On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence
For a Dirichlet series $F(s) = \sum^{\infty}_{n=0}a_n \exp \{s\lambda_n\}$ with the abscissa of absolute convergence $\sigma_a = 0$, let $M(\sigma) = \sup\{|F(\sigma+it)|:\;t \in {\mathbb R}\}$ and $\mu(\sigma) = \max\{|a_n| \exp(\sigma \lambda_n):\;n \geq 0\},\quad \sigma < 0.$ It is prov...
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| Дата: | 2008 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3202 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509249329692672 |
|---|---|
| author | Mulyava, O. M. Sheremeta, M. M. Мулява, О. М. Шеремета, М. М. |
| author_facet | Mulyava, O. M. Sheremeta, M. M. Мулява, О. М. Шеремета, М. М. |
| author_sort | Mulyava, O. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:06Z |
| description | For a Dirichlet series $F(s) = \sum^{\infty}_{n=0}a_n \exp \{s\lambda_n\}$ with the abscissa of absolute convergence
$\sigma_a = 0$, let $M(\sigma) = \sup\{|F(\sigma+it)|:\;t \in {\mathbb R}\}$ and $\mu(\sigma) = \max\{|a_n| \exp(\sigma \lambda_n):\;n \geq 0\},\quad \sigma < 0.$
It is proved that the condition $\ln \ln n = o(\ln \lambda_n),\;n\rightarrow\infty$,
is necessary and sufficient for equivalence of relations $\int^0_{-1}|\sigma|^{\rho-1}\ln M(\sigma)d\sigma < +\infty$
and $\int^0_{-1}|\sigma|^{\rho-1}\ln \mu(\sigma)d\sigma < +\infty,\quad \rho > 0,$ for each such series. |
| first_indexed | 2026-03-24T02:38:06Z |
| format | Article |
| fulltext |
UDK 517.537. 72
O. M. Mulqva (Ky]v. nac. un-t xarç. texnol.),
M. M. Íeremeta (L\viv. nac. un-t)
PRO NALEÛNIST| ABSOLGTNO ZBIÛNYX
U PIVPLOWYNI RQDIV DIRIXLE DO KLASU ZBIÛNOSTI
For a Dirichlet series F s( ) = a sn nn
exp λ{ }
=
∞∑ 0
with the abscissa of absolute convergence σa = 0,
let M( )σ = sup ( ) :F it tσ + ∈{ }R and µ σ( ) = max exp( ):a nn nσλ ≥{ }0 , σ < 0 . It is proved
that the condition ln ln n = o n(ln )λ , n → ∞ , is necessary and sufficient for equivalence of relations
σ σ σ
ρ
–
–
ln ( )
1
0 1
∫ M d < + ∞ and σ µ σ σ
ρ
–
–
ln ( )
1
0 1
∫ d < + ∞, ρ > 0, for each such series.
Pust\ M( )σ = sup ( ) :F it tσ + ∈{ }R , µ σ( ) = max exp( ):a nn nσλ ≥{ }0 , σ < 0 , dlq rqda
Dyryxle F s( ) = a sn nn
exp λ{ }
=
∞∑ 0
s abscyssoj absolgtnoj sxodymosty σa = 0. Dokazano, çto
uslovye ln ln n = o n(ln )λ , n → ∞ , qvlqetsq neobxodym¥m y dostatoçn¥m dlq ravnosyl\nosty
sootnoßenyj σ σ σ
ρ
–
–
ln ( )
1
0 1
∫ M d < + ∞ y σ µ σ σ
ρ
–
–
ln ( )
1
0 1
∫ d < + ∞, ρ > 0, dlq kaΩdoho tako-
ho rqda.
1. Vstup. Nexaj Λ = λn n( ) =
∞
0 — zrostagça do + ∞ poslidovnist\ nevid’[mnyx
çysel (λ0 = 0) , a S0( )Λ — klas rqdiv Dirixle
F s( ) = a sn n
n
exp λ{ }
=
∞
∑
0
, s = σ + i t, (1)
z abscysog absolgtno] zbiΩnosti σa = 0. Dlq σ < 0 nexaj M F( , )σ =
= sup ( )F itσ +{ : t ∈ }R , µ σ( , )F = max exp( )an n{ σ λ : n ≥ }0 — maksymal\nyj
çlen rqdu (1), ν σ( , )F = max n ≥{ 0 : an nexp( )σ λ = µ σ( , )F } — joho central\-
nyj indeks, a κn =
ln – ln
–
a an n
n n
+
+
1
1λ λ
.
