25

empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

Embed Size (px)

Citation preview

Page 1: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

��������� � ��� ������ ��� ���������������������� �"!$#&%('*),+.-0/

1�2*354567354589�:<;>=@?BADCFEG? HHI:<JLKNMIKPO@KPQRKTSPKPUWV HHI:N9�:<XYCFOZKTE[O]\�^�_`KPMaKTOZKTQRKPSTKPUWV HHI:bHI:dc]eR?gf�=Z?hCiU@?GOZUjE[\�klk�\�^�_RKNMaKTO@\F= HHI:nm`:dc]eR?po>qREGSPKT_R?GCF^rCFSTfF\�=@KNUseRk mHI:utR:dc]eR?pJLKT\FvRewCF^xUsKP^R?L?[yaqRCiUsKP\�^rzx{�|$}�~���� �m`:<;>=@KPkl?L��qRk�Q"?G=ZO �m`:N9�:dc]eR?�A�qR^R_RCFkl?[^xU�CFS&U@eR?G\�=Z?Gk�\iA�CF=ZKPU@eRkl?BUsKPE �m`:bHI:dc]eR?[=@?pCF=@?gKP^`�w^RKNUs?GSNV�klCF^xV�vR=ZKTk�?GO �m`:nm`:dc]eR?g_R?[^RO@KNUWV�\FA*v`=@KTk�?GO �m`:utR:<;>=@KPkl?[O]KT^�CF=ZKPUse`kl?BUsKTEvR=@\�fF=@?GOZO@KP\�^RO �tR:���\�^xU@KT^�qR?G_��w=sCFEBUsKP\�^RO ��I:���\a_RqRSTCF=]CF=ZKPU@eRkl?BUsKPE 9���I:N9�:���\�^RfF=@qR?[^REG?[O 9���I:bHI:���\a_RqRSTCF=YKT^xMF?G=@OZ?GO 9�9�I:nm`:dc]eR? ��eRKP^R?GOZ?���?GklCFKT^`_R?G=Yc]e`?G\�=Z?Gk 9�H�I:utR:dc]eR?gOZU@=@qREBUsqR=Z? \iA�U@eR?gk�qRSPU@KTvRSPKTEGCiUsKNM�?jf�=@\�q`v������G ¡��¢¤£ 9�m�I:b�I:<;>=@KPklKNUsKPMF?=Z\I\FU@O 9ht�`:�¥LqwCF_`=sCiU@KTELE[\�^Rf�=ZqR?G^RE[?GO 9���`:N9�:<o>qRST?[=G¦uO�EG=@KNUs?[=@KT\F^ 9���`:bHI:dc]eR?p§5?Gf�?[^R_R=Z? O¤VIk�Q¨\�S,Ci^R_�¥�qRCF_R=sC©UsKTEg��?GE[KTvR=Z\IE[KPUWV 9���I:<;(?GSTS.¦nO]?[y�qwCiUsKP\�^ 9���I:N9�:dc]eR?gf�=Z\�qRv�SªCh« 9���I:bHI:d¬W^xUs?[f�?G=�O@\FSTq`U@KT\�^RO HF��I:nm`:<�*KP^R_RKT^`f�UseR?�A�qR^`_wCFk�?G^xU�CFS5OZ\�STqIUsKT\F^ HF��`:dc]eR?p��KT?[k�CF^`^�­G?[UsC®A�q`^RE[U@KT\�^ HI9�`:N9�:d¯^RCFSPV�UsKPELEG\F^�U@KT^�qwCiU@KT\�^rCF^`_�A�q`^RE[U@KT\�^wCiS&?Gy�qwCiU@KT\�^r\FA*°"�²±�¢ HI9�`:bHI:���\�^R^`?GE[U@KT^Rf�U@eR?gvR=@KPkl?[O�CF^R_�UseR?g­G?[=@\�O�\FA*°"�²±�¢ HFm�`:nm`:dc]eR?p��KT?GklCF^R^�³�Vav¨\FU@eR?GOZKTO Hit��?[A�?G=Z?G^RE[?GO H��´�µ�¶¸·B¹iºg»½¼�¾�¿LÀ�Á©Â½Ã[Ã[Â�Ä Å

Page 2: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ���������� ���������9�:���� 6������&6

c]eRKPOpvRCFv¨?[=�KTOgCF^ ?! `vRCF^R_R?[_ M�?[=@OZKT\�^ \FA O@\�k�?�\FA�UseR?®ST?[E[U@qR=@?[OpfFKPM�?[^�C©UUseR?gOZqRklk�?G=�OZE�eR\I\FS(KP^�X \�ST\�fF^wC`: ¬W^�U@eR?GOZ? SP?GE[U@qR=@?[Oj« ?�f�C�MF?�Ci^�KP^�U@=@\a_RqRE!"UsKP\�^�U@\ MF?G=ZVlQwCFOZKTEj^�qRk�Q"?G= UseR?[\�=ZV$#RCFOZO@qRk�KT^Rf vR=sCiE[UsKPEhCFSPSPV�^R\�QwCiE&%afF=@\�qR^`_&:c]eR?LSP?GE[U@qR=@?[O�« ?G=@?�KP^xUs?G^`_R?G_�A�\�= f�=sCi_RqwCiU@?LOZU@qR_R?G^xU@O]KT^���C©Use�CF^R_�;>exVaO@KPEGOCF^R_�«�e`KTST?®Use`?�klCiUs?[=@KªCiS>KTO�E[\�k�vRST?BUs?GSNV OZUsCF^R_wCF=Z_�#*« ?�Us=ZKT?[_ Us\�k�C$%i?lU@eR?vR=@?[O@?[^�UsCiUsKP\�^�CFO]?GSP?Gk�?G^xU�CF=¤V�CFO�v"\�O@OZKTQRSP?F:' \�k�?g\FA(UseR?p?hCFOZKT?[=jv`=@\a\FA�O�CF=@?gKP^REGSPqR_R?G_ #w\FUseR?[=@OjCF=Z?p=Z?GSP?GfxCiU@?G_�Us\�?( I?G=)"

EGKPO@?GO*#RQRq`UYO@?[MF?G=@CFS&\FA,Use`?L_R?G?[v¨?[=YADCFE[U@O]CF=Z?LOZUsCiUs?[_�«�KPUse`\�q`UYv`=@\a\FA : ��\�O¤U�\FAUseR?gklCiUs?[=@KTCFS¨klChV�Q"?gA�\FqR^R_�KT^�EGSTCFO@OZKTELU@?( aUsOjO@qRE�e�CFO,+ 9*-,CF^R_.+um/-²:

HI:�0214351 8 1�6 1�781 4�9HI:N9�: �;:=<?>A@B<DC�EGFH>JIK>L<?>JMN>LOL>JP!Q�RTS ?>_R?G^R\iUs? QxV � �VU8W?W?W*X?YHZX?YHZX���Xh9$X½HZX?W?W�W![UseR?gOZ?[Uj\FA�KT^xUs?[f�?G=ZOG:c]eR? o>qREGSPKT_R?GCF^�vR=@\�v"?G=¤UWV]\�¬�A�};^����Use`?G^�A�\�=�CF^xV�z�#R«�?pEhCi^�«�=ZKPU@?

z��`_i}�|ba«�KPU@er=Z?GklCFKT^`_R?G=]�dcea;fhg }igaCF^R_ryaq`\FUsKP?G^xUj_I:lkm KPM�?[^ C vwCFKP=�\FALKP^xUs?GfF?G=@Olz�X�}l«�KNUse$}n^�<��# «�?rOsChV UsewC©U�}�_RKPMaKT_`?GO�z�#

_R?G^`\FUs?[_$CiO�}og(z�# KPA]Use`?�=@?GklCFKP^R_R?G=,a � ��#�U@ewCiU�KPO�KPAjz � }&_�A�\�= O@\�k�?KT^xUs?[f�?G=p_a: S ?g«�KPSTS5CiSTO@\�O@ChV�UsewCiUj}�KTO�C�_RKPMaKPO@\�=]\FA*zw:c]eR?pQwCFOZKTE�vR=Z\�v¨?[=ZU@KT?GO�\FA(Use`?p_`KPMaKTOZKTQRKPSTKPUWV®=Z?GSTCiUsKP\�^�CF=@?9�:>}qg©��A�\F=�CFSTS&^`\�^R­G?[=@\l}LCF^R_ 9rg©zlA�\�=�CFSPS5zw:HI:�c(=sCi^RO@KNUsKPMaKNUWV�\�},giz�CF^R_���g©}KPklvRSPKT?[OY��g©zw:m`:�KNA5sGgiz�CF^`_tsug©}jUseR?[^tsvgFz�{�|$}�~�A�\�=�CFSPS0KP^�U@?Gf�?[=@O]{KXs~":tR:�q`^RKPU@O�\]zwgR9 KPA CF^`_�\�^RSNV�KPA zl�yx®9F:�¬W^R_`?G?G_ #0O@KP^REG? ^R\�^R­[?G=Z\�KT^xUs?[f�?G=ZOeRC�MF?�CFQ`O@\�SPq`Us?gMiCFSTqR? CiU�ST?GCFOZU 9i#wUseR? \�^RSNV�O@\�SPq`UsKP\�^RO�\FA�U@eR?�?[yaqRCiUsKP\�^{7~���9pKT^�KT^xUs?[f�?G=ZO�CF=@?l�¸{KXs~R¢>� �¤9iXG9�¢�\�=p�zYp9iX?Yp9�¢½:

�I:>�w\�=�^R\F^R­G?[=@\�KT^xU@?Gf�?[=@OLz�X�}« ?�ewChMF?�Q¨\iUse z{g�}gCF^R_�}|g�z�KPA�CF^R_ \F^RSPVKNA�}]�}xgz7:¬W^R_R?G?[_�#7KPA(},giz®UseR?L« ?gEhCF^�«�=@KNUs?�z�� }s{�A�\�=]OZ\�k�?LKT^xUs?[f�?G=Y{K#RCF^R_

A�=Z\�k zug©}]«�?EhCF^�«�=ZKPU@?}]��z�~�A�\�=�O@\Fkl?�~�~ �]:>c]e�qROYzx{7~�� z®CF^R_OZKT^RE[?pzt^� ��Use`KTO�kl?GCF^ROY{7~®��9g«�e`KTE�e�A�\�=@E[?GO�{���x�9�:

HI:bHI: �,�H�|������:ZP���<*Pq@iC]���.C]EoFH>�IK>A<*C]�8R m KNM�?G^�C�vwCFKP=*\FA¨KT^xU@?Gf�?[=@O C��&�/�r��2�/�{�$�����4���/�YKPO�CF^�KP^xUs?GfF?G=jsl«�eRKPE�e�_RKNMIKP_R?GO�Q"\FUse0:��w\�=]KP^ROZUsCF^REG?gUse`?pE[\�kr"kl\F^�_RKPMaKTOZ\�=@O]\iA�t�CF^R_��®CF=@?LU@eR?gKT^xUs?[f�?G=ZO�x�9iX!xpHI:���8�NEH>JP�>JC]E���RL��Rq� f�=@?GCiUs?[OZU]E[\�klk�\�^�_RKPMaKPO@\�=��)�������$���,�)��� �/���/ (�¡�;����¢£ (�¤  (�¡�]z�X�};�4�¥���&�/�r�2�/�t�$�����4���/�Tst¦N§��¨�&§��4���|�?©i���|�$ª]���t¢�§� ��� (���� �¢J§��$¢«� �T¬�4�,�$��­��&�/�r�2�/�{�$�����4���/���)��z�X�}r¢J§� (�v¬2g/s]®¯±° ¼«²´³ µ(³ ¶B´·�¸D¹w³ µ!³ ¶BÂ�Ä

Page 3: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ��� �¯^ KT^`O@v"?GE[U@KT\�^ O@e`\�«�OgUseRCiUgUseR?�f�=@?GCiUs?[OZUpEG\Fklk�\�^ _RKPMaKPO@\�=ZO�\FAYt�CF^R_ �

CF=@?qxpHI:c]eR?�QwCFOZKTE�ADCFEBU®KTO�U@eR?�?( IKTO¤Us?[^REG?��D^R\iU®CFvR=ZKT\�=ZK�\�QxMaKT\�q`O�A�=@\�k \�qR=�_R?B��"

