Contributions to the theory of zeta-functions: the modular relation supremacy
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2015
|
| Schriftenreihe: | Series on number theory and its applications
10 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XII, 303 S. Ill. |
| ISBN: | 9789814449618 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
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| 007 | t| | ||
| 008 | 130718s2015 xx a||| |||| 00||| eng d | ||
| 020 | |a 9789814449618 |9 978-981-4449-61-8 | ||
| 035 | |a (OCoLC)903893732 | ||
| 035 | |a (DE-599)HBZHT017704181 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-703 |a DE-11 |a DE-384 |a DE-19 |a DE-824 | ||
| 084 | |a SK 180 |0 (DE-625)143222: |2 rvk | ||
| 100 | 1 | |a Kanemitsu, Shigeru |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Contributions to the theory of zeta-functions |b the modular relation supremacy |c Shigeru Kanemitsu & Haruo Tsukada |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2015 | |
| 300 | |a XII, 303 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Series on number theory and its applications |v 10 | |
| 650 | 0 | 7 | |a Zetafunktion |0 (DE-588)4190764-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Zetafunktion |0 (DE-588)4190764-4 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Tsukada, Haruo |d 1961- |e Verfasser |0 (DE-588)13744687X |4 aut | |
| 830 | 0 | |a Series on number theory and its applications |v 10 |w (DE-604)BV022244386 |9 10 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129532&sequence=000005&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129532&sequence=000006&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-026129532 | |
Datensatz im Suchindex
| _version_ | 1825578095369781248 |
|---|---|
| adam_text |
Contents
Pívf
а се лі і
1.
Prelude
1
1.1
Introduction
. 1
1.2
Eternal return or every
50
years
. 12
1.2.1
Pomcarć
recurrence theorem
. 12
1.2.2
Lerch-Chowla-Selberg formula
. 12
1.2.3
Kuopp-Hasse-Sondow formula
. 15
1.3
The
t heta-
transforma
ti
on formula
. 20
1.4
Summation
formulas
. 23
1.4.1
Poisson
summation formula
. 23
1.1.2
Generalization of the
Plana
summation formula
. 25
2.
Grocery of Special Functions
29
2.1
Formulas for the gamma function and their use
. 29
2.2
Zet a-funct
ions
. 33
2.3
Bessel functions
.
IV)
2.1
^-functions
. 38
2.5
Generalized hypergeometric functions
. 39
2.6
Fox //-functions
. 49
2.7
Formulas for Fox and
Meijer
functions
. 51
2.8
Special cases of G-functions
.
5G
3.
Unprocessed Modular Relations
61
3.1
The
#„'{? <->
H*'?
formula
. 61
IX
; (
hmtrihutiaiin
to the Theory of
Zcta-functions
3.1.1
The H^l
*->
FI{\S¡
formula
.
f>6
3.1.2
The Cl?2
++
Gfix formula
. 69
3.2
Dedekind
zeta-
function
1. 70
3.3
Transformation formulas for Lambert series
. 72
3.3.1
Lambert series
. 73
3.3.2
Lambert series and short character sums
. 78
3.3.3
Ramanujarrs formula leading to the eta-
transformation formula
. 80
3.3.4
A brief account of modular forms
. 84
3.3.5
The Ramanujan-Gumand formula
. 86
3.3.6
The reciprocity law for Dedekind sums
. 93
3.4
Koshlyakov's method [Koshl]
. 95
3.4.1
Dedekind
zet
a-function II
. 96
3.5
Koshlyakov's functions
. 97
3.5.1
Koshlyakov's X-fmictions
. 97
3.5.2
Koshlyakov's ^-function
. 101
4.
Fourier-Bessel Expansion
Ηλ
\ 4-> H0'2
107
4.1
Introduction
. 107
4.2
Stark's method
. 108
4.2.1
Murty-Sinha theorem
. 108
4.2.2
Stark's method
.
Ill
4.3
The main formula for modular relations
. 115
4.3.1
Specification of Theorem
4.4. 119
4.4
Dedekind jseta-function III
. 122
4.5
Elucidation of Koshlyakov's result in the real quadratic
case
. 127
4.6
Koshlyakov's A'-series
. 128
4.7
The Fourier-Bessel expansion G\'\
<->
Gq%
. 133
4.8
Bochner-Chandrasekharan and Xarasimhan formula
. . . 142
5.