Zrostannq funkci] F ∈ S0( )Λ perevaΩno vymirggt\ za dopomohog porqdku
ρ = lim
ln ln ( , )
lnσ
σ
σ↑ ( )0 1
M F
(dyv., napryklad, [1, s. 238] ). Za umovy 0 < ρ < + ∞ klas
zbiΩnosti oznaça[t\sq [2, 3] umovog
σ σ σρ–
–
ln ( , )1
1
0
∫ M F d < + ∞. (2)
V [3] dovedeno, wo qkwo ρ > 1 i ln ln n = o n(ln )λ , n → ∞ , to dlq toho wob rqd
(1) naleΩav do klasu zbiΩnosti, neobxidno, a u vypadku, koly poslidovnist\
( )κn [ nespadnog, i dostatn\o, wob ( – )––
λ λn nn 11
∞∑ ln+ +
an
nλ
ρ 1
< + ∞. U do-
vedenni c\oho rezul\tatu vaΩlyvym [ tverdΩennq, wo za umovy ln ln n =
= o n(ln )λ , n → ∞ , spivvidnoßennq (2) rivnosyl\ne [2, 3] spivvidnoßenng
© O. M. MULQVA, M. M. ÍEREMETA, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6 851
852 O. M. MULQVA, M. M. ÍEREMETA
σ µ σ σρ–
–
ln ( , )1
1
0
∫ F d < + ∞. (3)
Vynyka[ pytannq pro istotnist\ umovy ln ln n = o n(ln )λ , n → ∞ , u c\omu tver-
dΩenni, i cij problemi prysvqçeno danu stattg.
Ma[ misce taka teorema.
Teorema 1. Umova ln ln n = o n(ln )λ , n → ∞ , [ neobxidnog i dostatn\og
dlq toho, wob dlq koΩnyx ρ > 0 i F ∈ S0( )Λ spivvidnoßennq (2) i (3) buly
rivnosyl\nymy.
2. Dovedennq teoremy 1. Qk zaznaçeno vywe, dostatnist\ umovy ln ln n =
= o n(ln )λ , n → ∞ , dovedeno v [3]. Dlq dovedennq ]] neobxidnosti nam potribni
nastupni lemy.
Lema 1 [4]. Nexaj α : 1, + ∞[ ) → 0, + ∞[ ) i γ : 0, + ∞[ ) → 0, + ∞[ ) — ne-
vid’[mni neperervni zrostagçi do + ∞ funkci] i α x O+( )( )1 ~ α( )x pry x →
→ + ∞. Qkwo lim ( )/ ( )
n
nn
→ +∞
( )α γ λ > 1, to isnu[ pidposlidovnist\ λk
∗( ) posli-
dovnosti ( )λn taka, wo k ≤ α γ λ– ( )1
k
∗( ) + 1 dlq vsix k ≥ 1 i kj ≥
≥ α γ λ–1
k j
∗( )( ) dlq deqko] zrostagço] poslidovnosti ( )kj natural\nyx çysel.
Lema 2 [5, c. 115]. Qkwo ln n = o n( )λ pry n → ∞ , to abscysa σa abso-
lgtno] zbiΩnosti rqdu (1) obçyslg[t\sq za formulog σa = lim ln
n n na→∞
1 1
λ
.
Lema 3. Nexaj ρ > 0. Spivvidnoßennq (3) rivnosyl\ne spivvidnoßenng
σ λ σρ
ν σ( , )
–
F d
1
0
∫ < + ∞. (4)
Spravdi, oskil\ky [5, c. 182] ln ( , )µ σ F = ln (– , )µ 1 F + λν
σ
( , )– x F dx
1∫ , to
σ µ σ σρ–
–
ln ( , )1
1
0
∫ F d = σ σ λ µρ
ν
σ
–
–
( , )
–
ln (– , )1
1
0
1
1∫ ∫ +
d F dxx F =
= λ σ σν
σ
ρ
( , )
–
–
x F
x
dx d K
1
1
0
1∫ ∫ + = 1
1
0
1ρ
σ λ σρ
ν σ( , )
–
F d K∫ + , K1 ≡ const > 0,
tobto spivvidnoßennq (3) i (4) [ rivnosyl\nymy.