^RKPU@KT\�^w¢YCF^R_�?[O@OZ?G^xUsKTCFS�q`^RKTy�qR?[^R?GOZOL\FA�fF=@?hC©Us?GO¤ULE[\�klk�\�^�_RKNMIKPO@\�=ZOG: S ? «�KPSTS_R?G^`\FUs?�QxVgf�EG_,�Dz�X�}B¢(C�E�eR\�KPEG? \FARfF=@?hC©Us?GO¤U*E[\�k�kl\�^g_RKNMaKTO@\F=�#�«�eRKTE�e KTO,qR^RKPy�qR?KPA(=@?[y�qRKT=Z?gU@ewCiU�KNU�Q¨?gv"\�O@KNUsKPMF?$\���K�8C]����� ��R4��Rq� ��­�¢A¦l��� �/���/ (�¡������¢£  ¤  (�¡�*z�Xs}��B�$�r��¢�� ¤ �¡  �$¢£ &�(¢5�&�/�r�2�/��$�����4���/�������������� �����¢)���(� ¤ �]®����Z�!¢�§� (�!�2�/�& ����/�  �� �$�.�$ª ¦´�$­���� �]������¢)  ¤  (�¡�{KXs~ ���d¢�§��$¢*f�E[_,�.z�X�}B¢ ��z�{®| }�~�®X�?[ST\�« «�? «�KTSTS�OZ?G?]Cv`=@\a\FA7\FARU@eRKTO(«�eRKTE�e�f�KNM�?[O�CF^�?���EGKT?[^xU CFSTfF\�=@KNUseRk A�\F=

Q¨\iUse¡EG\FklvRqIUsKT^`f�UseR?�fFEG_ CF^R_¡�w^R_`KT^Rf�KP^�U@?Gf�?[=LO@\FSTq`U@KT\�^RO\FA UseR?�?[yaqRCiUsKP\�^f�EG_5�.z�X�}B¢ ��z�{®| }½~":S ?g�R=@OZUj_R?[=@KNM�?pC®A�?B« vR=Z\�v"?G=ZU@KT?[O]\FA�U@eR?gf�EG_ \

���8�NEH>JP�>JC]E���R���Rq� ���$���;�)������¢£  ¤  (�¡��z�Xs}d�4���&�&�]�(���2 �� ��f�EG_,�Dz�X�}B¢���9¯ qRO@?BA�qRS,EG=ZKPUs?[=@KP\�^�A�\�=]EG\FvR=@KPk�CFSPKPUWV�KTO

´�����o:.��R"!KR z�X�}¥�$�¡ ��&�&�]�!���2 T� �p�$�]�r�/��ª ­�� �p¢�§� (�¡ j�$�¡ p����¢£  ¤  (�¡��{KXs~2�����&§¢J§��$¢�z�{®| }½~���98®S ?g«�KPSTS¨^R?G?[_�Us\®qROZ?pU@eR?�A�\�STSP\�«�KT^Rf=\

´�����o:.��R�#�R ��$&%J�jz�X�}|�$�¡ r�&�&�]�!���2 q�$�]��zug©}½�d¢J§� (��zGg©�$®' $&%���z�X�}|�$�& ��&�&�]�(���2 ¥�$�]� ' �/¢�§u�$�����L�� ��d¢J§� (�t¢J§� (���l�]�¡���(���(¢�z�}|�$�����L�� &�

�$®) �&���)�*®�¬W^R_`?G?G_ #>KPA�z�X�}�CF=@?lEG\�v`=@KTk�?®UseR?[^ «�?�EGCF^ «�=@KNUs?�9�� z�{�|�}�~ A�\F=KT^xUs?[f�?G=ZO]{KXs~¨:>��qRSPU@KTvRSNVaKT^Rf�U@eRKTO�?[yaqRCiUsKP\�^�Q�V��«�?��w^R_

��� z&*h{"�>| ~�*�}½��W�²HI:N9�¢�w\�=�vwCF=¤Ur�.C�¢(#>«�?rCF=Z?�CFOZO@qRk�KT^Rf z�g�}½��CF^`_ OZ\ Q¨\FU@e O@qRk�k�Ci^R_RO�\�^ U@eR?��³ ' \FAp�²Ha:P9�¢YCi=@?g_RKPMaKPO@KTQ`ST?LQxV�z�#weR?[^REG?pO@\�KTOYU@eR?p§(³ ' #w^wCFk�?GSNV���:�w\�=>vwCF=¤U�DQ7¢!#�« ?jCF=@?jCFOZO@qRk�KT^`fgU@ewCiU zug©�]CF^`_lOZ\ z�}�g�~I}+* �]CF^`_lSPK4%F?B«�KTOZ?

O@KP^REG?p«�?�CFO@OZqRkl?pUsewC©U�}rg��/#7« ?�ewChM�?�z�}rg�z�{,*©� Ci^R_�Use�qROz�}g_`KPMaKT_R?[O�U@eR?§,³ ' \FAp�.HI:P9�¢YCF^R_rO@\�_RKPMaKT_`?GOYUse`? §,³ ' #w^wCFk�?GSNV���:HI:nm`: �,�H� �.-H@BOA>LFK��:=EV:�OJ��C�� >�P��H� R c]e`?�o>qREGSPKT_R?GCF^ CFSPf�\�=ZKPUse`k f�KNM�?GO�CF^?/��EGKP?G^xU®k�?[Use`\I_ A�\�=��R^R_RKT^`f U@eR?�f�EG_ CFO�« ?GSPS]CFO�A�\�=�E[\�k�vRq`UsKP^Rf {KXs~ O@\UsewC©Uf�E[_,�.z�X�}B¢]� z�{l| }�~¨:jc]eR?�k�?[U@eR\a_�KTOCFO�A�\FSTST\�«�O*\�¯OZO@qRk�?tg z�g�0 g }igb:¬�U�«�KTSPS�Q"?�EG\�^xMF?G^RKP?G^xU U@\�O@?BU,a21 k \ � g z�g #,CF^R_wa43r\b� g }ig : S ?�q`O@?®_RKPMaKPO@KT\F^«�KPU@er=Z?GklCFKT^`_R?G=�U@\�«�=ZKPU@?

z�� _ k }�| a k X �rcea k f g }ig8W

Page 4: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ���������� ���������¬�AHa k ����UseR?[^�}qg©z�CF^R_�f�E[_,�.z�X�}B¢ ��}�:��UseR?[=Z«�KPO@?$#`KPUs?[=sCiU@?Use`KTOYO¤Us?Gv�«�KPU@ezl=Z?GvRSTCFEG?[_�QxVr}LCF^`_�}�=@?Gv`SªCFE[?G_rQ�V�U@eR?g=@?[k�CFKP^R_R?[=Ta k Us\�«�=@KNUs?

}]�`_��¡a k | a��?X �rcea��jf�a k W��\�^xUsKP^aq`KT^Rf®KT^�U@eRKTO]ADCFOZeRKT\F^�#`« ?pfF?[U�CiA¸Us?[=���O¤Us?[vRO

a��/1��� _���a��/1 k |ba��$X �rcea��¥fna��/1 k W' KP^REG?�U@eR? OZ?Gy�qR?G^`EG? \FAp=@?[k�CFKP^R_R?[=@Oog }ig�� a43� a k a��� W�W?W�KPO�C

OZU@=@KTEBUsSNV�_R?[EG=Z?hCFOZKT^Rf�O@?[y�qR?G^RE[?l\iA ^`\�^Z"W^`?GfxCiU@KPMF?�KT^xUs?[f�?G=ZO�#&Use`KTOLvR=Z\IE[?GOZOpeRCFOUs\�Us?[=@k�KT^wC©Us?LKP^�C��w^`KPUs?g^�qRk�Q"?G=�\FA�OZU@?GvRO*#7OsChV�CiA¸U@?G=� ¡OZU@?GvRO�«�?pewChM�?

a�� 1��� _�� a��21 k |ba���X a��G^���CF^R_

a�� 1 k � _���� k a��S ?pE[SªCFKPk U@ewCiU

f�EG_(�Dz�Xs}[¢ � a��dW��\�=Z?G\�M�?[=�#wUseR?gvR=Z\IE[?GOZO�f�KNM�?[OjKP^xUs?GfF?G=@O]{KXs~lO@\®UsewC©U

fFEG_,�Dz�Xs}[¢ � z�{�|$}�~�Wc(\ O@?[?�UseRKPO�#Y« ?�«�KTSPS�O@eR\�« QxV _R?[O@E[?G^R_RKP^Rf KT^`_RqRE[U@KT\�^$\�^����� NX� .Y

9iX?W?W�W*X��ZX?Y 9�U@ewCiUa�� g�a���²HI:bH�¢

CF^R_�UseRCiUa�� � {���a�� 1 k | ~��4a���²HI:nmx¢

�L^REG?« ?�ewChM�?�U@eRKTO*#IU�C %aKP^Rf������ZX?Y 9L«�KPSTS"f�KPMF?�a���g/a43���}�CF^R_�a��2g a21 k �z�#wCF^R_�U�C$%�KP^Rf��*� ��KP^ �²HI:nmx¢�«�KTSPS0f�KNM�?�a�� � { 3½z�| ~ 3�}�:�w\�=����� r« ?E[ST?GCF=@SNV�ewChM�?�a���g/a�� CF^R_2a��p� 9?a��*|¡�$a��21 k :>�w\�=����� �Y 9«�?]eRC�MF?pa�� 1 k � _��ia��f�KPMaKP^RfQ"\FUser�.HI:bH�¢*CF^R_r�.HI:nmx¢(KP^�UsewCiU�EhCFOZ?F:�¯�O@O@q`klKP^Rf

«�?r%�^R\�«�UsewC©U¥a��{gZa���CF^R_{a��{g�a��/1 k #¨«�?®qROZ?ra��/1���h_���a���1 k |�a���U@\��w^R_UsewC©U�a���g�a��/1�h: �wqR=¤UseR?[=�#7CFOZO@qRk�KT^`f®« ?¥%�^R\�« �²HI:nmx¢ A�\�= ���!��f�KPMF?GOa��p� {"��a��/1 k | ~#�(a��� {"��a��/1 k | ~#�F�La��/1� Y _���a��/1 k ¢��~#�(a��/1��|���{"�pY _��[~#�G¢za���1 k

UsewC©U�KTOp�²Ha:bmx¢�e`\�ST_ROYA�\F= �*�$�rY 9�«�KNUse�{"�/1 k ��~#�/#w~#�/1 k � {"�pY _��G~��i:

Page 5: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ��� �oN `CFklv`ST?$\�z$����#�}����Z\�c(\�EG\FklvRqIUs?�f�EG_5�.��X½�F¢(# « ?rEhCF=Z=ZV \�q`U®U@eR?

A�\�STSP\�«�KT^Rf OZUs?[vRO�\�p� 9 *©�]|$m�g� 9 *�m�| Hmp� 9 *©H]| 9Hg� H *x9qW

c]e�qRO>f�EG_5�.��X½�F¢>��9�:�c�\L�w^`_�KT^xUs?[f�?G=*O@\�SPq`UsKP\�^RO(\FA"�i{ |��i~�� 9i#F« ?]vR=@\aEG?[?G_QwCFE&%x«�CF=@_RO*\

9��m�Y 9 *©H��m�Y �²��Y 9 *�mx¢ � H *�m�Y �� H *a�.��Y 9 *���¢NY �g� H *���Y m.*©�

CF^R_�Use�qRO�« ?pA�\�q`^R_�UseR?gO@\FSTq`U@KT\�^�{�� HZ#R~®�VYm�Us\l�i{�| �i~���9�:¯^�?( `CFk�KT^wCiU@KT\�^�\FA*UseR?�o q`EGSTKP_R?hCi^ CFSPf�\�=@KNUseRk O@eR\�«�OU@ewCiUjUseR? ^aq`k�Q¨?[=

\FA�OZUs?[vROKPO�CiUjkl\FOZUH�ST\�f � g }igh| 9L�����DSP\�f>klKP^&�&g z�g X g }ig ¢Z¢(#`U@ewCiUjKTO]Use`?pE[\�kr"vRST?! IKPUWV�KPO OA>LEK��:�� KT^�U@eR?g^�qRk�Q"?G=�\FA�QRKPU@Oj^`?G?G_`?G_�U@\®=@?Gv`=@?GOZ?G^xUU@eR?gKT^Rv`q`Uh:�����8�/@B>A<*� ��R���R ) �¡�/�$ q¢�§��4�; &�(¢L���|�$¢) r�/� ¢�§� q� ��� '  (�r�)���(¢) £����®³�KP^xU?\ ' eR\�« U@ewCiU�a���c a���1����HI:

�����8�/@B>A<*� ��R���R §��/¦ ¢�§��$¢��/�  o� �$��� �/¢��(� ¤ ��� �T� �$��¢Lª ­������]�¡�/�$  ¢�§��4�. &�(¢L����|�$¢) �®³�KP^xU?\�c�C$%i? z�X�}�Us\ Q"?¡EG\�^`O@?GE[q`UsKNM�? �*KPQ¨\�^RCFEGE[K^�qRk�Q"?G=ZOG: c]e`?GO@? CF=Z?

