Limit theorems for Lerch zeta-functions with algebraic irrational parameter ; Lercho dzeta funkcijų su algebriniu iracionaliuoju parametru ribinės teoremos

English
65 Pages
Read an excerpt
Gain access to the library to view online
Learn more

Informations

Published by
Published 01 January 2010
Reads 9
Language English
Report a problem

sciences,
UN
IVERSITY
Dan
ut

e
Regina
Ph
(01P)
WIT
ALGEBRAIC
TIONAL
ARAMETER
ctoral
Genien
20

H
e
IRRA
LIMIT
P
THEOREMS
Do
F
dissertation
OR
ysical
LER
mathematics
CH
Vilnius,
ZET
09
A-FUNCTIONS
VILNIUSPh
Laurin£
9
Scien
.
in
Univ
w
sciences,
as
consult
Dr.
out
ta
ork
as
2003200
y
tic
-
scien
w
carried
t:
at
tic
’iauliai
an
Univ
Prof.
ersit
Habil
y
An
.
nas
The
ik
dissertation
(Vilnius
w
ersit
as
,
prepared
ysical
ext
Mathematics
ernally
01P)
.
Them
U
S
UNIVERSITET
AS
Dan
ut

disertacija
Matematik
20
IRA
P
U

TEOREMOS
e
(01P)
Regina
ALGEBRINIU
Genien
CIONALIUOJU

ARAMETR
e
RIBIN
LER
ES
CHO
Daktaro
DZET
Fiziniai
A
okslai,
FUNK
a
CIJU
Vilnius,

09
SU
VILNIAmatematik
rengta
2003
2009
metais
’iauli

u
ziniai
-
Habil.
An
La
in£ik
(Vil
Univ
univ
Prof.
ersite
Dr.
te.
tanas
Disertacija
ur
ginama
as
ekstern
niaus
u.
ersitetas,
Mokslinis
mokslai,
k
a
onsultan
01P)
tas:
DisertacijaTheorem
33
.
.
Pro
.
.
.
of
parameter
.
1.4.
.
.
.
limi
the
in
.
3.
.
.
.
.
.
.
.
.
.
.
.
.
o
.
.
.
.
.
.
.
.
y
.
.
with
.
y
theorem
mean
.
.
statemen
.
.
.
on
.
.
.
.
.
.
.
.
.
.
.
.
.
t
.
irrational
.
t
.
.
7
.
.
.
6
v
.
.
.
.
.
.
.
.
t
.
.
of
.
.
.
analytic
.
.
of
.
Le
.
.
.
.
.
.
.
.
.
.
.
.
.
of
.
.
.
.
.
.
.
65
.
.
.
the
.
.
.
.
.
.
.
1.1
.
.
.
.
.
.
.
A
.
the
.
h
Principal
.
.
t
.
.
.
.
.
.
.
.
.
.
y
.
.
.
.
Case
.
hlet
.
.
.
36
.
.
.
.
.
.
.
2.5.
.
.
A
.
.
.
.
.
.
in
.
the
.
irrational
.
3.1.
.
theorem
.
3.2.
.
t
Chapter
.
.
.
complex
Appro
.
.
wit
.
.
.
.
of
.
.
.
.
.
.
main
54
.
.
.
.
.
.
.
.
.
60
1.2.
.
.
.
.
.
m
.
i
.
.
.
.
.
.
.
.
.
erget
.
ries
.
.
.
.
.
ximation
.
.
.
.
Approbation
.
.
.
.
.
.
.
.
24
.
of
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter
.
o
.
theorem
.
plane
.
the
.
with
.
.
.
.
.
The
.
a
.
theorem
.
.
..
.
.
.
.
.
.
2.2.
.
theorem
.
.
v
.
.
.
.
.
.
.
.
.
Outline
.
.
34
sis
absolutely
.
t
.
.
and
.
.
.
.
.
.
Appro
A
mean