Zaverßennq dovedennq teoremy 1 . Prypustymo, wo umova ln ln n =
= o n(ln )λ , n → ∞ , ne vykonu[t\sq, tobto isnu[ take δ ∈ (0, 1), wo
lim
ln ln
lnn n
n
→ +∞ δ λ
> 1. Todi za lemogF1 z α( )x = ln ln x i γ ( )x = δ ln x , x ≥ 1, isnu[
pidposlidovnist\ λk
∗( ) poslidovnosti ( )λn taka, wo k ≤ exp λ
δ
k
∗( ){ } + 1 dlq
vsix k ≥ 1 i kj ≥ exp λ δ
k j
∗( ){ } dlq deqko] zrostagço] poslidovnosti ( )kj natu-
ral\nyx çysel.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO NALEÛNIST| ABSOLGTNO ZBIÛNYX U PIVPLOWYNI RQDIV DIRIXLE.… 853
Qkwo λn ∉ λk
∗( ), poklademo an = 0, a z metog skoroçennq zapysu v otryma-
nomu rqdi Dirixle zaminymo λk
∗
na λn . OderΩymo rqd Dirixle (1), de poslidov-
nist\ ( )λn taka, wo ln n ≤ λδ
n + 1 dlq vsix n ≥ 1 i ln nj ≥ λδ
n j
dlq deqko]
zrostagço] poslidovnosti ( )nj natural\nyx çysel. Poslidovnist\ ( )nj moΩe-
mo vvaΩaty takog, wo
ln ln
ln
n
j
j → ∞ pry j → ∞ i nj +1 > 2nj dlq vsix j ≥ 1.
Nexaj ( )qk — zrostagça do 0 poslidovnist\ vid’[mnyx çysel i mj = nj +[ ]1 2/ .
Poklademo n0 = 0, an0
= 1, an = 0 dlq vsix nj < n < mj ,
anj +1
= exp –qk n n
k
j
k k
λ λ
+( ){ }
=
∏ 1
0
, j = 1, 2, 3, … , (5)
i
an = a qn j n nj j
exp –λ λ( ){ }, mj ≤ n < nj +1, (6)
tobto otrymu[mo rqd Dirixle
F s∗( ) = a s a sn n n n
n m
n
j
j j
j
j
exp exp
–
λ λ{ } + { }
==
∞ +
∑∑
1 1
0
. (7)
Z (5) i (6) lehko vyplyva[
ln – ln
–
a an n
n n
j j
j j
+
+
1
1
λ λ
=
ln – ln
–
a an m
m n
j j
j j
λ λ
=
ln – ln
–
a an n
n n
+
+
1
1λ λ
= qj , mj ≤ n < nj +1.
Tomu qkwo qj ≤ σ < qj +1, to ν σ, F∗( ) = nj +1 i µ σ, F∗( ) = an nj j+ +{ }1 1
exp σ λ .
Zvidsy vyplyva[
σ λ σρ
ν σ( , )F
q
d∗∫
1
0
= σ λ σρ
ν σ( , )F
q
q
j j
j
d∗
+
∫∑
=
∞ 1
1
=
= λ σ σρ
n
q
q
j
j
j
j
d
+
+
∫∑
=
∞
1
1
1
≤ 1
1 1
1
1ρ
λ ρ
+ +
+
=
∞
∑ n j
j
j
q . (8)
Z inßoho boku, dlq vsix dosyt\ velykyx j
M q Fj ,
∗( ) ≥ a qn j n
n m
n
j
j
exp λ{ }
=
+
∑
1
= ( – ) ,n m q Fj j j+
∗( )1 µ ≥ K nj2 1+ , (9)
K2 ≡ const > 0.
