\{
\sum
0<\gamma \leq T
\beta - 1/2>\delta
+
\sum
0<\gamma \leq T
| \beta - 1/2| \leq \delta
+
\sum
0<\gamma \leq T
\beta - 1/2< - \delta
\right\}
\bigm| \bigm| \bigm| \bigm| \beta - 1
2
\bigm| \bigm| \bigm| \bigm| . (24)
In view of zero free regions of the Lerch zeta-function we have that | \beta - 1/2| \ll 1. Then the
expression (24) together with Proposition 2 proves Lemma 6.
Proof of Theorem 1. Similarly, as in the proof of Theorem 1 in [21], we can write
x
1
2
\sum
0<\gamma \leq T
xi\gamma =
\sum
0<\gamma \leq T
x\beta +i\gamma +
\sum
0<\gamma \leq T
(x1/2+i\gamma - x\beta +i\gamma ). (25)
We find
x
1
2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum
0<\gamma \leq T
xi\gamma
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum
0<\gamma \leq T
x\beta +i\gamma
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
\sum
0<\gamma \leq T
\bigm| \bigm| \bigm| x1/2+i\gamma - x\beta +i\gamma
\bigm| \bigm| \bigm| .
By inequality
\bigm| \bigm| \mathrm{e}\mathrm{x}\mathrm{p}(y) - 1
\bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
y\int
0
\mathrm{e}\mathrm{x}\mathrm{p}(t) dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq | y| \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, \mathrm{e}\mathrm{x}\mathrm{p}(y)\} ,
where y is a real number, we get
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
THE ZEROS OF THE LERCH ZETA-FUNCTION ARE UNIFORMLY DISTRIBUTED MODULO ONE 1179
| x1/2+i\gamma - x\beta +i\gamma | = x\beta
\bigm| \bigm| \bigm| \bigm| \mathrm{e}\mathrm{x}\mathrm{p}\biggl( \biggl( 1
2
- \beta
\biggr)
\mathrm{l}\mathrm{o}\mathrm{g} x
\biggr)
- 1
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \beta - 1
2
\bigm| \bigm| \bigm| \bigm| | \mathrm{l}\mathrm{o}\mathrm{g} x| \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
x\beta , x1/2
\bigr\}
.
Then by Lemma 6 and formula (3) we have
1
N(\lambda , \alpha , T )
\sum
0<\gamma \leq T
| x1/2+i\gamma - x\beta +i\gamma | \leq
\leq
| \mathrm{l}\mathrm{o}\mathrm{g} x| \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
x\beta , x1/2
\bigr\}
N(\lambda , \alpha , T )
\sum
0<\gamma \leq T
\bigm| \bigm| \bigm| \bigm| \beta - 1
2
\bigm| \bigm| \bigm| \bigm| \ll 1
\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T
.
This, formula (25), and Proposition 1 give that
1
N(\lambda , \alpha , T )
\sum
0<\gamma \leq T
x1/2+i\gamma \ll 1
\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T
.
Let x = zm with some positive z \not = 1 and m \in \BbbN . Then, after dividing the previous formula by
x
1
2 , we deduce
\mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
1
N(\lambda , \alpha , T )
\sum
0<\gamma \leq T
\mathrm{e}\mathrm{x}\mathrm{p}(im\gamma \mathrm{l}\mathrm{o}\mathrm{g} z) = 0.
Then Weyl’s criterion (Lemma 3) implies that the sequence of numbers \gamma \mathrm{l}\mathrm{o}\mathrm{g} z/2\pi is uniformly
distributed modulo 1.
Theorem 1 is proved.
References
1. M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables,
National Bureau of Standards, Wiley-Intersci. Publ., New York (1972).
2. A. Akbary, M. R. Murty, Uniform distribution of zeros of Dirichlet series, Anatomy of Integer, CRM Proc. Lecture
Notes, 46, 143 – 158 (2008).
3. P. D. T. A. Elliott, The Riemann zeta function and coin tossing, J. reine und angew. Math., 254, 100 – 109 (1972).
4. K. Ford, K. Soundararajan, A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function,
II, Math. Ann., 343, 487 – 505 (2009).
5. A. Fujii, On the uniformity of the distribution of zeros of the Riemann zeta function, J. reine und angew. Math., 302,
167 – 205 (1978).
6. R. Garunkštis, The universality theorem with weight for the Lerch zeta-function, New Trends in Probability and
Statistics, vol. 4 (Palanga, 1996), VSP, Utrecht (1997).
7. R. Garunkštis, A. Laurinčikas, On zeros of the Lerch zeta-function, Number Theory and Its Applications, S. Kanemitsu,
K. Gyory (eds.), Kluwer Acad. Publ., 129 – 143 (1999).
8. R. Garunkštis, A. Laurinčikas, The Lerch zeta-function, Integral Transforms Spec. Funct., 10, 211 – 226 (2000).
9. R. Garunkštis, J. Steuding, On the zero distributions of Lerch zeta-functions, Analysis, 22, 1 – 12 (2002).
10. R. Garunkštis, A. Laurinčikas, J. Steuding, On the mean square of Lerch zeta-functions, Arch. Math., 80, 47 – 60
(2003).
11. R. Garunkštis, J. Steuding, R. Šimėnas, The a-points of the Selberg zeta-function are uniformly distributed modulo
one, Illinois J. Math., 58, 207 – 218 (2014).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1180 R. GARUNKŠTIS, T. PANAVAS
12. R. Garunkštis, J. Steuding, Do Lerch zeta-functions satisfy the Lindeloff hypothesis?, Anal. and Probab. Methods
Number Theory, Proc. Third Intern. Conf. Honour of J. Kubilius (Palanga, Lithuania, 24 – 28 September 2001), TEV,
Vilnius (2002), p. 61 – 74.