The
Ewald
Expansion or the Incomplete Gamma Series
145
5.1 Ewald
expansion for
zet
a-functions with a single
gamma factor
. 145
5.1.1
Confluent hvpergeometric series imply
incomplete gamma series.
Ewald
expansions
. 148
5.1.2
Bochner-Chandrasekharan formula as
И\і <* Н\І.
150
Contents xi
5.2 Atkinson-Berndt Abel
mean
. 154
5.2.1
Landau's exposition
. 156
5.2.2
Screened Coulomb potential
. 158
6.
The Riesz Sums
169
6.1
Various modular relations
. 169
6.1.1
Riesz sums
. 169
6.1.2
Improper modular relations as Riesz sums
. 172
6.1.3
The H^'l
<->
H*£ formula
. 173
6.1.4
The
Aj;¿
<-► #?;£
formula
. 174
6.1.5
Katsurada's formula combined
. 175
6.1.6
Linearized product of two
zet
a-functions
. 178
6.2
Modular relations in integral form
. 185
6.2.1
Integration in the parameter
. 185
6.2.2
Generaìization
of Ramanujan's integral formula
, 188
6.3
Integrated modular relations
. 191
6.3.1
The Hardy-Littlewood sum
. 192
6.3.2
The Hill
++ #0$
formula
. 194
6.3.3
Arithmetical Fourier series
. 196
6.3.4
Riemann's legacy
. 206
7.
The General Modular Relation
211
7.1
Definitions
. 211
7.2
Assumptions
. 213
7.3
Theorem
. 216
7.4
The Main Formula (basic version)
. 220
8.
The
Hecke
Type Zeta-functions
225
8.1
Statement of the formula
. 225
8.1.1
The bilateral form
. 227
8.1.2
The Bochner modular relation:
ej;?
<-►
G&J
. . - 229
8.2
The Riesz sums or the first J-Bessel expansion:
CÌ;? <r>
GlQ-°2
. 229
8.3
The partial sum formula: G^ ^
GW . 230
8.4
The Fourier-Bessel expansion: G11
о
G0'2
. 231
8.5
The
Ewald
expansion: G\*%
о
G\A2
. 231
8.6
The Bochner-Chandrasekharan formula:
Н\%
++
Н\1 .
. 232
8.7
The G\\ ^ G\\ formula
. 232
xii
Contributions to the Theory of
Zeta-
functions
8.8
The second J-Bessel expansion: G2')2 ^
^і'з
. 232
8.9
The
#2,2 ** #1,3
formula
. 234
8.10
The second
iŕ-Bessel
expansion:
(3?;° <->
G^'J
.
235
8.11
The G\'\
о
Gl'I
formula
.
236
8.12
The
Gl'I
^ Gl\l formula
. 236
8.13
The
GÌf3
++
Gl]l formula
. 237
8.14
The
Gì]]
<->·
Gl'I
formula
. 237
8.15
The
Gì}
*+
GIÌ
formula
. 238
8.16
The Gjji!? ^>
¿¿;ξ
formula
. 239
9.
The Product of Zeta-functions
241
9.1
The product of zeta-functions
. 241
9.1.1
Statement of the Main Formula
. 241
9.1.2
Wilton's Riesz sum: G^
+*
G\%
. 243
9.2
Powers of zeta-functions
. 253
9.2.1
Statement of the Main Formula
. 253
9.2.2
The G%'°N
о
G^2°7v formula
. 257
9.2.3
The
G*íi;°+1
+>
O^VÍiTç+^+i formula
.
263
10.
Miscellany
267
10.1
Future projects
. 267
10.1.1
Rankin-Selberg convolution
. 267
10.1.2
Maass forms
. 269
10.1.3
G-functions of two variables
. 270
10.1.4
Plausible general form
. 273
10.2 Quellenangaben. 275
10.2.1
Berndt-Knopp
book and
Berndt
's
series of papers
275
10.2.2
Corrections to "Number Theory and its
Applications"
. 276
Bibliography
279
Index
301
Series
on Number Theory and Its Applications
- -
Vol.