_R?[�R^R?G_ =@?[EGqR=ZO@KPMF?GSNV CFO�� 3�� ��#�� k � 9i#CF^R_ A�\�= 0 9 QxV � ��� k �� �5|�� � 1 k : ' eR\�«$U@ewCiU>Use`?j^�qRk�Q¨?[=>\FA0O¤Us?[vRO�A�\F=�EG\�k�vRq`U@KT^Rf�fFEG_,��� � � k X�� �F¢KTO ���©¢®CF^R_�U@ewCiU� �� ST\�f�� ����ST\�fR�@� 9�|�� ��¢Z��H�¢�CiO� �� � :HI:utR: �,�H���v>LC��5�H:�E]P�>AEK�G����-H:�P�>LC]E z�{�| }�~®� � R m KPMF?G^�KT^xU@?Gf�?[=@O�z�X�}�X���#«�?�«�KPO@e�Us\��w^R_����¢)  ¤  (� O@\�SPq`UsKP\�^ROYU@\®U@eR?g?Gy�qwCiU@KT\�^

z�{®|$}�~®����W¯OjUseR?p?( `CFk�vRST?ptF{�|$�F~�� 9�KPSTSPqROZU@=sCiU@?GO�#wOZqRE�e¡O@\FSTq`U@KT\�^RO�^R?[?G_¡^R\iU?! `KPOZU�#O@KP^REG?�U@eR?��j³ ' KTO�\I_R_�«�eRKPST?�UseR?�§,³ ' KPO�?[MF?G^����r\�=@?]f�?[^R?G=@CFSTSNVi#�KNA7f�EG_5�.z�X�}B¢_R\a?GO^`\FUL_RKNMaKT_R? �gU@eR?G^ U@eR?G=Z?�«�KTSPS,Q¨? ^R\�KT^xU@?Gf�?[=O@\FSTq`U@KT\�^RO[:�¬�U�UsqR=Z^RO\Fq`UUsewC©UjU@eRKTOj_RKNMIKPO@KPQRKTSPKPUWV�EG\�^`_RKPU@KT\�^rKTO]Use`?�\�^RSNV�\�QRO¤Us=ZqRE[U@KT\�^rUs\®UseR? ?( IKTO¤Us?G^`EG?\FA7O@\FSTq`U@KT\�^RO*#�CF^`_�\�^REG? UseRKPO*\�QRO¤Us=ZqRE[U@KT\�^�MiCF^`KTO@e`?GO�Use`?G^�Use`?G=@?YCF=@?YKP^`�w^RKNUs?[SPVk�Ci^�V�KT^xUs?[f�?G=�OZ\�STq`U@KT\�^`O�\���K�8C]����� ��R ��R"! §� d  �����$¢A�¨�/�rz�{�| }�~�� �d§��/�,����¢£  ¤  (�¥���/ª ��¢L�¨�/���;� ���$�]��/��ª ­d� ��f�EG_,�Dz�X�}B¢�gi� ®

Page 6: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ���������� ���������%J�j¢�§� (�¡ ��4�¥�/�  ����/ª �Z¢L�¨�/����{ 3�Xs~ 3�¢�¢�§� (�v¢J§� (�¡ ��$�¡ ����4� ����¢£ (ª ­��|�$��­,����¢)  ¤  (�

���/ª �Z¢L�¨�/�����D�$�]��¢�§� (­2�$�& ��$ª�ª ¤ ���$ (� ' ­{"��� { 3*| � }

f�EG_,�Dz�X�}B¢ X ~��j� ~(3 Y � zfFEG_,�Dz�Xs}[¢

¦N§� (�¡  �o� �Z���r�/�$ (�q�$ª�ª�����¢£  ¤  (�¡��®c]eR?®vR=Z\I\FA \FA Use`KTOLUse`?G\�=Z?Gk KPOLM�?G=¤V ?hCFO¤V¡\�^RE[?�«�?d%�^R\�« U@eR?®?( IKTO¤Us?G^`EG?

\FA \�^R?�O@\�SPq`UsKP\�^&:®c�\��w^R_�CrOZ\�STqIUsKT\F^�#&�w=@O¤U�q`O@?®UseR?lo>qREGSPKT_R?GCF^�CiSTf�\�=ZKPU@eRkUs\®O@\FSPM�?�U@eR?g?Gy�qwCiU@KT\�^rz���| }���� f�E[_(�.z�X�}B¢]CF^R_�UseR?[^rUsC$%F?

{ 3Y��� �f�E[_,�.z�X�}B¢ X ~(3 ��� �

f�E[_,�.z�X�}B¢ W

m`:�����1�� 6�� ��6 6 � 8m`:N9�: �,�H�� -HEHFH:��o�ZE]P�:�OpP��H�8C������ C� �:=�/>�P��5�o�8P�>A@=R ¯y�]�!���2 �KPOC�^wCiUz"qR=sCiS ^�qRk�Q"?G=�� 9�«�eRKTE�e eRCFO�^R\�vR=@\�v"?G=�_RKPMaKTOZ\�=@O���U@ewCiU�KTO�^`\ _RKPMaKPO@\�=ZO\FUse`?G=YUseRCF^ x���CF^R_tx®9�¢½:c]eR?pO@?[yaq`?G^RE[?�\FA�vR=@KPkl?[O]KTOYU@eaq`O�� k � H�X�� �Y� mZX����Y� ��X?W?W?WB:

´�����o: ��RL��R %J���.�4�p�]�!���2 ¥¦N§��¨�&§G�$�����L�� &�,���]�&���(���(¢����gi}½�4�´¢J§� (� ��¢«§��/�¢)�2�$�����L�� ��/�  ;�)��¢�§�  �*���(¢)�/�¡�����{gi}��/���g©�$®¬W^R_R?[?G_�#*KPA���KTOpvR=@KPkl?®Ci^R_���_R\I?[O�^R\iU�_RKNMaKT_R?�}�Use`?G^ CFqIUs\�klCiUsKPEhCFSPSPV��

CF^R_r}LCF=Z?gEG\�vR=ZKTk�?$#IeR?G^RE[?pU@eR?g=@?[O@qRSNUjA�\FSTST\�«�O�A�=Z\�k §5?Gk�klC®HI: �a:S ?�E[SªCFKPk U@ewCiUg?[MF?G=¤V KT^xUs?[f�?G= 9�KPOgCrv`=@\a_RqRE[Up\FA vR=ZKTk�?GO�\�¬W^`_R?G?[_�#

9�KTOL?[KPUse`?G=�C�vR=ZKTk�?$#¨KT^ «�eRKPE�e EGCFO@?�« ?®CF=Z?�_R\�^R? #5\�=L_R?[EG\�k�v"\�OsCFQ`ST?$\ �� za}�#w«�KPU@e�z�X�} 9�: ¬W^�Use`? STCiU@U@?G=]EhCiO@?$#w« ? eRC�MF?�z�X�}�f� CF^`_ CF=Zf�qRKT^`fQxV�KT^R_`qRE[U@KT\�^�#`Q¨\FU@erz�Xs}LCF=@?gvR=Z\I_`qRE[U@Oj\iA*vR=ZKTk�?GO�CF^`_re`?G^RE[? OZ\lKPO ���za}�:' KP^REG?�?[MF?G=¤VlKP^xUs?GfF?G= KTO CgvR=@\a_RqREBU�\FA0v`=@KTk�?GO*#xUseR?BV®CF=@?]U@eaq`O�U@eR?�Q`qRKTSP_RKT^Rf

QRST\aE&%�O�\iA CFSTS&U@eR?�KP^xUs?GfF?G=@O[:�¬W^�ADCFEBU?#"kl\�=Z?gKTO�Us=@q`?q"�UseR?pADCFE[U@\�=@KP­hCiU@KT\�^�KP^�U@\vR=@\a_RqREBUsO�\FA�vR=@KPkl?[O]KTO.���������� (\���K�8C]����� ��R4� �.�wqR^R_RCFkl?[^xU�CFS5c]eR?[\�=@?[k�\FA*¯=@KNUseRk�?[U@KTEh¢ R � �$ (�!­j�]�$¢ ��� �$ªB� �Z�r�'  (���4�jqR^RKPyaq`?GSPV.�� &�&�/��������� ' ª4 �����¢)���q�]�¡���(���(¢��)�T�]�!���2 T���/¦l (�¡��®�����8�/@B>A<*� ��R���R §��/¦¢�§��$¢�� �´¢J§� H�]�!���2  �*���(¢£�/�!� ���$¢A�¨�/���)�T�T���$���p�)�D����¢)  ¤  (�¡��4� ¤ ���$ (� ' ­ z��"!��$#&% ')(��$�]� }��*!*��+,% ')(p¢J§� (�

f�E[_,�.z�X�}B¢ �*- ��.0/21 %3#&% ')(54 +,% ')(6( WS eRKPST? « ?;%�^R\�«�UsewCiU?[MF?G=ZV�KT^xUs?[f�?G=�ADCFE[U@\�=@O�KT^xUs\�ClvR=@\a_RqREBU�\FA vR=ZKTk�?GO�#

C�QRCFO@KPEpv`=@\�QRSP?Gk KPOje`\�« Us\®�w^R_�U@eRKTO]ADCFEBUs\�=ZKT­GCiUsKP\�^�?/��EGKP?G^xUsSNV�:���qR=Z=@?[^�U@SPV$#UseR?[=@?rKTO®^R\o%a^`\�«�^ CFSTf�\F=@KPU@eRk<«�eRKTE�e «�KTSTS]fFKPM�?�U@eR?�v`=@KTk�?�ADCFE[U@\�=@KP­hCiU@KT\�^\FA�CF^ KT^xUs?[f�?G=�KP^ C¡^aq`k�Q¨?[=�\FA�OZUs?[vRO�«�eRKTE�e KTO�v"\�SNVI^`\�klKTCFS KT^ U@eR?�KT^RvRqIU�Dy�qwCF^xU@qRk CFSPf�\�=@KNUseRk�OYCiO@KT_`?�¢B:

Page 7: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ��� �� � 6 7 6���� ��\�k�vwCF=ZKTO@\F^rQ"?[UW« ?G?[^����¸{¨¢]CF^R_�§,K²��{¨¢½:{ � ��{¨¢ +u§5K���{¨¢)-=Y�����{¨¢9h� ��# �F�I9i#ntx��� ���it9h� k 3 tx���Z#n�x��HZ#b�I9�9 m�#N9��Ft9h��k � m��Z#b�F�x�Z#n�`9�HZ#n�`9�� m���#bHF��m9h��k�� mZ# HF�it=#b�itR9i#b���F��#n���xH m`9ht=#n�����9h��k� HF�F��#bHFm���#nmFtR9i#n��m�mZ#b�xHF� m�#bHI9ht=#n��mxH

m`:bHI: �,�H�8��� :=���w>LE�� EH>JP?�ZO�Qb�.:=E�Q �H�/>A�o�Z< R���K�8C]����� ��R ! �.o>qRE[STKT_w¢ R ! §� (�¡ ��$�¡ ¥���4� ����¢£ (ª ­��|�$��­��]�!���2 &��®) �&���)�*®�¯�=@f�qR? QxVy�¡  �(���(¢L�¨�n�B��� ' ����� �(�Z�r\ ¬�A�U@eR?G=Z?¡«�?[=@? �R^RKPU@?GSPV k�Ci^�VvR=@KPkl?[O�#�OsChVg\�^`SPV�� \FAIU@eR?GkG#�UseR?[^ A�\�=Zk UseR? KP^�U@?Gf�?[=�� � � k * � � * * * * *2���l|l9�:¬�UpKTO�?GKNUseR?[=pCrvR=ZKTk�?�\�=g_R?[EG\�k�v¨\FOsCFQRSP?F: ' KP^REG?�� KTO�f�=Z?hCiU@?G=�UseRCF^�CiSTS*U@eR?vR=@KPkl?[O � k X?W?W�W�X�����#YKPU�EhCi^R^R\FU�Q¨?�C vR=ZKTk�?F: ³j\�«�?BM�?G=*#��dE[ST?hCi=@SPV ST?hChMF?GO=@?[k�CFKP^R_R?[=l9�\F^ _RKPMaKPO@KT\F^ QxV ?hCFE�e \FA]U@eR?�ChMiCFKPSªCFQRSP?®vR=@KPkl?[O�� �¨#>CF^R_ Use�qROQ¨?[KT^Rf�_RKNMaKTO@KPQRST?�QxVg^R\�vR=@KPkl? #©EGCF^R^R\iU�_`?GEG\Fklv"\�O@? KP^�U@\jC�v`=@\a_RqRE[U(\FAIvR=ZKTk�?GO �S ?�U@eaq`O�Ci=@=@KNM�?pCiUjC�EG\F^�U@=sCF_`KTE[U@KT\�^0:�����8�/@B>A<*� ��R�#�R §��/¦o¢�§��$¢�¢J§� (�¡ l���4� ����¢) (ª ­��|�$��­´�]�(���2 &�j�)�D¢J§�  �?�/�(� t �5|m�®m`:nm`: �,�H�nFK�ZEH<?>�P!Q C� �5�/>L�.��<$R ¯jA¸Us?G=q%�^R\�«�ST?[_Rf�?�U@ewCiUpUseR?[=@?�Ci=@?�KT^`�w^�"KPU@?GSPV klCF^xV�vR=ZKTk�?GO*#x\�^R?]EhCi^�U@=ZV�U@\gCFO@OZ?GOZO�U@eR?GKP=>_R?G^ROZKPUWVF: m CiqRO@O =@?[EG\�q`^�U@?G_UsewC©UYKT^�9��F�xHZ#ICFO]C Q¨\�V�\iA 9��Z#aeR?gCF=@=ZKPMF?G_�CiU�UseR?LE[\�^��Z?[E[Usq`=@?LUseRCiU�U@eR?L_R?[^Z"O@KNUWV�\FA*vR=@KPkl?[O]^R?hCF=Y{�KTOjCFQ"\�q`Ug9���SP\�f*{¡Ci^R_�OZ\lKNA�« ?g_R?G^`\FUs?pQxV�����{¨¢�U@eR?^�qRk�Q"?G=�\FA�vR=@KPkl?[O]qRv�Us\®{� ��{¨¢T\ ���2U \ � �dc {�[UseR?[^����¸{¨¢ KTO CFOZVak�v`Us\iUsKTEGCFSTSNV�?Gy�qwCiS¨Us\ UseR?LSP\�fxCF=ZKPU@eRklKPEYKT^xU@?Gf�=@CFS¨#af�KPMF?G^�A�\F={ H®QxV