13. R. Garunkštis, R. Tamošiūnas, Symmetry of zeros of Lerch zeta-function for equal parameters, Lith. Math. J., 57,
433 – 440 (2017).
14. E. Hlawka, Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafunktionen
zusammenhängen, Österr. Akad. Wiss., Math.-Natur. Kl. Abt. II, 184, 459 – 471 (1975).
15. A. Laurinčikas, The universality of the Lerch zeta-function, Lith. Math. J., 37, 275 – 280 (1997).
16. A. Laurinčikas, R. Garunkštis, The Lerch zeta-function, Kluwer Acad. Publ., Dordrecht (2002).
17. M. Lerch, Note sur la fonction \scrK (z, x, s) =
\sum \infty
k=0
e2k\pi ix(z + k) - s , Acta Math., 11, 19 – 24 (1887).
18. Y. Lee, T. Nakamura, Ł. Pańkowski, Joint universality for Lerch zeta-functions, J. Math. Soc. Japan, 69, 153 – 161
(2017).
19. N. Levinson, Almost all root of \zeta (s) = a are arbitrarily close to \sigma = 1/2, Proc. Nat. Acad. Sci. USA, 72, 1322 – 1324
(1975).
20. H. G. Rademacher, Fourier analysis in number theory, Symp. Harmonic Analysis and Related Integral Transforms
(Cornell Univ., Ithaca, N.Y., 1956), Collected Papers of Hans Rademacher, vol. II (1974), p. 434 – 458.
21. J. Steuding, The roots of the equation \zeta (s) = a are uniformly distributed modulo one, Anal. and Probab. Methods
Number Theory TEV, Vilnius (2012), p. 243 – 249.
22. R. Spira, Zeros of Hurwitz zeta-functions, Math. Comput., 136, 863 – 866 (1976).
23. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., rev. by D. R. Heath-Brown, Oxford Sci. Publ.,
Clarendon Press, Oxford (1986).
24. H. Weyl, Sur une application de la théorie des nombres à la mécaniques statistique et la théorie des pertubations,
Enseign. Math., 16, 455 – 467 (1914).
25. H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann., 77, 313 – 352 (1916).
Received 20.02.18,
after revision — 22.11.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
|
| id | umjimathkievua-article-893 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:59Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9a/603dc752a5c0ae360c26d9204efa339a.pdf |
| spelling | umjimathkievua-article-8932025-03-31T08:46:40Z The zeros of the Lerch zeta-function are uniformly distributed modulo one The zeros of the Lerch zeta-function are uniformly distributed modulo one Garunkštis, R. Panavas, T. Garunkštis, R. Garunkštis, R. Panavas, T. Lerch zeta-function zero distribution uniform distribution Lerch zeta-function zero distribution uniform distribution UDC 511.311We prove that the ordinates of the nontrivial zeros of the Lerch zeta-function are uniformly distributed modulo one. UDC 511.311 Нулi дзета-функцiї Лерха, рiвномiрно розподiленi за модулем 1 Доведено, що ординати нетривiальних нулiв дзета-функцiї Лерха рiвномiрно розподiленi за модулем 1. Institute of Mathematics, NAS of Ukraine 2021-09-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/893 10.37863/umzh.v73i9.893 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 9 (2021); 1170 - 1180 Український математичний журнал; Том 73 № 9 (2021); 1170 - 1180 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/893/9103 |
| spellingShingle | Garunkštis, R. Panavas, T. Garunkštis, R. Garunkštis, R. Panavas, T. The zeros of the Lerch zeta-function are uniformly distributed modulo one |
| title | The zeros of the Lerch zeta-function are uniformly distributed modulo one |
| title_alt | The zeros of the Lerch zeta-function are uniformly distributed modulo one |
| title_full | The zeros of the Lerch zeta-function are uniformly distributed modulo one |
| title_fullStr | The zeros of the Lerch zeta-function are uniformly distributed modulo one |
| title_full_unstemmed | The zeros of the Lerch zeta-function are uniformly distributed modulo one |
| title_short | The zeros of the Lerch zeta-function are uniformly distributed modulo one |
| title_sort | zeros of the lerch zeta-function are uniformly distributed modulo one |
| topic_facet | Lerch zeta-function zero distribution uniform distribution Lerch zeta-function zero distribution uniform distribution |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/893 |
| work_keys_str_mv | AT garunkstisr thezerosofthelerchzetafunctionareuniformlydistributedmoduloone AT panavast thezerosofthelerchzetafunctionareuniformlydistributedmoduloone AT garunkstisr thezerosofthelerchzetafunctionareuniformlydistributedmoduloone AT garunkstisr thezerosofthelerchzetafunctionareuniformlydistributedmoduloone AT panavast thezerosofthelerchzetafunctionareuniformlydistributedmoduloone AT garunkstisr zerosofthelerchzetafunctionareuniformlydistributedmoduloone AT panavast zerosofthelerchzetafunctionareuniformlydistributedmoduloone AT garunkstisr zerosofthelerchzetafunctionareuniformlydistributedmoduloone AT garunkstisr zerosofthelerchzetafunctionareuniformlydistributedmoduloone AT panavast zerosofthelerchzetafunctionareuniformlydistributedmoduloone |