.
.
.
.
.
.
.
.
.
.
.
.
.
of
duction
.
.
.
.
.
kno
.
.
.
.
.
.
.
.
39
.
limit
.
he
.
functions
.
h
.
i
.
.
.
.
.
statemen
,
limi
.
space
.
.
.
of
.
v
.
.
Defended
.
A
.
.
.
t
.
thesis
48
for
in
.
.
h
.
.
.
algebraic
.
.
.
.
.
.
3.4.
.
Theorem
.
.
.
.
.
.
1.1.
.
.
.
of
.
.
.
m
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
y
8
.
limit
.
.
.
torus
.
problem
.
.
.
Metho
.
.
.
ds
.
n
Notation
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Diric
.
se
.
.
.
.
.
.
.
.
.
20
.
Appro
.
in
.
mean
.
.
.
.
8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1.5.
.
of
.
Theorem
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
28
.
2.
.
j
.
in
.
limit
.
on
.
complex
.
f
.
r
.
Lerc
.
zeta-functions
16
algebraic
.
parameters
publications
.
.
.
.
.
tro
2.1.
.
statemen
.
of
.
join
.
limit
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
33
.
A
.
t
.
on
No
.
.
.
.
.
.
.
elt
.
.
.
.
.
.
.
.
.
.
.
.
.
16
.
.
.
of
.
.
.
the
2.3.
.
of
.
con
Aims
ergen
.
Diric
.
series
.
.
6
.
.
.
.
.
.
.
problems
.
.
.
.
2.4.
.
ximation
.
the
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ctualit
.
.
.
.
.
.
.
.
.
.
38
.
Pro
.
of
ts
2.1
.
.
.
.
16
.
.
.
c
.
.
.
wledgmen
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter
.
A
.
theorem
.
t
.
space
.
analytic
.
for
.
Lerc
.
zeta-function
.
algebra
.
c
.
parameter
.
.
.
.
.
47
.
The
.
t
.
the
.
t
.
the
.
of
.
functions
.
47
.
Case
.
absolutel
.
con
.
ergen
8
series
16
.
.
.
1.
.
results
.
limit
.
.
.
on
.
the
.
he
.
.
.
plane
3.3.
.
ximation
the
the
.
.
rc
.
.
.
zeta-function
.
.
.
h
.
.
.
irrational
.
.
.
.
.
.
.
.
.
.
50
.
Pro
.
of
.
3.1
.
.
.
.
.
.
.
.
.
.
17
.
.
.
The
.
.
.
t
.
.
.
the
.
.
.
theore
.
.
Conclusions
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Bibliograph
17
.
.
.
A
.
History
.
theorem
.
.
.
the
.
the
.
.
.
7
.
.
.
and
.
.
.
.
.
.
.
a
.
ten
In
.
.
.
.
.
results
.
.
.
.
.
.
61
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
19
.
1.3.
.
Limit
.
theorems
.
for
.
absolutely
.
con
.
v
5
hlet
Con
Ω
rΩzeta-function.
as
ortan
prop
[64]
analytic
In
for
duced
The
]
pro
and
Sev
icu-
form
b
,
er
ze
ta
F
c
of
,
the
as
a
d.
the
form
of
are
rc
of
elsewhere.
satised
for
loss
n
supp
e
.
to
w
,
in
is
,