Vyberemo qj = – λ δ ρ
n j +1
–
. Todi z (9) otryma[mo
ln ,M q Fj
∗( ) ≥ ln nj +1 + ln K2 ≥ λδ
n j +1
+ ln K2 = 1/ qj
ρ
+ ln K2 ,
tobto spivvidnoßennq (2) ne vykonu[t\sq, oskil\ky z n\oho vyplyva[,
wo
ln (M σ , F) = o 1 σ ρ( ), σ ↑ 0 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
854 O. M. MULQVA, M. M. ÍEREMETA
Qkwo
ρ
ρ + 1
< δ, to z ohlqdu na umovu
ln ln
ln
n
j
j → ∞, j → ∞ , z (8) otrymu[mo
σ λ σρ
ν σ( , )F
q
d∗∫
1
0
≤
ρ
ρ
λ δ ρ ρ
+ +
+
=
∞
∑1 1
1 1
1
n
j
j
– ( ) ≤ 2
1
1
1
1 1
1ρ ρ ρ δ+ +
+
=
∞
∑
(ln )( ) – /njj
< + ∞,
tobto spivvidnoßennq (4), a za lemogF3 i spivvidnoßennq (3) [ pravyl\nym.
Zalyßylos\ dovesty, wo rqd (7) ma[ nul\ovu abscysu absolgtno] zbiΩnosti.
Oskil\ky qk ↓ 0 , k → ∞ , to z (5) vyplyva[
lim
ln
j
n
n
a
j
j
→∞
+
+
1
1
λ
= lim
–
–j
k n nk
j
k
j
n n
q
k k
k k
→∞
=
=
+
+
( )
( )
∑
∑
λ λ
λ λ
1
1
0
0
= 0,
a z (6) dlq mj ≤ n < nj +1 ma[mo
ln an
nλ
≤
ln an
n
j
λ
+ qj ≤
ln an
n
j
j
λ
+ qj → 0, j → ∞.
OtΩe, lim
ln
n
n
n
a
→∞ λ
= 0 i, oskil\ky ln n ≤ λδ
n + 1, n ≥ 1, δ ∈ (0, 1), za lemogF2
σa = 0.
TeoremuF1 dovedeno.
3. ZauvaΩennq ta dopovnennq. Spoçatku zauvaΩymo, wo z ohlqdu na ana-
loh nerivnosti Koßi µ σ( , )F ≤ M F( , )σ zi spivvidnoßennq (2) vyplyva[ spivvid-
noßennq (3) i, otΩe, zadaça zvodyt\sq do znaxodΩennq umov na Λ, za qkyx z (3)
vyplyva[ (2). Tomu çerez S0( , )Λ ρ poznaçymo klas rqdiv iz S0( )Λ , dlq qkyx
vykonu[t\sq (3) zi zadanym ρ ∈ (0, + ∞). Budemo hovoryty, wo β ∈ Lnz , qkwo
funkciq β dodatna, neperervna, zrosta[ do + ∞ na (0, + ∞) i [ povil\no zmin-
nog, tobto β( )cx ~ β( )x pry x → + ∞ dlq koΩnoho c ∈ (0, + ∞). U klasi
S0( , )Λ ρ pravyl\nym [ nastupne tverdΩennq.
Teorema 2. Dlq toho wob dlq koΩno] funkci] F ∈ S0( , )Λ ρ vykonuvalos\
spivvidnoßennq (2), neobxidno, wob ln n = O nλρ ρ( )+( )1
, n → ∞ , i dosyt\, wob
ln n ≤
λ
β λ
ρ ρ
ρ
n
n
( )
( )
+
+( )
1
1 1 , n ≥ n0, de funkciq β ∈
Lnz
taka, wo ln ( ( ))β e o x1 1+( ) =
= lnβ ex( ) + O( )1 , x → + ∞, i dx
x xβ ρ( ) +
∞
∫ 11
≤ + ∞.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO NALEÛNIST| ABSOLGTNO ZBIÛNYX U PIVPLOWYNI RQDIV DIRIXLE.… 855
Dovedennq. Neobxidnist\. Prypustymo, wo umova ln n = O nλρ ρ( )+( )1
, n →
→ ∞, ne vykonu[t\sq, tobto isnu[ l ∈ Lnz taka, wo lim
ln
( ) ( )n n n
n
l→∞ +( )λ λ ρ ρ 1 > 1.