§,K²��{¨¢T\b��� ��

s"!ST\Ff#!

¬W^rU@qR=@^ #w§5K��¸{¨¢]eRCFOjCF^�CiOZVaklvIUs\FU@KTE�?( IvwCi^RO@KP\�^§5K²��{¨¢ � {

SP\�f>{ |{

�DST\Ff {¨¢ � | * * *�| � � {�DSP\�f>{¨¢ � | ��� {

��ST\�f�{¨¢ � � k ¢c(\ E�eR?[E&% U@eR? O¤Us=Z?G^RfFU@e \FA�§5K��¸{¨¢rCFO�CF^ CFvRvR=Z\� IKPk�CiU@KT\�^ U@\$� ��{¨¢(#L« ?

?( `CFk�KT^R?jc�CFQRST? 9��¸«�=@KPU@KT^Rf�+ ~8-�\b� KT^xUs?[f�?G= vwCF=ZU \FA,~`¢B:�¯�O�KPO�O@?[?G^�A�=Z\�k UseRKPOU�CFQ`ST?$#I§5K��¸{¨¢ KTOYC�=@?GklCF= %iCiQRSPV®f�\a\a_�CFvRvR=Z\? `KPk�C©UsKT\F^lUs\%����{¨¢�KP^�UseRKPO�=@CF^Rf�?i:

Page 8: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ���������� ���������¯O]C�k�?hCiO@qR=Z?L\FA5Use`?�yaqRCFSTKNUWVl\iA5UseR?�CivRvR=@\? IKTklCiU@KT\�^�#�^R\iUs?LUseRCiU�U@eR?L«�KT_`U@e\FA0U@eR?]UseRKP=@_�EG\�SPqRk�^lKPO CiQ¨\�qIU�CgewCFSNA0\iA0Use`?j«�KP_`Use�\FA0U@eR?jO@?GE[\�^R_�\F^R?$#�UsewC©UKTO]Us\lOsChV�UsewC©UjU@eR?�=Z?GklCFKT^`_R?G=�KTOjCFv`vR=@\? IKTklCiUs?[SPV�OZy�qwCF=@? =@\a\FUj\FA�U@eR? klCFKP^Us?[=@k��c]eR?rOZUsCiUs?[kl?[^�U�U@ewCiU ���¸{¨¢ �<§5K��¸{¨¢�KPOr%�^R\�«�^ CFO�U@eR?r; =ZKTk�?���q`k�Q¨?[=

c]eR?G\F=@?Gkr:*¬�U(«�CFO*vR=Z\�M�?G_�KP^l9h������Q�V ³�CF_wCFklCF=@_�CF^R_ _R?�STC���CFSPS��?[? ;�\�qROZO@KP^�#QxV�qROZKT^Rf�UseR?���KT?[k�Ci^R^�­[?[U�C�A�q`^RE[U@KT\�^&:�c]eR?�?Gk�vRKP=@KTEGCFS OZUsCiUs?[kl?[^�U�k�CF_`?CFQ"\�MF? A�=Z\�k UseR?�_RCiU�C�KT^ c�CFQRSP?l9�CiO�Us\lUseR? k�CFfF^RKPU@qR_R?p\FA>UseR? =@?[k�CFKP^R_R?[=KT^®UseRKPOYCFv`vR=@\? IKTklCiUsKP\�^®KTO CpA�\�=Zk \FA&U@eR?E[?GST?[QR=sC©Us?G_��jKP?GklCF^R^l³jVav¨\iUseR?[O@KTO*#O@?[? ' ?GE[U@KT\�^��`:bHI:

m`:utR:�� �/>A�o�Z<�>AE{:=�/>JP��5�.�BP�>L@ �H��C]������<?<?>JC]E5<$R ¯�^®KTk�v¨\�=¤U�CF^xU>KPO@OZqR?]KTO�U@eR??( IKTO¤Us?[^REG? \FApvR=ZKTk�?GO�KP^ C f�KPMF?G^ CF=@KNUseRk�?[U@KTErvR=@\Ff�=@?[O@O@KP\�^�\ m KPMF?G^ z$CF^R__� 9i#�Us\��R^R_ C SªCF=Zf�?�v`=@KTk�? � «�KPUse ��� z k�\I_v_a: ��ST?hCi=@SPV$#�KT^�O@\�k�?KT^RO¤U�CF^`EG?GO�KPU�EhCF^`^R\FU�Q¨? _R\�^R? #�OsChVpUseR? vR=@\�fF=@?GOZO@KP\�^�UiHZX@t=X���X��ZX?W?W?W&[]E[\�^xU�CFKP^RO^R\LSTCF=@f�?�vR=ZKTk�?GO�CFO>CFSPSav`=@KTk�?GO(?( IE[?Gv`U HCi=@?�\a_R_&:*§5K4%F?B«�KTOZ?$#�KNA7z Ci^R_;_jewChM�?CpE[\�k�kl\�^�ADCiE[Us\F=Ds 9�UseR?[^�KPU _`KPMaKT_R?[O ?[M�?[=ZV�?GSP?Gk�?G^xU�\FA0U@eR?�vR=@\Ff�=@?[O@O@KP\�^z�X�z| _�X�zL| H _lW�W?W�CF^R_rO@\�Use`?G=@?pCF=Z?�^R\�vR=@KPkl?[OYKT^�KNU �D?( IE[?Gv`U@KT^Rf�v¨?[=@ewCivROKPA�z��hs�KTOvR=@KPkl?h¢B: S ?�O@eR\FqRST_ U@eaq`O�=@?GO¤Us=ZKTE[UpC©U@Us?[^xUsKT\F^�Us\�U@eR?�EhCiO@?�UsewC©Uz Ci^R_b_ CF=Z?�EG\ "�vR=@KPkl?i: ¬�U�U@qR=@^`O�\�q`U�UseRCiU�Use`KTO�KPO�UseR?�\F^RSPV�\�QROZU@=@qREBUsKP\�^Us\®UseR?p?( IKPOZUs?[^REG? \FA�v`=@KTk�?GO]KP^�Ci=@KPU@eRk�?[UsKPE�vR=@\Ff�=@?[O@O@KP\�^�#7CFO�«�CFO�v`=@\�M�?[_ QxVJLKT=ZKTE�eRSP?[U KT^ 9���m��I:�¬W^ ADCFEBU�UseR?[=@?rCF=Z?�CF=@QRKNUs=@CF=@KPSPV SªCi=@f�?lvR=@KPkl?[O�KT^ ?BM�?G=¤VvR=@\Ff�=@?[O@O@KP\�^�^R\FUj?( IE[STqR_R?[_�QxV�OZqRE�e�=Z?hCFOZ\�^RKT^`f=\���K�8C]����� ��R � �.JLKP=@KTE�e`ST?[UG¦nO�c]eR?G\�=Z?Gk�¢ R �H�/�,_ 9�$�]�w�$��­�z �&�/�4�]�!���2 ¢)�q_ � ¢�§� (�¡ ��$�& q���4� ����¢£ (ª ­d�|�$��­¥�]�!���2 &�;�)��¢J§�  �?�/�!� zL| �Z_�®�L^R?pEhCF^rUs=ZV�U@\lfFKPM�? CF^�CF=@f�q`kl?[^�U�A�\�=]U@eRKTOjCFSP\�^Rf�UseR? STKP^R?GO]\FA>o q`EGSTKP_&¦uO

CF=@fFqRkl?[^xU�A�\�=�U@eR?�?( IKTO¤Us?[^REG?�\FAjKT^I�w^RKPU@?GSNV klCF^xV vR=@KPkl?[O��Dc]eR?[\�=@?[k m`:ut�¢B:c]eRKTO «�\�= %�OjKP^�C�A�?[«�EhCiO@?GO]\iA�OZk�CiSTS]_Z#wCi^R_�A�\�=YOZ\�kl?O@v"?GEGKTCFS0v`=@\�f�=Z?GOZO@KT\F^ROO@qRE�e�CFO0�� 9 k�\a_G_�#xQRq`U U@eRKTO>SPKT^R?]\FA0CiUZU�CFE&%�eRCFO ^R\FU VaKT?GSP_R?G_�JLKP=@KTE�e`ST?[UG¦nOUseR?[\�=@?[k�KT^�KNUsOYA�qRSPS0A�\�=@E[?F:¯ y�qwCF^xUsKNU�CiU@KPM�?�MF?G=@OZKT\�^ \FAYJLKT=@KPE�eRST?BUh¦uOUse`?G\�=Z?GkdKTOLU@eR?�;>=@KTk�?®��q`k�Q¨?[=

c]eR?G\F=@?Gk A�\�=�CF=ZKPU@eRkl?BUsKPE vR=Z\�f�=Z?GO@OZKT\�^`O�#0«�eRKPE�e CFO@OZ?G=ZU@OLUsewC©ULA�\�=�� I?[__ 9i#&?BM�?G=¤V vR=@\Ff�=@?[O@O@KP\�^ z k�\a_G_lewCFOgCFO¤Vaklv`U@\FUsKPEhCFSPSPV�Use`?�OsCFk�?�_R?[^RO@KNUWV¡\FAvR=@KPkl?[OG:�c]e�qRO�OZ?[U@U@KT^Rf�� ��{�¡_�X�za¢�\ � �2U �oc {�\ ��� z k�\I_v_8[8#0« ?�ewChM�?A�\�=�zlE[\�vR=ZKTk�?LUs\d_

���¸{�¡_�X�za¢ � 9 �L_F¢ §,K²��{¨¢TW

Page 9: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ��� �

tR: 1g2*354 1 3 �,6���� � ���04 1 2�3(8��\�^xUsKP^�qR?G_rA�=sCFEBUsKT\F^RO�CF=@?�?! IvR=@?[O@O@KP\�^RO�O@qRE�e�CFO

9 | 9H�| 9

mj| 9t

W

S ?g^R\�« OZUsq`_`V�UseR?[k�OZVaO¤Us?GklCiU@KTEhCiSTSPVF:m KPM�?[^ KP^xUs?GfF?G=@Ogz 3d~ �p#�z k X�z ��X4* * * 0 9i#,EG\�^`O@KT_`?G=LUse`?®�R^RKPU@?�E[\�^xUsKP^aq`?G_

A�=sCFEBUsKP\�^+ z 3 �z k X?W?W�W�X�z��´-K\ � z 3*| 9

z k |9

z ��| 9: : : | 9

z��

W

c�\�EG\FklvRqIUs?�UseRKPOYA�=sCFEBUsKP\�^�#w\�^R?g_R?B�w^R?[OKT^xU@?Gf�?[=@O�����#�_�� QxV�UseR?g=Z?GEGq`=@O@KP\�^� 0 9�¢(\