10
CONTRIBUTIONS TO THE
THEORY OF ZETA-FUNCTIONS
The Modular Relation Supremacy
This volume provides a systematic survey of
almo
all the equivalent assertions to the function
equations
—
zeta
symmetry
—
which zeta-function
satisfy, thus streamlining previously publishei
results on zeta-functions. The equivalent relation
are given in the form of modular relations in Fc
Н
-function series, which at present include all tha.
have been considered as candidates for ingredients
of a series. The results are presented in a clear and
simple manner for readers to readily apply without
much knowledge of zeta-functions.
This volume aims to keep a record of the
1
50-year-
old heritage starting from Riemann on zeta-
functions, which are ubiquitous in all mathematical
sciences, wherever there is a notion of the norm. It
provides almost all possible equivalent relations to
the zeta-functions without requiring a reader's deep
knowledge on their definitions. This can be an ideal
reference book for those studying zeta-functions. |
| any_adam_object | 1 |
| author | Kanemitsu, Shigeru Tsukada, Haruo 1961- |
| author_GND | (DE-588)13744687X |
| author_facet | Kanemitsu, Shigeru Tsukada, Haruo 1961- |
| author_role | aut aut |
| author_sort | Kanemitsu, Shigeru |
| author_variant | s k sk h t ht |
| building | Verbundindex |
| bvnumber | BV041154122 |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)903893732 (DE-599)HBZHT017704181 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041154122 |
| illustrated | Illustrated |
| indexdate | 2025-03-03T13:02:21Z |
| institution | BVB |
| isbn | 9789814449618 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026129532 |
| oclc_num | 903893732 |
| open_access_boolean | |
| owner | DE-703 DE-11 DE-384 DE-19 DE-BY-UBM DE-824 |
| owner_facet | DE-703 DE-11 DE-384 DE-19 DE-BY-UBM DE-824 |
| physical | XII, 303 S. Ill. |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | World Scientific |
| record_format | marc |
| series | Series on number theory and its applications |
| series2 | Series on number theory and its applications |
| spelling | Kanemitsu, Shigeru Verfasser aut Contributions to the theory of zeta-functions the modular relation supremacy Shigeru Kanemitsu & Haruo Tsukada New Jersey [u.a.] World Scientific 2015 XII, 303 S. Ill. txt rdacontent n rdamedia nc rdacarrier Series on number theory and its applications 10 Zetafunktion (DE-588)4190764-4 gnd rswk-swf Zetafunktion (DE-588)4190764-4 s DE-604 Tsukada, Haruo 1961- Verfasser (DE-588)13744687X aut Series on number theory and its applications 10 (DE-604)BV022244386 10 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129532&sequence=000005&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129532&sequence=000006&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Kanemitsu, Shigeru Tsukada, Haruo 1961- Contributions to the theory of zeta-functions the modular relation supremacy Series on number theory and its applications Zetafunktion (DE-588)4190764-4 gnd |
| subject_GND | (DE-588)4190764-4 |
| title | Contributions to the theory of zeta-functions the modular relation supremacy |
| title_auth | Contributions to the theory of zeta-functions the modular relation supremacy |
| title_exact_search | Contributions to the theory of zeta-functions the modular relation supremacy |
| title_full | Contributions to the theory of zeta-functions the modular relation supremacy Shigeru Kanemitsu & Haruo Tsukada |
| title_fullStr | Contributions to the theory of zeta-functions the modular relation supremacy Shigeru Kanemitsu & Haruo Tsukada |
| title_full_unstemmed | Contributions to the theory of zeta-functions the modular relation supremacy Shigeru Kanemitsu & Haruo Tsukada |
| title_short | Contributions to the theory of zeta-functions |
| title_sort | contributions to the theory of zeta functions the modular relation supremacy |
| title_sub | the modular relation supremacy |
| topic | Zetafunktion (DE-588)4190764-4 gnd |
| topic_facet | Zetafunktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129532&sequence=000005&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026129532&sequence=000006&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV022244386 |
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