Todi za lemogF1 isnu[ pidposlidovnist\ ( )λk
∗
poslidovnosti ( )λn taka, wo k ≤
≤ exp ( )
( )
λ λ
ρ ρ
k kl∗ ∗ +( ){ }1
dlq vsix k ≥ 1 i kj ≥ exp ( )
( )
λ λ
ρ ρ
k kj j
l∗ ∗ +( ){ }1
dlq deqko]
zrostagço] poslidovnosti ( )kj natural\nyx çysel. Pokladagçy an = 0, qkwo
λn ∉ ( )λk
∗
, qk u dovedenni teoremyF1, otrymu[mo rqd Dirixle (1), de poslidov-
nist\ ( )λn taka, wo ln n ≤ λ λ ρ ρ
n nl( ) ( )( ) +1 + 1 dlq vsix n ≥ 1 i ln nj ≥
≥F λ λ
ρ ρ
n nj j
l( )
( )( ) +1
dlq deqko] zrostagço] poslidovnosti ( )nj natural\nyx çy-
sel. Poslidovnist\ ( )nj moΩemo vvaΩaty takog, wo nj +1 > 2nj dlq vsix j ≥ 1
i
1
1
0 l n
j j
j
( )λ
+
=
∞∑ < + ∞.
Nexaj ( )qk — zrostagça do 0 poslidovnist\ vid’[mnyx çysel, a koefici[nty
an vyberemo, qk u dovedenni teoremyF1. Dlq otrymanoho takym çynom rqdu (7)
zalyßagt\sq pravyl\nymy ocinky (8) i (9).
Vyberemo teper qj = – ( )
– ( )
λ λ
ρ
n nj j
l
+ +( ) +
1 1
1 1
. Todi z (9) ma[mo
ln ,M q Fj
∗( ) ≥ ln nj +1 + ln K2 ≥ λ λ
ρ
n nj j
l
+ +( ) +
1 1
1 1
( )
– ( )
+ ln K2 =
= 1 qj
ρ
+ ln K2 ,
tobto spivvidnoßennq (2) ne vykonu[t\sq, a z (8) oderΩu[mo
σ λ σρ
ν σ( , )F
q
d∗∫
1
0
≤
ρ
ρ λ+
+=
∞
∑1
1
11 l nj j
( )
< + ∞,
tobto spivvidnoßennq (4), a za lemogF3 i spivvidnoßennq (3) [ pravyl\nym.
Nareßti, oskil\ky l — povil\no zrostagça funkciq, to ln n ≤
≤ λ λ ρ ρ
n nl( ) ( )( ) +1 + 1 = o n( )λ , n → ∞ , i, qk u dovedenni teoremyF1, σa = 0.
Neobxidnist\ umovy ln n = O nλρ ρ( )+( )1
, n → ∞, dovedeno.
Dlq dovedennq dostatnosti vykorysta[mo nastupnu lemu.
Lema 4 [3]. Qkwo lim
ln
( )n n n
n
→∞ λ γ λ
≤ h0 < + ∞, de γ — dodatna neperervna
spadna do 0 na 0, +∞[ ) funkciq taka, wo t tγ( ) ↑ + ∞, t ≥ + ∞, to dlq koΩno-
ho ε > 0 isnu[ stala K( )ε > 0 taka, wo dlq vsix σ < 0
M F( , )σ ≤ µ σ
ε
ε σ
ε
γ ε σ
ε ε
ε
1 1 1
1
2
0
2+
+ + +( )
+
, exp
( )
( )–F
h
K .
NevaΩko pereviryty, wo za umov teoremyF2 vykonu[t\sq umova lemy 4 z
γ ( )x = 1
1 1 1 1x x/( ) /( )ρ ρβ+ +( ) , x ≥ x0, i h0 = 1. Wob znajty γ – ( )1 t , potribno rozv’q-
zaty rivnqnnq
ln /( )x1 1ρ+ + ln /( )β ρx1 1+( ) = ln( )1 t . (10)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
856 O. M. MULQVA, M. M. ÍEREMETA
Oskil\ky β — povil\no zrostagça funkciq, to ln ( )β x = o x(ln ), x → +∞, i to-
mu rozv’qzok c\oho rivnqnnq ßuka[mo u vyhlqdi ln /( )x1 1ρ+ = ln( )1 t – α ( )1 t , de
α ( )1 t = o tln( / )1( ) , t → 0. Pidstavlqgçy cej vyraz u (10), otrymu[mo α ( )1 t =
= ln exp ln( / ) – ( )β α1 1t t{ }( ) = ln exp ln( / )β 1 t{ }( ) + O( )1 , t → 0. OtΩe, ln /( )x1 1ρ+ =
= ln( / )1 t – ln ( / )β 1 t + O( )1 i γ – ( )1 t = e
t t
O( )
( / )
1
11β ρ( ) + , t → 0. Todi za lemog 4 z ohlq-
du na povil\ne zrostannq funkci] β ma[mo
ln ,M Fσ( ) ≤ ln ,µ σ
ε1 +
F +
K1
11
( )
/
ε
σ β σρ ρ( ) + , K1( )ε = const > 0.