��� � z�� ��� 1 k | ��� 1�_�� � z��´_�� 1 k |b_�� 1�

«�KPU@e��+1 k � 9i# � 3���z 3�#�_21 k ����#�_/3Y��9�:>c]eR?GOZ? O@CiUsKPOZA¸V�U@eR?g=@?[SªCiU@KT\�^RO���´_�� 1 k Y ��� 1 k _�� � �)Y 9�¢ � 1 k

CF^R_���l_�� 1�´Y ��� 1�¡_��$� �zYp9�¢ � z�� W

�L^�U@eR?G^rO@e`\�«�OjUsewC©U+uz 3 �z k X?W�W?W�X�z��´-"� ����� _�� W

c]eR?�KT^`�R^RKPU@?�OZKTk�vRST?�EG\�^xUsKP^�qR?G_ A�=sCFEBUsKT\F^ + z 3 �z k X�z ��X?W?W�W - KTO�U@eR?®STKTk�KPU\FAUseR? �ZE[\�^xM�?[=@f�?[^�U@O�� ���Y� _���:�o�M�?G=¤V�KP=@=@CiUsKP\�^wCFS�� eRCFOpC�qR^RKPy�qR?lE[\�^xUsKP^aq`?G_A�=sCFEBUsKP\�^�?( IvwCF^`O@KT\F^&:oN `CFklv`ST?$\ S ?r«�KTSTS]\FQ`U�CFKP^ U@eR?�E[\�^xUsKP^aq`?G_ A�=@CFE[U@KT\�^ ?( IvwCF^`O@KT\F^ \FALU@eR?

y�qwCF_R=@CiUsKPE�KT=Z=sCiU@KT\�^RCFSTKNUWV � m�\ ' KP^REG?�� m STKP?GO�Q"?[UW« ?G?[^�9�Ci^R_ HZ#�« ?�«�=ZKPU@?� mp� 9 | 9��©{ k «�KPUse

{ k �9

� m�Y 9 �� m�| 9

Hc]e�qROj{ k � 9 | 9��©{"�j«�KNUse

{"��� H� m�Y 9 � � m�| 9

Page 10: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

Å� ���������� ���������CF^R_�Use�qRO�« ? EGCF^�«�=@KPU@?�{"�Y� H�| 9��©{���«�KPUse

{��]� 9� m�Y 9 � { k W

c]eR?YvR=Z\IE[?G_Rq`=@?YewCiO�Use�qRO*E[VaEGSP?G_�QwCFE&% CF^R_ «�?]klChVgEG\�^xUsKP^�qR?YKPU�KT^`_R?[�w^`KPUs?[SPVUs\��w^R_�UseRCiU � m®ewCFO]Use`?��� (�!�¨���$�¨� EG\F^�U@KT^�qR?[_rA�=@CFE[U@KT\�^�?( IvwCi^RO@KP\�^

� mp� 9�| 99 | 9

Hj| 9: : :

�h+ 9 G9iX½HZXh9$X½HZX?W?W�W±-¨� + 9 9iX½H�-�W

c]eRKTO�KTO�OZ\�kl?BUseRKP^Rf \FA C =sCi=@KPUWV �KNU�Usq`=@^RO�\�qIU�UsewC©U�CF^ KT=Z=sCiU@KT\�^wCiSjeRCFO�Cv¨?[=@KP\I_RKPE�E[\�^xUsKT^�qR?[_$A�=@CFE[U@KT\�^ ?! `vRCF^RO@KP\�^ KPA�CF^R_ \�^RSNV KNA�KNU®KTO®C �����B�$� �$¢L�¨����!� �$¢L�¨�/�]�$ª ��¢L­ #wUseRCiUKPO�\iA�UseR?pA�\�=@k aj| ± � s=#"«�KNUseua/X�±�=sCiU@KT\�^wCiS,CF^R_ s �9CF^�KT^xUs?[f�?G=]«�eRKPE�e�KPO]^R\FUjC�v"?G=ZA�?[E[U�O@y�qwCF=Z?F:c]eR?�E[\�^xM�?[=@f�?[^�U@O�f�KPMF?�MF?G=¤V�f�\a\a_�=sC©UsKT\F^wCFS CFvRv`=@\? IKTklCiUsKP\�^RO�Us\ �D\ S ?

ewChM�? 9H

9_��l_�� � k

f g � Y ���_�� g8f9

_��D_�� � kW

c]eR?rEG\�^xMF?G=@fF?G^xUsO ����� _�� CF=Z?�UseR? �ZQ"?GO¤U � =sCiU@KT\�^RCFSYCFv`vR=@\? IKTklCiUsKP\�^RO�U@\�D#LKP^ U@eR? A�\�STSP\�«�KP^Rf O@?[^RO@?[O�\ ¬�A �7� _ OsCiU@KTO¤�w?GObg �eY �"� _�g;f 9��FH _ � Use`?G^�"� _�� ����� _�� A�\�= O@\�k�? ¡: ��\F=@?G\�MF?G=�#>A�\�= 9i#*KNA]�.f _{c _�� CF^R_�"� _�^� ���Y� _�� U@eR?G^bg � Y��"� _�g hg �tY � �Y�/_��qg :

�I:�� 2�� � 7 � � � ��1 4�� � 6w4 1 ��I:N9�: �,C]EK�]�2-K��E5@i��<$R m KPM�?[^   9i#�« ?�OsChV�U@ewCiU�UW« \ KT^xUs?[f�?G=ZO®z�Xs}lCF=Z?EG\�^`f�=@qR?[^xU�k�\a_RqRST\® #R«�=@KNU@Us?[^rCFOjz � } k�\I_r #wKPA*  g©z,Y }�:��\�^RfF=@qR?[^REG? kl\a_RqRSP\   KTO�CF^ �Z?[yaq`KPMiCFST?[^REG? =Z?GSªC©UsKT\F^�� \�^ UseR? OZ?[Ur\FA

KT^xUs?[f�?G=ZO�#RU@ewCiU�KTO�CiO�A�\FSTST\�«�OYKPklk�?G_`KªCiU@?GSPV®A�=Z\�k U@eR?g_R?[�w^`KPUsKP\�^�#9�:>z � z k�\I_r HI:>z � } k�\I_r dKPA�CF^R_r\�^RSNV�KNA*}�� z k�\I_r m`:>z � } k�\I_r <CF^R_r}�� � k�\a_� dKPklvRSPKT?[O]z � � k�\a_� m KPM�?[^�  �9$#(?BM�?[=ZV KT^xUs?[f�?G=gKPOpE[\�^Rf�=ZqR?G^xUpUs\rvR=@?[EGKPO@?GSNV \�^R?®\FA UseR?l 

KT^xUs?[f�?G=ZOlU©��Xh9$X?W?W?W(X� eY�9$[8#xCFO*KTO�OZ?G?G^®QxVp«�=@KPU@KT^Rfz��`_i  |�a�«�KNUse�=Z?GklCFKT^Z"_R?G=j�dcearf   : c]e�qRO�_R?G^R\iUsKT^`f�QxV � �G  � Use`?pOZ?[U�\FA�EG\�^RfF=@qR?[^REG?pEGSTCFO@OZ?GO\FA"KT^xUs?[f�?G=ZO kl\a_RqRSP\g o#x«�?�OZ?G?]UsewC©U�C�E[\�klv`ST?[U@?YO@?BU�\FA0=Z?GvR=Z?GOZ?G^xU�CiU@KPMF?GO�EGCF^Q¨? U�C$%i?G^®Us\�Q"?�U©��Xh9iX?W�W?W*Xs `Yr9$[8#�UseR\FqRf�e®\FUse`?G=>E�eR\�KPEG?[O CF=@?]?[yaqRCFSTSNV MiCFSPKT_&:c]eR?gO@?BUj\FA(EG\�^RfF=@qR?[^REG?gEGSTCFO@OZ?GO�k�\I_Rq`ST\� dCFSPO@\®CF_`klKNUsO]CFSPf�?GQ`=sCFKPE\�v"?G=)"

CiUsKP\�^ROj\FA�CF_R_`KPUsKP\�^ Ci^R_¡k�qRSNUsKTv`STKTEGCiUsKP\�^&:Yc(\�O@?[?�Use`KTO�#¨\�^R?�^`?G?G_`OLUs\�E�eR?[E&%UsewC©U]KPA(z � z�� k�\I_r dCF^`_�}�� }�� k�\a_�  U@eR?G^rzj| }�� z���| }�� kl\a_r 

Page 11: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ���ÅZÅ

� � 6�7 6�� � ��qRSPU@KTvRSPKTEGCiUsKP\�^lk�\a_RqRST\�t� 9 H m

� � � � �9 � 9 H mH � H � Hm � m H 9

� � 6�7 6�� � ��qRSPU@KTvRSPKTEGCiUsKP\�^lk�\a_RqRST\��� 9 H m t

� � � � � �9 � 9 H m tH � H t 9 mm � m 9 t Ht � t m H 9

CF^R_pSTK %F?[«�KPO@?�z�*N} � z � *N} � kl\a_� #©U@ewCiU(KTO5U@eR? O@qRk��ivR=Z\I_Rq`E[U(\FARE[\�^Rf�=ZqR?G^`EG?EGSTCFO@OZ?GO�KTO]KP^R_R?Gv"?G^`_R?G^xU�\FA�U@eR?gE�eR\�KPEG?g\FA�=@?Gv`=@?GOZ?G^xU�C©UsKPMF?GO[:oN `CFklv`ST?$\ ' ?G?gU�CiQRST?[OLH®CF^R_�m�A�\�=]Use`? k�q`SPUsKPvRSTKPEhCiU@KT\�^�U�CFQ`ST?GOjk�\I_Rq`ST\®t

CF^R_��I:c]eR?�?( IKPOZUs?[^REG?�\FA¨CF_`_RKPU@KT\�^�Ci^R_®k�qRSPU@KTvRSPKTEGCiUsKP\�^gOsCiU@KTOZA¸V UseR?]q`O@qwCFS`SªCh«�O>\FA

KT^xUs?[f�?G= CF=ZKPUse`kl?BUsKTE #&UsewC©U�KPO E[\�klk�q`U�CiU@KPMaKPUWV$#(CFO@OZ\IE[KªCiU@KPMaKPUWV$#(_`KTOZU@=@KPQRq`UsKNMaKPUWV?[U@EF: c]eRKTO]KPO]A�\�=@klCFSTSNV®?( IvR=Z?GO@OZ?G_¡QxV�O@C�VaKP^Rf�U@ewCiUL� �G  �$KTOjC��J�!��� ¤� :�I:bHI: � C�F -HOL:=�v>LE�I]�8�/<���<$R ¯^�KT^xUs?[f�?G=jz�KPO¥�����$ (�(¢L� ' ª4 ¥�2���(��ª4�  �KNA�U@eR?G=Z?KTO�CF^ KP^�U@?Gf�?[=L}gO@\�U@ewCiULza} � 9 kl\a_r  : S ?�OsChVrUsewC©U�}�KTOCF^ KT^xMF?G=@OZ?�\FAzlk�\I_`qRST\® #w_R?G^R\iUs?G_�QxV�} � z 1 k�kl\a_r  :�w\�=*KP^ROZUsCF^RE[?L9YKTO*CFSP«�ChVaO�KT^xM�?[=ZU@KTQRSP?YCF^R_�9 1 k � 9 k�\I_r  :*¬W^xM�?[=ZUsKPQRKTSPKPUWV

k�C %F?GOjO@?[^RO@?pA�\�=�EG\�^`f�=@qR?[^REG? EGSTCFO@OZ?GOjkl\a_RqRSP\l  ��«�exV ��¢B: S ?g«�KTSTS5_`?G^R\FU@?UseR?gOZ?[Uj\FA�KT^xM�?[=ZUsKPQRST?�=Z?GOZKT_RqR?gE[SªCFOZO@?GO�k�\I_`qRST\® dQxV �W� �G  � ¢Z£h:oN `CFklKP^RKT^`f®U@eR?�k�qRSPU@KTvRSPKTEhC©UsKT\F^lU�CiQRST?[O��.H�¢!#*�.m�¢!#7«�? O@?[? UseRCiUk�\a_RqRST\®t=#

UseR?rKT^xMF?G=ZU@KTQRSP?�=@?GOZKT_Rq`?�E[SªCFOZO@?GO�CF=@? 9iX�m¡«�KPU@e m 1 k � m�k�\a_rt «�eRKPST?rCFSTS^R\�^R­[?G=Z\r=Z?GOZKT_RqR?®E[SªCFOZO@?GOpk�\I_Rq`ST\���CF=Z?�KP^xM�?G=¤UsKPQRST? #5CF^R_�H 1 k � m�k�\a_ �Z#m 1 k � H kl\a_��®Ci^R_�t 1 k � t k�\I_��I:¯OlUseR?[O@?¡?( `CFk�vRSP?GO�KP^R_RKTEGCiUs? #]CF^ KP^�MF?G=ZO@?¡k�\I_Rq`ST\  o#j\FAgKPUl?( IKTO¤UsO*#�KPO

�Z��������  kl\a_RqRSP\�  :�¬W^`_R?G?[_�#wKPA(z�}�� 9 k�\I_r dCF^R_�z���� 9 k�\I_r  Use`?G^qRO@KP^Rf E[\�klk�q`U�CiU@KPMaKPUWV Ci^R_ CFO@OZ\IE[KªCiU@KPMaKPUWV \FAgk�\a_RqRSªCi=lk�q`SPUsKPvRSTKPEhCiU@KT\�^ « ?ewChM�?