pro
dev
[50].
satises
the
giv
[12],
[46].
the
a
con
Diric
osed
at
ourier
ha
while
in
the
theory
ofs
b
rst
es,
n
tin
based
pa
nce
f
func
zeta-function
case,
,
generalit
is
e
in
that
in
Lerc
so
Lerc
on,
func
Hurwitz
in
o
enden
ion
and
y
all
h
function
complex
to
ctualit
with
is
Chapter
[49]
functional
with
in
16
tion
other
theory
see
see
sum
h
ell
et
in
function
using
ving
si
p
Berndt
zeta-function
series
can
the
L
pap
arious
is
for
oisson
y
In
lar
pro
and
this
,
wn.
y
of
teresting
en
con
A
rt
[1]
uation
a
h
and
I
tial
some
y
er
function.
b
this
um
without
,
of
then
y
prim
w
analytic
can
dened,
ose
t
There
duction
h
imp
the
except
The
reduces
h
v
ta-
the
tion
zeta-functi
as
ze
tro
a
indep
-funct
tly
b
[52]
not
[53].
duct
or
zeta-function
of
lassical
the
Lerc
erties
is
statistical
y
oted
a
is
A
[46]
Euler
5
and
.
function
the
[47],
equation
v
[46
]
en
hand,
is
[
func
parameters
of
also
The
the
also
found,
ula,
mation
whic
Euler-Maclaurin
-functions.
as
is
w
no
tegration
meromorphic
tour
hl
the
with
ofs
The
mple
simple
[4]
or
prop
ole
B.C.
and
metho
b
to
Riemann
F
for
uses
therefore,
er
spaces
the
example,
applied,
v
ula
cases,
summation
measures
P
as,
[65],
probabilit
.
e
eral
ergence
of
tro
Let
it
ariable.
the
On
e
form
v
equation
con
kno
.
The
If
pro
k
w
a
giv
e
i
,
[52].
the
pro
n
in
w
is
of
on
sense
transformation
the
ula
in
diere
theorems
dieren
limit
equation
is
b
an
the
en
tion
tire
6
In
s = σ +it L(λ,α,s)
λ∈R α∈R 0<α≤ 1 σ > 1
∞ 2πiλmX e
L(λ,α,s) =
s(m+α)
m=0
λ∈Z L(λ,α,s)
∞X 1
ζ(s,α) = , σ > 1,
s(m+α)
m=0
s = 1 Res ζ(s,α) =s=1
1 λ6∈ Z L(λ,α,s)
0<λ< 1
s L(λ,α,s) 0<λ < 1
Γ(s) πis
L(λ,α,1−s) = (exp{ −2πiαλ}L(−α,λ,s)+
s(2π) 2
πis
exp{− +2πiα(1−λ)}L(α,1−λ,s)).
2
L(λ,α,s)
L(λ,α,s)
L(λ,α,s)
L
L(λ,α,s)h
[47],
his
o
tal
elopmen
of
comes
T
ys
in
in
the-
Matsumoto,
probabilit
remained
.
the
of
algebraic
[67],
pla
can
Lerc
probabilistic
irrational
of
the
theorems
i
b
is
[
go
the
D.
In
o
and
Steuding
limit
of
role
e
probl
that
on
vi£i
alge-
is
theorem
Macaitien
T
la
linear
imp
of
as
[7],
y
ob
of
v
h
t
momen
Last
as,
a
others
erio
[41],
of
Ho
r
of
i
problem
3
gap
y
zeta-function
K.
s
ze
pro
as
ms
duct.
This
R.
or-
H.
.
’leºevi
foll
the
e
as
m
alue
system
e
endence
ends
a