Oskil\ky σ ε
σ β σ
σρ
ρ ρ
–
–
( )
/
1
1
0 1
11∫ ( ) +
K
d = K dx
x x
1 11
( )
( )
ε
β ρ+
∞
∫ < + ∞, to z ostan-
n\o] nerivnosti i spivvidnoßennq (3) vyplyva[ spivvidnoßennq (2).
ZauvaΩymo, wo umovy teoremyF2 zadovol\nq[ funkciq β( )x = ln( )x + 1 . Qk
vydno, u teoremiF2 neobxidna umova ne zbiha[t\sq z dostatn\og. Pravdopodibnym
[ nastupne tverdΩennq.
Hipoteza. Dlq toho wob dlq koΩno] funkci] F ∈ S0( , )Λ ρ vykonuvalos\
spivvidnoßennq (2), dosyt\, wob ln n = O nλρ ρ( )+( )1
, n → ∞.
1. Bojçuk V. S. O roste absolgtno sxodqwehosq v poluploskosty rqda Dyryxle // Mat. sb. –
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| id | umjimathkievua-article-3202 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:38:06Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ca/6c3cc0254bbd59b8ca07fa62a234c3ca.pdf |
| spelling | umjimathkievua-article-32022020-03-18T19:48:06Z On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence Про належність абсолютно збіжних у півплощині рядів Діріхле до класу збіжності Mulyava, O. M. Sheremeta, M. M. Мулява, О. М. Шеремета, М. М. For a Dirichlet series $F(s) = \sum^{\infty}_{n=0}a_n \exp \{s\lambda_n\}$ with the abscissa of absolute convergence $\sigma_a = 0$, let $M(\sigma) = \sup\{|F(\sigma+it)|:\;t \in {\mathbb R}\}$ and $\mu(\sigma) = \max\{|a_n| \exp(\sigma \lambda_n):\;n \geq 0\},\quad \sigma < 0.$ It is proved that the condition $\ln \ln n = o(\ln \lambda_n),\;n\rightarrow\infty$, is necessary and sufficient for equivalence of relations $\int^0_{-1}|\sigma|^{\rho-1}\ln M(\sigma)d\sigma < +\infty$ and $\int^0_{-1}|\sigma|^{\rho-1}\ln \mu(\sigma)d\sigma < +\infty,\quad \rho > 0,$ for each such series. Пусть $M(\sigma) = \sup\{|F(\sigma+it)|:\;t \in {\mathbb R}\},$ $\mu(\sigma) = \max\{|a_n| \exp(\sigma \lambda_n):\;n \geq 0\},\quad \sigma < 0,$ для ряда Дирихле $F(s) = \sum^{\infty}_{n=0}a_n \exp \{s\lambda_n\}$ с абсциссой абсолютной сходимости $\sigma_a = 0$. Доказано, что условие $\ln \ln n = o(\ln \lambda_n),\;n\rightarrow\infty$ является необходимым и достаточным для равносильности соотношений $\int^0_{-1}|\sigma|^{\rho-1}\ln M(\sigma)d\sigma < +\infty,$ $\int^0_{-1}|\sigma|^{\rho-1}\ln \mu(\sigma)d\sigma < +\infty,\quad \rho > 0$, для каждого такого ряда. Institute of Mathematics, NAS of Ukraine 2008-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3202 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 6 (2008); 851–856 Український математичний журнал; Том 60 № 6 (2008); 851–856 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3202/3150 https://umj.imath.kiev.ua/index.php/umj/article/view/3202/3151 Copyright (c) 2008 Mulyava O. M.; Sheremeta M. M. |
| spellingShingle | Mulyava, O. M. Sheremeta, M. M. Мулява, О. М. Шеремета, М. М. On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence |
| title | On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence |
| title_alt | Про належність абсолютно збіжних у півплощині рядів Діріхле до класу збіжності |
| title_full | On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence |
| title_fullStr | On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence |
| title_full_unstemmed | On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence |
| title_short | On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence |
| title_sort | on conditions for dirichlet series absolutely convergent in a half-plane to belong to the class of convergence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3202 |
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