Table of contents for Zeta functions of groups and rings / Marcus du Sautoy, Luke Woodward.


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1  Introduction .. .. . ...                             ....... .
1. . A Brief History of Zeta F' nctions .  .. . .. . . .    .  1
1  1.1  Euler, R eiemann  .........     ..          ....   L
L1 .2  D irichlet  ..... ...  ..   .   ... .  .  .... .. . ..   3
1.L3  Dedekind ... .   ..           ......          .     4
1 1.4  Artin, Wel ....  ..
S1.5 Birch. Swnnerton-Dyer ...         .
1.2  Zeta Functions of Groups ... ..........        .... ...   6
1.2.1 Zeta Functions of Algebraic Groups. ..  . . . .   ... .  7
1.2.2  Zeta Functions of Rings ........         .. 9
1 ..3  Loca  F nctional Equations  .. . .. ...... ..... .  10
1.24  lTniform ity    ..  ......   ...              ..    1
1.125  Analytic Properties . . ....        ...   .. .. . 12
1.3  p-A dic  Integorals  ........  ... ....  ..... ... .  . .... .  .  14
1.4 Natural Boundaries of Euler Products . . .  ..    .. . . .  16
2   Nilpotent Groups: Explicit Examples ........
2.1  Calculating Zeta Functions of Groups .......      ...... 21
.2   Calculating Zeta Functions of Lie Rings ....  ..  . . . ... 23
2.2.1 Constructing the Cone Integral .. ....  . . ..   . . . 23
2.2.2  Resolution          . ...... ...... ...  .. .....  25
2.2.3 Evaluating Monomial Integrals . ..   .    ...... 31
2.2.4  Sunming the Rational Functions . . .. .    .      3.  2
2.3  Explicit Examples  . .... . ... .. .    ...... .         32
24   Fee Abelian Lie Rings .   ........... 33
2.5 Heisenberg Lie Ring and Variants            . . . . .  . . . . .  34
.6   Grenham's Lie Rings  ..  ...... ....  . .        ......  38
2.7 Free Class-2 Nilpotent Lie Rings ........     .......     40
2, 7.  Three  Generators...... ....... .  ..  .  .   ......  40
"2,7.2 r   Generators  . ....   .  ......              .  41
.8 The 'Elliptic Curve xample                  . .     ..   . 42
2     he  .Class                               . .. .mp   .....  43
h e     , ixrr,at( Cl ss Lie Ring 1  va t         . . . . . .
2 1 Lie Ri ngs with Large Abelian ldeas . ... .             48
..   .  . ..  .  a  .  1
23. i    -V a , imal Class Lie Rings -.kt and Fi  .  . ..... 52
2 ,14 iNpotent Le Algebras of Dimensin nt Lie Algebras of Diension 7 .    . ..     ..  .. 62
Si.e MH Rings ....                                      ...  69
3.   i ro ucion       . .                    .        ..  . .  69
3.2  Proof of The orem  3.1  . ...........  . .......  ...    71
S-.1 Chnoosi-g a Basis fr . (/) ...                 .
32.2  Determ.ining1 the Conditions        ......          2
A2 13      . . .  "  *   "  '  o  '  . ..