for
ed
algebraic
created
e
ry
analytic
zeta-functions
of
B.
Metho
in
based
h
r
on
the
v
w
d
in
theorems
who
theory
t.
the
probabilistic
with
the
rational
for
ere
of
A.
y
Garunk²tis,
and
Steuding
new
[16],
on
3]
of
[44],
Euler
[50],
of
ev
metho
complicated
approac
irrational
erned
op
yner
no
the
thesis,
,
the
alue
Lerc
hi
lled.
y
e
[56]-[63],
aim
tic
is
A.
e
-funct
o
a
the
those
a
tuden
pro
k
t
ait
an
Th
parameter
[36],
sp
He
are

wing:
the
pro
Belo
limit
complicated
c
J.
plane
h
algebraic
ut
of
of
irrational
J.
2.
o
v
V.
t
e
complex
[11],
collection
v
zeta-functions
e
parameters.
y
pro
mo
limit
t
space
ory
for
H.
on
vi
with
e
a
t
Pro
and
theorems
univ
the
the
.
zeta-
a
ell
[6],
t
parameters
eak
[33]
of
matical
The
ere
con
r,
Prokhoro
rst
elemen
jec
go
pro
applied.
and
for
ed
Lerc
An
zeta-function
limit
transcenden
with
and
orems
parameter
dea
w
zeta-functions.
obtained
,
y
fteen
Laurin£ik
applic
R.
ears
K.
function
J.
a
zeta-functions.
ti
[12],
p
[19],
is
2
d
,
of
[43],
dev
[46],
with
[48],
t
[51].
y
w
Boh
er,
v
most
's
case
ds
algebraic
h.
new
distribution
an
Jo
en
n
till
[
w.
b
the
4]
this
the
in
B
theory
pro
the
Bagc
h
r
is
[2],
Aims
arithme
probl
Matsumoto
m
of
The
J.
of
v
thesis
[66],
to
ta
v
Laurin£ik
probabilistic
nature
the
[40]
re
ions
of
nd
Lerc
prop
zeta-function
s
is
bac
ofs.
ts
in
them.
with
Ka£insk
tan
to
imp

and
theory
irrational
[35],
ys
Bohr.
The
R.
ecied
us,
ems
£ien
the
prevised
o
e
1.
More-
o
I.
v
the
a
v
theorem
Lerc
the
[3],
o
v
plex
Ignata
for
dep
irrational

with
distribution
the

with
zeta-function
braic
[27]-[32],
parameter
zeta-functions
.
Gen
T
of
pro
[25],
e
b
join
Garbaliauskien
limit
a
on
e
plane
describ
a
R.
of
on
h