} � } *x9 � } *�z���� �Dza}B¢ *���� 9 *���� � kl\a_r CF^R_�Use�qRO�} � � k�\a_� <CFO�E[SªCFKPkl?[_&:

Page 12: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

Å� ���������� �����������\�=Z?G\�M�?[= U@eR?�v`=@\a_RqRE[U�\FA0KT^xM�?[=ZU@KTQRSP?]EGSªCiO@O@?[O�KPO>OZUsKPSTSRKP^xM�?G=¤UsKPQRST?i:�c]e�qRO�« ?

f�?[U�C EG\�k�k�q`UsCiUsKNM�?�fF=@\�qRv O¤Us=ZqRE[U@qR=@?�\�^ U@eR?�O@?BU��W� �G  � ¢Z£�\FAjKT^xM�?[=ZU@KTQRSP?=@?[O@KT_`qR? EGSªCiO@O@?[O�k�\a_RqRST\   : S ?¡«�KTSPSLeR?G^`EG? =@?[A�?[=�Us\ �W���[  ��¢ £ CFO�U@eR?���Zª ¢L� �]ª �¨� �$¢L���$  ¤ �¡�2����2���(��ª4�p  :¯ E[=@KNUs?G=ZKT\�^ A�\�=�KP^xM�?G=¤UsKPQRKTSPKPUWV¡k�\I_`qRST\� o#>CiO « ?GSTS�CFO�Ci^ CFSTfF\�=@KNUseRk A�\F=

�w^R_RKP^Rf�UseR?gk�\I_`qRSªCF= KT^xM�?[=@OZ?$#7KTO]f�KNM�?[^�QxV ´�����o: #�RL��Rq� �  &�& &�&���$�!­2�$�]�|�����e�(�¨ (��¢j�&�/�]�$��¢L�¨�/�;�?�/�,�$�{����¢)  ¤  (�jz{¢)�'  ¥�����$ (�!¢A� ' ª4 ��2���(�Zª4�   �4�q¢J§��$¢�z�$�]��  �$�¡ ;�&�&�]�!���2 �� f�E[_,�.z�X� �¢���98®¬W^R_R?[?G_�#`z�CF^R_�  Ci=@?�EG\�v`=@KTk�?]KPA&CF^R_�\�^RSNV�KPA"«�?�EhCF^�O@\FSPM�?�z�{j|  �~�� 9

�.§5?Gk�klC®HI:ntx¢(#"CF^R_�UseR?pSªC©U@Us?[=]?Gy�qwCiU@KT\�^rKTO�?[yaq`KPMiCFST?[^xUjU@\lOZ\�STq`QRKTSPKPUWVl\FA*U@eR?EG\�^`f�=@qR?[^REG? z�{ � 9 k�\I_r  :¯O Ci^�KPklk�?G_`KªCiU@? E[\�=@\�SPSªCF=¤Vi#�«�?�OZ?G?]UsewC©U>KPA" �KPO5�]�!���2 �U@eR?G^�CFSPS`^R\F^R­G?[=@\

=@?[O@KT_`qR?�E[SªCFOZO@?GO kl\a_RqRSP\�  CF=@?lKT^xM�?[=ZU@KTQRSP?$#(O@KT^`EG?lU@eR?�\�^`SPV _RKNMIKPO@\�=ZO \iA� CF=@?l9�CF^`_¡  CF^R_�O@\�U@eR?�\�^RSNVrCFSPU@?G=Z^wCiUsKNM�?gUs\�fFEG_,�Dz�Xs  ¢]� 9�KT^�U@eRKTOjEhCiO@?KTO]f�E[_,�.z�X� �¢ �  o#7CF^`_�U@e�qROz � ��kl\a_r  :��\�=Z?G\�M�?[=�#�CFO�_R?[O@EG=ZKTQ"?G_$KT^ �²HI:nmx¢&# Use`?�o>qRE[STKT_`?hCF^$CFSTfF\�=@KNUseRk vR=@\�MaKT_R?[O

A�\�=]CF^r?/��E[KT?G^xUjk�?[Use`\I_�\FA(�w^R_RKP^Rf�C�O@\�SPq`UsKP\�^�\FA*zx{�|$ r~®��9$#RUsewCiU�KPO]\FAv¨\FSPVa^R\�k�KªCFSwU@KTk�?LKT^�Use`?�KT^RvRqIU?#wCF^`_rU@eaq`O�Ci^r?���EGKT?[^xU�k�?[U@eR\a_�A�\F=Y�w^R_RKP^Rfkl\a_RqRSTCF=�KP^xM�?G=ZO@?[OG:���8�NEH>JP�>JC]E #�R ��R � ��ª4 (��� �p¢£�/¢A�¨ (��¢i� ��� �(¢L�¨�/� �D  ¢T�4�p¢�§� p� ��� '  (���)�l�����$ (�(¢L�� ' ª4 ¥�¡ &�(�L�(�� ��(ª �/�&�� &�¥�2���(�Zª4�g �®¯O]« ?gOsCh« CFQ"\�M�? #w\FA �rKPOYvR=@KPkl?�UseR?[^�CFSPS0^R\F^R­G?[=@\�=@?GOZKT_Rq`?�EGSªCiO@O@?[O�kl\a_

��CF=@?gKT^xMF?G=ZU@KTQRSP?gCF^R_�Use�qRO � �"¢>� �|Y 9pKT^�U@eRKTO�EhCiO@?F:�����8�/@B>A<*� #�R���R §��/¦�¢J§��$¢��?�/�l�]�!���2 ���¦l �§��$�$  � � � ¢ � � � Y � ��1 k�� �,09/$$®�I:nm`: �,�H�®���5>LEH��<�� ����.:�>LE5FK�8�j���K��C������ R m KPM�?[^p{ k�\I_ N#�« ? f�?BUC�vwCiKT=p\FA]=@?[O@KP_RqR?�E[SªCFOZO@?GO���{�kl\a_ wX@{ k�\a_ &¢pKP^ ��� ���$� �� (�Y:�¬�AwX� CF=Z?pE[\�vR=ZKTk�?$#`U@eR?G^�«�?gklChV�=@?[EG\�M�?[=j{ kl\a_ ¡A�=@\�k UseRKPOjvRCFKT=[:���K�8C]����� #�R ! �²�Y�*cj¢ R %J� wX �$�¡ ��&�&�]�!���2 d¢�§� (�2�?�/�gz�X�}�¦l �� �$�o���/ª �$ ¢J§� ��&�/� ¤ ���� (� �&  { � z k�\I_

{ � } k�\I_ �²�I:N9�¢¢J§� ����/ª �Z¢L�¨�/�w�4� �Z�������� q�2���(��ª4� «®) �&���)�*®�c(\lOZ?G?pqR^RKPyaq`?G^R?[O@O�#"^R\FU@?�UsewCiUjKNA�{KX@{ �0CF=Z?pOZ\�STqIUsKT\F^RO]\FAp�.�I:P9�¢(#RUse`?G^Q¨\iUse CF^`_ _RKNMIKP_R?�{Y { � CF^R_ OZKT^RE[? wX� Ci=@?�E[\�vR=ZKTk�?�Use`KTOlA�\F=@EG?[O g�{�Y�{ ��#RU@ewCiU�KPOY{ ��{ � kl\a_ *:

Page 13: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ����� � 6 7 6�� � �L=@_R?[=@O�\FA�?GSP?Gk�?G^xUsO�k�\I_��

z 9 H m t\�=@_5�.z�X½��¢ 9 t t H

c]eR?®k�?[Use`\I_ \FA O@\�SPq`UsKP\�^ KPO�EG\�^RO¤Us=ZqRE[U@KPM�? \ ' KT^RE[? wX� CF=@?�E[\�vR=ZKTk�?$# KTO®KT^xMF?G=ZU@KTQRSP?�kl\a_RqRSP\ *: JL?[^R\FU@?�QxV�� U@eR?�KP^xM�?G=ZO@?�\FA kl\a_ N#�CF^R_STK %F?[«�KPO@?¡SP?[U�� Q¨?¡U@eR? KT^xM�?[=@O@? \FA kl\a_ ¡: c]eR?G^ C OZ\�STqIUsKT\F^ \iA U@eR?OZVaOZU@?Gk \FA�EG\F^Rf�=@q`?G^RE[?GO��.�I:N9�¢�KTO]f�KNM�?[^�QxV

{��� �� 5zL| ���} k�\I_ {W¬W^R_R?[?G_�#¨kl\a_RqRSP\ «�? ewChM�?� �� �� 9 k�\a_ «�e`KTST? � � k�\a_ CF^R_Use�qRO�{��$ �� 0z�| ��r} � 9 *�z�| �.*���} ��z k�\I_ w#"CF^R_rSTK4%i?[«�KPO@?�{ ��}kl\a_ *:�����8�/@B>A<*� #�R�#�R %J��{�� z k�\I_rm� { � }�k�\a_ � � { � � k�\I_�� � �]�l{kl\a_ 9��x�=®�����8�/@B>A<*� #�R���R §��/¦ ¢�§��$¢ � � wX� �$�¡ ¥�&�&�]�(���2 �¢�§� (��{��4�������$ (�!¢A� ' ª4 ��2���$��Zª4� �� �,�$�]�v�/��ª ­|� ����¢D�4�q�����$ (�!¢L� ' ª4  ' �/¢�§G�2���(�Zª4� �$�]�2�2���(��ª4� ®�����8�/@B>A<*� #�R���R §��/¦o¢�§��$¢�� � wX� o�$�¡ ��&�&�]�!���2 l¢J§� (� � 5¢ � �r¢ � &¢�®�����8�/@B>A<*� #�R���R §��/¦n¢�§��$¢��?�/�Y  9d¦l j§��$�$  �. �¢��   ! ' � � 9 Y 9��)�"¢¦N§� (�¡ ¥¢J§� p�]�¡���(���(¢D�4���/�$ (���$ª�ª��]�(���2 &�q�$�����L�$��� ¤  �®�I:utR: �,�H�v<(P��2-5@$P -K����C� DP��K�v��-5O�P�> �NOL>L@i:�P�>�I��r����C�- � �����G ¡��¢Z£ R ¬W^lUseRKPOO@?[E[UsKP\�^�#�«�?�«�KTSTS7KP^�MF?GO¤UsKTf�CiUs?jUseR?�OZUs=ZqRE[U@qR=@?\FA,k�qRSNUsKTv`STKTEGCiUsKP\�^�KP^lUse`?Lf�=@\FqRv\FA�KT^xM�?[=ZUsKPQRST?�=Z?GOZKT_RqR?gE[SªCFOZO@?GO�k�\I_`qRST\®  :�����8�/@B>A<*� #�R ��R %J��z�X�};~ � �G  �h¢J§� (�o¦l r�|�$­t�&�/������¢) �z�*�} k�\a_�  �������ST\�f>z�ST\�f�}B¢ ' ��¢p�&�� (� �$¢A�¨�/����®¯ A�qR^R_wCFk�?G^xUsCFS(CFO@v"?GEBULKTO�Use`? v"\�«�?[=\�v"?G=@CiUsKP\�^�#7\FA�=@CFKTOZKT^RflCF^�?GST?[kl?[^xU

Us\®C�v"\�« ?G=*\�z��� z � k�\I_r  :�¬�U]U@qR=@^`O]\�q`U]UseRCiU]UseRKPO]KTOYE[\�k�vRq`U�C©UsKT\F^wCFSTSNV?hCFO¤V]\�����8�/@B>A<*� #�RL��� �.JLKNMIKP_R?�CF^`_rE[\�^Ry�qR?[=�¢ R §��/¦ ¢J§��$¢r¦l � �$�V�&�/������¢) �z �kl\a_r  ����$¢´�2���(¢ ���DSP\�f��&��ST\�f�  ¢ � ¢r (ª4 (�2 (��¢¨�$�!­ ' ��¢p�&�� (� �$¢A�¨�/����®���8�NEH>JP�>JC]E #�RL����R"! §� ¥�/� �� (�q�)��z2~ �����G ¡��¢ £ �4��¢J§� �ª4  �/�(¢ ����¢£  ¤  (� � 0 9�?�/��¦N§��¨�&§�z � � 9 kl\a_r �®S ?p_`?G^R\FU@?pU@eRKTO]KP^xUs?GfF?G=]QxV�\�=@_5�.z�X�  ¢½:

�����8�/@B>A<*� #�RL����R�� ���B  ���|¢±� ' ª4 &�T�)�T�/� �� (�¡�T�)�D�$ª�ª� (ª4 (�2 (��¢�� �2���(��ª4� �ZX���XG9�98®c]ewCiUj\�=Z_,�.z�X� �¢Y?( IKTO¤UsO�KTO]f�qRCF=sCF^xU@?G?G_�QxV�\

Page 14: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ���������� ���������� � 6 7 6�� � �L=@_R?[=@O�\FA�?GSP?Gk�?G^xUsO�k�\I_r�

z 9 m � �\�=@_5�.z�X��x¢ 9 H H H

� � 6 7 6�� � �L=@_R?[=@O�\FA�?GST?[kl?[^xUsO�kl\a_¡9�9z 9 H m t � � � � � 9h�\�=Z_,�.z�Xh9F9�¢ 9 9�� � � � 9�� 9�� 9�� � H

���K�8C]����� #�RA�� �.�w?G=Zk�CiUz" o>qRSP?G=�¢ R �H�/�q�$��­�z�~ �W� �G  � ¢Z£;¦l �§��$�$ z�� % ��( � 9 kl\a_r  W

¬W^rvwCF=ZU@KTE[qRSªCF=*#I« ? OZ?G?�UseRCiUj\�=@_,�Dz�X�  ¢Tc �D  ¢½:>¬W^rADCFEBUjkl\�=Z?�KTOYU@=@qR? \� ��C��«C]<?>�P�>LC]E #�RA� !KR��  (¢�z|~ �W� �G  � ¢ £ '  ¥�����$ (�!¢A� ' ª4 ��2���(�Zª4�p .®��$ ���B������ jz � � 9 k�\a_� .® ! §� (��\�=Z_,�.z�X� �¢��$�����L�� &� ��®' $ %(�|���$�(¢L�¨����ª �$���(\F=@_,�Dz�Xs  ¢��$�����L�� &� �. �¢�®

) �&���)�*® S =@KNUs? ��� _*\�=Z_,�.z�X� �¢j|yaZ#j«�KPU@e �yc a f \�=@_5�.z�X�  ¢½: c]eR?G^9 � z � ���.z�� � %�� 4 ��(.¢�� *Fz� � 9�� *Fz�� z� k�\I_r  : ' KT^RE[?�a�f�\�=@_,�Dz�X�  ¢!#UseRKPO]A�\�=@E[?GO�a�� �lQxV�Use`? k�KT^RKPk�CiSTKPUWV�\FA*\�=@_5�.z�X�  ¢!#"U@ewCiU�KTO�\�=Z_,�.z�X� �¢qg �":;*CF=¤U �DQ7¢�A�\�SPST\�«�O�A�=Z\�k�vwCF=¤U �.Cx¢YCi^R_��w?[=@klCiU "Wo q`ST?G=[:�����8�/@B>A<*� #�RL�#�R��  (¢ � ^� HZX�� '  o�{�]�!���2 ����$�]� �&�/���(�L�� (� ¢�§� .�� &�(���|�$ª� ' �/�� �9h�2$u  ©*���$���(�¨�/���)�,¢J§� ;� �$¢A�¨�/�]�$ª(9��)�H® ! §��4��  ©*���$���(�¨�/�{�4���� (�(�¨���$�¨�|�)�¢J§� ��?�/�!�<9��)� � ��W z k W?W�Wsz��}¦N§� (�¡ �� �� (� �/¢£ &�2¢J§� ��r�������|�$ª��� (�!�¨���B® �5�/������(¢±�$� �& ��p9��Fm �d��Wnm�m�m * * *��d��W m � � � 9�$ ��9���� �d��W 9ht�HF�x�F� � � �d�2$ �9��I9�9j����W �F� � � � H $� (¢)��®

§��/¦�¢�§��$¢�� � \�=Z_,�¤9��ZX��"¢�®��� (�  (� �$ª � �/ �®�I:b�I:�� �/>A�.>JP�>�I]����C�C�P�<$R c]eR?gklC/ IKTklCFS"\�=@_R?[=�\FA�Ci^rKP^xM�?G=¤UsKPQRST?�?[ST?Gk�?G^xU�KPO �D  ¢B: S ?�«�KTSPS*OsChV�UsewCiU�z ~ �����G ¡��¢¤£pKPO�C��]�!���r��¢A���$ ��¡���/¢j�2���(��ª4�®  KPA\�=@_5�.z�X�  ¢>� �. �¢B:¯^ ?! RCiklKP^wCiUsKP\�^ \FA U@eR?�U�CFQ`ST?GO�\iA \F=@_R?[=@O�\FA�?GSP?Gk�?G^xUsO�k�\a_RqRST\��Z#,� #P:N:N:

KT^R_`KTEhC©Us?GO UsewCiU�U@eRKTO�O@\�k�?[U@KTk�?GO�_R\a?GO®KP^R_R?G?[_ ewCFvRv"?G^�# Use`\�qRf�e «�eR?[^ U@eR?kl\a_RqRSPqROYKPO���U@eR?gk�C� `KPk�CiS0\�=Z_R?G=]KPOH�=@CiUseR?[=]UsewCF^�t�� �.�x¢½:c]eR?gA�\�STSP\�«�KT^Rf�U@eR?G\F=@?Gk ?( IvRSªCiKT^ROYU@eRKTO]?[klvRKP=@KPEhCFS"�w^R_RKP^Rf=\

���K�8C]����� #�RA� ��R %J���b�4�;�,�]�!���2 ,¢J§� (�{¢�§� (�& ,�4���,�]�!���r��¢A���$ ,�¡���/¢l�2���(��ª4��H®�w\�=]EG\Fklv"\�O@KNUs?Lk�\a_RqRSTKA#IKNU�KTO]=Z?GSªC©UsKPMF?GSNV�=sCi=@?LUs\®ewChMF? vR=ZKTk�KPU@KPM�?�=@\a\FUsO*\

�����8�/@B>A<*� #�RL� � R ��$ §��/¦ ¢J§��$¢´� �� Hu¢�§� (� � 5¢¥�4�r (�$ (�]®' $,%J� wX� �Hb�$�¡ G�&�&�]�(���2 2¢�§� (�n¢�§� (�¡ ��4�2� ���]�!���r��¢L���$ ��¡���/¢��2���(��ª4�

«®

Page 15: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ����

¬�UgUsqR=Z^ROg\�q`U�U@ewCiU�Use`?G=@?�KTOgC�vR=ZKTk�KPU@KPM�? =@\a\FU�k�\I_`qRST\�  KPA CF^`_ \�^`SPV�KPA  � HZXst®\�=�  � � � X½H)� � «�eR?G=Z?���KTO�Ci^r\a_R_rvR=@KPkl?i:S eRKPST? c]eR?[\�=@?[k �I:N9�� f�qwCF=@CF^xUs?G?[O�UseR? ?( IKPOZUs?[^REG? \FA�C vR=@KPklKNUsKNM�?�=Z\I\iU

kl\a_RqRSP\�CgvR=@KPkl? #�\�^R?�_R\a?GO ^R\FU´%�^R\�« \FA0CF^�?/��EGKP?G^xU CFSTfF\�=@KNUseRk UsewC©U f�KPMF?G^Cr�DSªCi=@f�?h¢ vR=@KPkl?���#`�w^R_RO�C�vR=@KPklKNUsKNM�?=Z\I\iU�kl\a_RqRSP\ �5:¬W^ Use`KTOLE[\�^xUs?( aU�#0UseR?[=@?�KPOLC�EG\F^��Z?GEBUsqR=Z?�\FA o>klKPS(¯�=ZU@KT^�A�=Z\�k U@eR?�9���HF�`¦uO

«�eRKTE�e CFk�\�^Rf�\iUseR?[=LUseRKP^Rf�OLO@C�VaOgUsewC©ULUseR?®^�qRk�Q"?G=pH�KPO�C�vR=@KPklKNUsKPMF? =Z\I\iUkl\a_RqRSP\�KP^`�w^RKNUs?[SPV k�Ci^�V vR=ZKTk�?GO�#�CF^R_ Use`?�OsCFk�?�KPO Us=ZqR?�A�\�=�CF^xV�KP^�U@?Gf�?[=zt^�}x Ci^R_�^`\FUjC�v¨?[=ZA�?GEBUOZy�qwCF=@?�"�OZ?G?gUseR?gO@q`=ZM�?BVrQ�V���CFk ��qR=ZUWV{+ t$-5A�\F=A�qR=ZU@eR?G=]_`?[U�CiKTSTO[:c]eR?gKTk�v"\�=ZUsCF^REG?\FA,UseR?g^R\FU@KT\�^�\iA�v`=@KTk�KPU@KPMF?=@\a\FU@O]EG\�k�?GOYA�=Z\�k U@eR?LA�\�S "

ST\�«�KT^`f�\�Q`O@?G=¤MiCiUsKP\�^�\��w\�=�� ~ �W� �G  � ¢Z£?#*U@eR?lA�qR^`E[UsKP\�^�{ �� � � kl\a_r ewCFO�v¨?[=@KP\I_®?( `CFEBUsSNV®\�=@_,���]X�  ¢�QxVl; =Z\�v"\�O@KNUsKT\F^��I:P9Gt=#aCF^R_�KP^lvRCF=ZU@KTEGq`SªCF=>KNA��KTO]C�vR=ZKTk�KPU@KPM�?j=@\a\FU]k�\a_RqRST\� dU@eR?G^�U@eRKTO]v"?G=ZKT\a_�KTOYvR=Z?GE[KTO@?[SPV �. �¢B: c]e�qROKPA���KTO�C�vR=ZKTk�KPUsKNM�? =@\a\FU�k�\I_Rq`ST\�  U@eR?G^ A�\�=gCF^xV¡z{~��W� �G  � ¢Z£�«�?�k�ChV«�=@KNUs?�z���� � k�\I_�  «�KPUse { q`^RKTy�qR?rkl\a_RqRSP\ �D  ¢B: ' KP^REG?�Use`?G=@?rCF=Z? �D  ¢�KT^xM�?[=ZUsKPQRST?�=@?[O@KP_RqR?GO�CF^R_�U@eR?�O@CFk�?l^�qRk�Q"?G= \FAp��^R?GE[?GO@O@CF=@KPSPV KT^xMF?G=ZUz"KTQRSP?�¢ v"\�«�?[=@O�� � kl\a_� o#R« ?g�R^R_�\ ´�����o: #�RL� ��Rq� �w�����$ (�(¢L� ' ª4 ; (ª4 (�2 (��¢���~ �����G ¡��¢Z£r�4���q�]�!���r��¢L���$ ¥�¡���/¢�2���(�Zª4�g  � �,�$�]�v�/��ª ­|� �; (�$ (�(­� (ª4 (�2 (��¢�z2~ �W� �G  � ¢ £ � �$� '  ¥¦«�!��¢L¢) (�o�/�z ��� � k�\I_r �®c]eRKPO�SP?Gk�k�CCFSTSP\�«�O*qRO>Us\gEG\�^xMF?G=ZU k�\I_Rq`SªCF=D���Zª ¢L� �]ª �¨� �$¢A�¨�/�gKP^®U@eR?�f�=@\FqRv

�W� �G  � ¢Z£*KP^�U@\k�\I_`qRSªCF=´�B�B�$��¢L�¨�/�LKP^®� � �D  ¢��T#xCFOZO@qRk�KT^Rfj«�? ewChM�?]CjvR=@KPkd"KPU@KPM�?=@\a\FU?#ROZKT^RE[?gKPA*z ��� � CF^R_�} ���Use`?G^ z *�} �� � � k�\a_�  :¬�A�z ��� � kl\a_r o#�«�?�«�KTSPS «�=@KNUs?�{y\b� ¬W^R_� F�.z�X� �¢B: S ?rklChV�U@eRKT^=%

\FA�¬W^`_� i�.z�X�  ¢�CFO�C}�$�4���(�¡ (¢£ {ª4� ¤ �$�!��¢J§��r#LOZKT^RE[?�QxV UseR? CFQ"\�M�?�=Z?hCFOZ\�^RKP^Rf¬W^R_� i�Dza}B¢ � ¬W^`_� i�.za¢]| ¬W^R_� F�D}[¢ kl\a_ �. �¢B: m KNM�?[^��]#�{ KPU�KPOr?GCFOZV U@\EG\�k�vRq`U@?�� � k�\I_� o#0U@eR?G=Z?�KTOL^`\�%a^`\�«�^ ?/��E[KT?G^xUgk�?[U@eR\I_¡\iA _`?[Us?[=@k�KT^Z"KT^Rf�{ � ¬W^`_� i�.z�X� �¢ A�=@\�k z�#�� CF^`_   : c]eRKTO�KPO�%�^R\�«�^$CFO�Use`?�$�4���(�& (¢£ ª4� ¤ �$�!��¢J§�� �]�¡� ' ª4 (��:

�`:�� ��� � � �`4 1 �n�02*3�� � �,673K�&678�`:N9�: � -5OJ�8��� <@i�/>JP��8�/>JC�E R §5?BU � Q"?rCF^ \I_R_ vR=ZKTk�?F: S ?�«�KTSPS�OZU@qR_`V U@eR?EG\�^`f�=@qR?[^REG?

