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
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| 007 | t | ||
| 008 | 130718s2015 a||| |||| 00||| eng d | ||
| 020 | |a 9789814449618 |9 978-981-4449-61-8 | ||
| 035 | |a (OCoLC)903893732 | ||
| 035 | |a (DE-599)HBZHT017704181 | ||
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| 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 |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-026129532 | ||
Datensatz im Suchindex
| _version_ | 1804150557599858688 |
|---|---|
| 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 | 2024-07-10T00:40:49Z |
| institution | BVB |
| isbn | 9789814449618 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026129532 |
| oclc_num | 903893732 |
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| 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 |
| work_keys_str_mv | AT kanemitsushigeru contributionstothetheoryofzetafunctionsthemodularrelationsupremacy AT tsukadaharuo contributionstothetheoryofzetafunctionsthemodularrelationsupremacy |