, 2,.3 Co Ii,nstrlucting the Zeta Functuion .......      74
.2 ..   ..ransforning the Conidtions .  ...              7 4
3.2.5  Dvcing the FP inctional Equation              .   75
SExplici  Examiples t  .t    .         . ........        77
3. 4 . lu o t ie n t s   o f Jta i.! . . .  .. .  . . . . . . . . . .. . . . ... . .  7 8
3,L I  C tients of tr4, 'Z)                              78
.2     oitiing All Siubrings ........           ... ....  82
4    ,Lo  'inct-nioal Equations .... ..                         83
-4.2  Al ebraic  G rou-ips  .. .. .. .. . . .............  ...   83
,O  Nilpoltent G'our_p and Lie Rings  .                  83
4.4  T he Co( i(c"ure                      ..                84
S.          Cases Known to 111Hold ..........  .          . 86
4)  i  Speci al  se of the Conjecturt  . ..... ....         87
6,   i  ro jectvsation  .     . . ....                   88
I,6-2  R,esolution .  ..                                89
.6 3  Manpulatng the Cone Sums . ...                    91
, 64 (1 Cns and Schemes         .    .. .... 93
C6 .5  Quasi-Good Sets              ..            ...   95
,6-6  Quas-Go od Sets: The Monaomial Case           ... 97
47     piwatios of Conjectue 45    .......                   98
4 S  CountM, Siubrings a,nd p-Subrings ... .... 102
4.9  Couning Ideals and p- deals    .  .........           . 103
19 1 Heights, Cocentral Bases and the --Map  ...... .... 104
49.92  Properiy  (t         .              . .   .  ... .  107
L93  Lie Rings Without t)  . .              ..     119
r5  Natural Boundaries 1: Theory .      . .......              12
5.1 A Natural Boundary for S'-    - - -    -... 121
5.2 Natural Boundaries for Euler Prodcts  .... .     .   . 123
5.21  Practic lities  ..             ..   .   .     ... .
5.2 2 Disntinuishing Types I, I aind II ......         L6
13 Avoiding the Rinmann l ypothesis   .   .   .     ...    139
5.4 1Ai Local Zeros on or to the Left of a(s) =--  ....  . 142
"  1 Using Riemann Zeros   .              .    ...143
5,412 Avoiiding  ational Independence of Riemnann Zeros . 145
5.43    Continuation witil Finitely Malny Rieranna Zeta
Functions  .  . .......               .  . . . .   149
t 4A  Bfinite Products of Rienmann Zeta Functions  ..   150
6  Natural Boundaries II: Algeb aic Groups         . ..       15.
0.1  I[n roduction  .                 .   . .. ... . .>      )
6.2    - G024.. of Tvpe B    ...   ..... 159
6   C     S -  S  Of ' vpe , or G  - GO  of Iftpe D,  . .   t 6
6.311     G Pz1 of Type     ........             ..... 162
"6 3.2 G  GO-' of Tv e D            . . . ., 165
7   Natural Boundaries 1I Nilpotent Groups .           . . . ,69
,  Itrlodiuctrion       .  ..                              69
72 Zeta Functions with Meromorphic Continuation        ..... 169
7 3 Zeta Functions with Natural Boundaries    .           . 170
7.3 1  'ype I                   ........          .  .  171
F.o2 i fp  4  .    ..         .   ..               .,.  1(71
7 ,3 )  pe      . T                    .            .. 171
{ 'TI vTPe 1 1                                         1 an
,   Other  'pes  . . ........ . . .    .  .   ........   177
S.4.1 Types Ia, and Ilb . . ....... 177
7,4 1 2  Tpes IV V and VI                   .. 177
A   Lrge Polynomials                    ....17
A,    C ..i. Cgwtnin Ideals .17 9
A2 9A Co. ing All Subrings     ....                      1.80
A3 't Counting All Si rings . ..                           180
AA. Li- Counting Ideals     . . . .                      . 181
SV         Co. uing i eals  ... ..          ..     .     182
A,6 9 i2, Ciounting All SIubrings  .            . .
A-7 9187, Counting Ideals     ..     ......                184
4A,    5 7(.  ( L Countinig ideals  . .. .. .  . . .  .    186
A,, 9 C1457B Counting Ideals.  . . .                   . .  87
A.1 0i3 (', Counting ideals ... . .       .         ...... 188
ZA.10 tn(), Counting Idea,                                 1,8ls8
/s. l 1 7s )  o n i g  I e l       .,..,.,....,.           8
Factorisation of Polynomlals Associated
t Classica Grup                          .......191



Library of Congress subject headings for this publication: Group theory, Functions, Zeta, Rings (Algebra)Noncommutative algebras