with
b
irrational
[55]
3.
e
o
the
v
probabilistic
a
dern
theorem
v
the
the-
of
ws.
functions
of
indep
attractiv
the
ha
Cassels
Bohr,
result
ng
algebraic
w
parameter
ortan
.
Jessen
ds
applications
ofs
classical
limit
the
are
A.
on
ersalit
analytic
of
theory
Lerc
tne
function
its
erties
zeta-functions
e
o
m
Win
y
theory
Therefore,
w
this
as
researc
the
h
heory
directi
w
on
con
has
ergence
a
probabilit
large
measures.
inuence
metho
i
of
n
tour
dev
tegration,
elopmen
v's
t
and
of
ts
math-
er-
ematics.
dic
Probabilistic
are
lim
Also,
it
7
and
compare
L(λ,α,s) λ α
α
α
L(λ,α,s) λ∈ (0,1) α
L(λ,α,s)
α
L(λ,α,s)
α
{log(m+α) : m∈C } α0D.
the
of
n
ppro
2.
ta-
eak
D.
-func
ab
the
equation
o
A
a
the
re
parameter
w
compl
Lerc
on
a
fo
c
o
[21].
functions
of
y
tional
b
zeta-function
main
li
the
the
in
eak
forgotten.
new.
obtained
y
square
of
results
Lerc
the
f
the
complex
if
r
the
the
t
v
for
he
Defended
of
obtained
mate
Lerc
an
algebraic
algebraic
thesis
param-
functional
History
sis
a
join
F
theorem
time
theorem
h
algebraic
tion
of
v
of
w
v
in
results
h
babil
asymptotic
v
the
in
h
are
tioned
plane
zeta-function.
y
of
ze
estigations
t
c
with
te
o
stim
plane
v
theorems
tegral
r
in
the
ti
Lerc
asym
T
zeta-function
T
time.
ears
Let,
in
the
space
in
analytic
results
for
for
xi-
the
a
with
using
the
with
irra-
irrati
h
nal
parameter
eter
1.
.
.
of
All
problem
A
nd
limit
results
t
or
with
long
mit
,
in
Lerc
in
ze
are
func
sense
[20]
sense
ed
w
impro
irrational
e
con
as
w
Only
ergence
1987,
elt
Klusc
pro
[37]
con
the
it
form
,
for
measures
mean
ergence
of
the
Klusc
Limi
of
ex
men
probabilit
The
for
h
obtained
the
h
theory
measures
in
ta
v
of
i
tions
probabilisti
the
d
algebraic
ul
v
y
the
for
rst
h
[39]
irrational
results
parameters.
e
3.
o
A
The
limit
in
theorem
for
in
expansion
the
if
sense
o
of
p
w
the
eak
as
con
e
v
.
ergence
w
of
y
probabilit
later,
y
ga
measures
8
ula
No
α
L(λ,α,s)
α
L(λ,α,s)
α
L(λ,α,s)
L(λ,α,s)
T Z
1T logT σ = ,2 2|L(λ,α,σ +it)| dt∼ 1Tζ(2σ,α) <σ < 1,
2
0
→∞ δ
∞Z
2 −δt
|L(λ,α,σ +it)| e dt.
0
L(λ,α,s)
T > 0
1tν (...) = meas{t∈ [0,T] :...},T Texistence
ol-
h
measure.
.
Then
,
of
is
asserti
tran-
random
,
Let
theorems
a
the
Theorem
e
is
olynomials
w
Denote
bilit
[46]).
from
on
ary
and
,
jection
me
on
Throughout
ultipl
is
on
plane
to
B.
Theorem
b
measure
co
e
akly
for
[15]
form
able
form
giv
in
c
rational
et
the
the
d.
e.
of
a
that
de
e
co
a
the
ability
By
pr
the
tion
oin
ansc
torus
the
Ab
ar
This
dissertation
we
ose
as
the
.
.
only
a
the
ts
space
the
Ho
ra-
er,
ose
ortan
no
to
ges
n
follo
the
as
h
n
the
y
Theorem
(see
case
andom
tal
A.
follo
asur
theorems
there
of
b
e
unit
Borel
,
for
ariable
First
plex-v
,
Then
space
,
e
e
e
space
e
to
de
pro
ob
for
some
space
e
Tikhono
limit
theorem,
the
duct
dots
and
endental.
wise
on
the
tr
probabilit
the
top
ameter
group.
complex
es
p
measure
satised
probabilit
that
akly
for
.
Supp
5
w
.
Theorem
In
function
A,
that
the
place
of
suc
limit
distribution
for
ecien
y
supp
obtained.
tional
w
to
v
with
it
y
imp
p
t
we
applications
In
kno
that
a
the
explicit
.
of
wing
limit
c
Suc
o
the
b
of
is
measure
vari-
in
en
A
onver
the
also
of
is
scenden
Theorem
or
e
proba-
L
ws
r
limit
is
in
the
space
if
analytic
Denote
b
y
of
the
xe
circle
.
on
Then,
i.
sets
dened
arbitr
v
b
alued
transcenden
com
the
is
is
dene
w
or
recall
and
me
n
W
F
ndi
,
ther
ordinate
and
the
exists
of
recall
where
pr
.
ne
e
ability
b
written.
.
asur
.
ob
the
result.
v
with
in
of
o
to
the
class
tal
functions.
wing
pro
follo
top
the
ogy
implies
p
[46]
t
of
m
2
ication
such
innite-dimensional
that
y
the
is
pr
compact
ob
ological
ability
elian
me
Therefore,
asur
the
e
giv
.
exists.
2
Haar
c
the
onver
y
ges
9
where
t
0<λ< 1
L(λ,α,s) B(S)
S
tP (A) =ν (L(λ,α,σ +it)∈A), A∈B(C).T T
1σ > α 0 < α≤ 12
P (C,B(C))
P P T →∞T
PT
P
α
γ C γ ={s∈C :|s| = 1}
∞Y
Ω = γ ,1 m
m=0
γ = γ m∈Cm 0
Ω1
(Ω ,B(Ω ))1 1
m (Ω ,B(Ω ),m )1H 1 1 1H
ω (m) ω ∈ Ω γ m∈C1 1 1 m 0
1σ > ω ∈ Ω1 12
∞ 2πiλmX e ω (m)1
L (λ,α,σ,ω ) = .1 1 σ(m+α)
m=0
L (λ,α,σ,ω )1 1
(Ω ,B(Ω ),m )1 1 1H
α P(s)6≡ 0
P(α) = 0
α
PT
L (λ,α,σ,ω ) T →∞1 1