{ � � z k�\I_ ��.�`:N9�¢«�eR?G=Z?Lz�KPO KP^�MF?G=¤UsKTQ`ST?�kl\a_RqRSP\ �5:��L^R?LCiSTf�\�=ZKPU@eRk A�\�= _R?[EGKP_RKT^RfgUse`?O@\FSTqRQRKPS "KPUWV�\FA�U@eRKTO]E[\�^Rf�=ZqR?G^`EG?pKTO]f�KNM�?G^rQxVro>qRST?[=G¦uOjE[=@KNUs?G=ZKT\�^ \� ��C��«C]<?>�P�>LC]E ��RA� �.o>qRST?[=G¦uO�EG=@KNUs?[=@KT\F^7¢ R�! §� (�& w�4�.� ���/ª �Z¢L�¨�/� �)��{ � � zkl\a_ � � �q�$�]�G�/��ª ­d� �jz % ' 1 k (�� � � 9 k�\I_��H®

Page 16: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

Å� ���������� ���������) �&���)�*®�¬W^R_`?G?G_ #0KPA>z��{ � k�\I_ � U@eR?G^ z % ' 1 k ( � � � { ' 1 k � 9 kl\a_ �5:j�7\F=UseR?l=@?BM�?G=ZO@?�_`KT=@?[E[U@KT\�^�#(«�?lk�ChV qROZ?lUse`?�?( IKPOZUs?[^REG?�\iA�C�vR=ZKTk�KPU@KPM�?�=Z\I\iU �kl\a_ ��Us\�Us=¤V�CF^R_rO@\�SNM�?�Use`?pE[\�^Rf�=ZqR?G^`EG?��.�`:N9�¢�QxV�«�=@KNUsKT^`f

z �� � k�\a_���X �|c }�f�|Y 9¬�U «�KTSTS`O@q��EG?�Us\pO@eR\�« U@ewCiU�} � H�� kl\a_ ��Y 9jO@KT^`EG?�UseR?[^�{ ���� k�\I_ �O@\�SNM�?[O U@eR?jEG\�^RfF=@qR?[^REG?��.�`:N9�¢½: ' qRQRO¤UsKNUsq`U@KT^Rfpz �� � k�\I_ �lKT^�z % ' 1 k (�� � � 9kl\a_ �rf�KPMF?GO

������ � ��9 k�\I_��GWX V�;>=Z\�v¨\FO@KPU@KT\�^��I:P9GtLUseRKPO*A�\�=@E[?GOj� �pY�9�¢Z}B��Hp� ��kl\a_ \F=@_,����X �"¢�CF^`_�OZKT^RE[?��KPO�C�vR=ZKTk�KPU@KPM�?�=@\a\FU?#`\�=@_,���]X��"¢>� ��Y 9 CF^R_�«�?g�w^R_�UsewC©U}�� ��k�\a_ HZ#UsewC©U�KTO�}]� H��lCFO�=Z?Gy�qRKT=Z?G_&:¯OjCF^rKTk�kl?[_RKªC©Us?E[\�^ROZ?Gy�qR?G^`EG?$#"«�?gOZ?G?gUsewCiU�A�\�=���\a_R_�#�Yp9 ��{ � k�\I_ �KPA¨CF^R_®\�^`SPV KPA � � 9 k�\a_�t=#xOZKT^RE[?p�zYp9�¢ % ' 1 k ( � � � 9�vR=@?[EGKPO@?GSNV�KP^�UsewCiU EGCFO@?i:�`:bHI: �,�H� ´�8����E5FK���v<*QK�wM«C]ON:�EHF� -H:=FH��:�P�>L@l2�Z@B> �H�/C�@B>JP!Q«R �w\�=0��^�H�CF^ \a_R_¡vR=@KPkl?�CF^R_ z�KT^xM�?[=ZU@KTQRSP?�kl\a_RqRSP\ ��#¨UseR?�§5?[f�?G^R_`=@?®OZVak�Q"\�S�� �'��KTO]_R?B�w^R?[_ CFO �

z��� �

|�9 z ��{ � kl\a_ �Yp9 \iUseR?[=Z«�KTOZ?

¬�U¡KTO�O@\�k�?[U@KTk�? E[\�^xM�?G^`KT?G^xU Us\ ?! aUs?G^`_�U@eR? _R?B�w^RKPU@KT\�^�Us\ KT^`EGSTq`_R? ^R\�^�"KT^xM�?[=ZU@KTQRSP?L=@?[O@KT_`qR?GOjQxV�=@?Gy�qRKP=@KP^Rf�� �' � � ��KPA��g©zw:X�?[ST\�« CF=Z?�O@\�k�?gO@KPklvRSP?LvR=Z\�v¨?[=ZU@KT?GO]\iA�U@eR?p§,?[f�?G^`_R=@?pOZVak�Q"\�S¨\9�:�¬�A*z � }�kl\a_ ��U@eR?G^�� �' � ��� �' � :HI: � � �' � � |�9�: X�\FU@e$Use`?GO@?rA�\�STSP\�« A�=Z\�k UseR?rM�?[=ZV _R?[�R^RKPU@KT\�^ \FALU@eR?§5?[f�?G^R_`=@?gOZVak�Q"\�S²:

m`:>o>qRSP?G=[¦nO�EG=ZKPU@?G=@KP\�^�EhCi^�Q"?�=@?BA�\�=@k�q`SªCiU@?G_�Us\®=@?GCF_�z��� � z % ' 1 k ( � � k�\I_ �

c]e`KTOgf�KPMF?GOpC�EG\�k�vRq`UsCiUsKP\�^wCFSPSPVr?/��EGKP?G^xU kl?BUseR\a_ \FA��w^`_RKT^Rf�U@eR?�§5?!"fF?G^R_R=Z?�OZVak�Q"\�S¨# CFO®«�?rEhCF^ EG\FklvRqIUs?rz % ' 1 k (�� � k�\I_ �$KT^ ����ST\�f � �7¢O¤Us?[vROG:¯ O@KPklvRSP?�EG\�^ROZ?Gy�qR?[^REG?gKTO�C�=@q`ST?�A�\F= «�eR?[^wY 9LKPOYC�OZy�qwCF=@?kl\a_RqRSP\

��\ �Y 9� � � �)Y 9�¢ % ' 1 k (�� � �

|�9iX � � 9 k�\a_rtY 9$X � � m k�\a_rt

Page 17: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ���Š�

tR: �rqRSPU@KTvRSPKTEhC©UsKPMaKNUWV�\ �z�}� � �

�z� �

�}� �

c]e`KTOYA�\�SPST\�«�OYA�=@\�k o>qRST?[=G¦uOjE[=@KNUs?G=ZKT\�^ ���\�=Z? vR=Z\FA�\�qR^`_�KPO�U@eR?pEG?GSP?GQR=@CiUs?[_�STC�« \FA ¥LqwCi_R=sCiU@KTEp��?GEGKPvR=@\aEGKNUWVi#wE[\�^��Z?GE!"UsqR=Z?G_rQxV�o>qRST?[=�Ci^R_�vR=@\�M�?[_�QxV m CFqRO@O*\���K�8C]����� ��R4��R %�����^� _ �$�¡ ��$�4�(¢L��� �(¢T���B�q�]�(���2 &��¢J§� (��

�_ �

�_� � � �zYp9�¢ � ��� � � ���

¯^�CF_R_`?G^R_Rq`k KTOYUse`?py�qwCi_R=sCiU@KTE�E�ewCi=sCFEBUs?G=j\FA�HZ\�H� � � �)Y 9�¢ % ' � 1 k (�� � |�9iX � � x�9 k�\I_r�

Y 9$X � � xgm k�\I_r�oN `CFklv`ST?$\�c�\ KTSPSTqRO¤Us=sC©Us?YUse`?�v"\�«�?[=�\FA¨UseR?jSªCh« \iA&y�qwCF_R=@CiUsKPE�=Z?GE[KTvR=Z\IE[KPUWVi#

«�?�«�KPSTS�E[\�k�vRq`Us?�U@eR?�§5?Gf�?[^R_R=Z?rO¤Vak�Q¨\FS ���' � A�\F=�vR=@KPkl?[O � ^�dHZX��I: S ?ewChM�? �

�� � � � � � � �zYp9�¢ % ' 1 k ( � � � % � 1 k (�� � ��� � � � W

c�\EG\�k�vRq`U@?�� ' ��� #i« ?Y\�^RSNV ^`?G?G_�U@\�%�^R\�« UseR? =@?GklCFKP^R_R?G=�\FA �®C©A¸Us?G=�_RKPMaKPO@KT\F^QxVr��Ci^R_�UseR?[^�E�eR?[E&%�U@ewCiU]U@eR?gKT^xM�?[=ZUsKPQRST?�OZy�qwCF=@?[Ok�\I_`qRST\®�®CF=Z?�9�CF^`_�tR:c]eRKTO�f�KPMF?GOjUsewCiU ���' � � 9�KNA � � x�9 kl\a_���CF^R_ ��' � � Y 9 A�\�=�� � xpHkl\a_��I:

�I:,� 6 787� 8�6 �����R4 1 2*3m KPM�?[^�C�v"\�O@KNUsKNM�?�KT^xU@?Gf�?[=�s�#R^`\FU�C®v"?G=¤A�?GEBUOZyaqRCF=@? #7« ?�«�KTOZerU@\��w^`_rU@eR?

KT^xUs?[f�?G=]OZ\�STqIUsKT\F^RO]\FA(UseR?g?Gy�qwC©UsKT\F^{ � Y sx~ � � 9¥W

c]eRKTO?Gy�qwCiU@KT\�^ «�CFO�OZU@qR_RKP?G_¡O@?BM�?[=sCFS>E[?G^xUsqR=ZKT?[OgCFf�\R:g¬�UsOOZUsq`_`V¡KT^�kl\a_R?G=Z^UsKPkl?[O�«YCFO�E�ewCFk�vRKP\�^R?G_ QxV �w?G=Zk�CiU��D;�?[STS]ewCF_ ^R\FU@eRKT^`f U@\ _`\ «�KNUse UseRKPO?Gy�qwCiU@KT\�^ #wCF^R_r\�«�?[O�KNU]Q¨?[KT^Rf®^wCikl?[_�CiA¸U@?G=]eRKPk U@\lo>qRSP?G=s¢B:�����8�/@B>A<*� ��RL��R §��/¦ ¢J§��$¢�� �Gs}�4�.��� (���? &�(¢d�������$�¡ ¢J§� (� ¢�§�    ���=�$¢L�¨�/�{ 1 sx~ � � 9d§��/�r�/��ª ­ � ����¢) (ª ­|�|�$��­�����¢£  ¤  (�����/ª ��¢A�¨�/����®�LQxMIKP\�qRO�OZ\�STq`U@KT\�^`O�CF=@?¡��{KXs~`¢�� �¨x�9iX���¢(#>EhCiSTST?[_ UseR?w¢L�!�����L�$ª´���/ª ��¢A�¨�/���B:

c]eR? ^R\�^xUs=ZKPMaKªCiSO@\FSTq`U@KT\�^RO�EG\�k�? KT^ y�qwCF_`=@qRvRSP?GO*\ ¬�A��¸{KXs~R¢lCF=@?¡OZ\�STq`U@KT\�^`OUseR?[^�OZ\�CF=@?��¨xL{KX!x�~R¢½: S ?j«�KPSTSwOsChV®UseRCiUYCpO@\�SPq`UsKP\�^®KTO«�����(��¢A���$ �KPA¨{KXs~ �`:

Page 18: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

Å� ���������� ���������� � 6 7 6�� �¥' \FSTq`U@KT\�^RO�\FA�{ � Y Hi~ � � 9

~��9 HH 9�Hm �F�t tx���� HFmx�i�� 9�m��F���� ���x�i�xH� t��F�F��mxH� H��itFt�HI9��9�� 9��F�F�Ft�t�HF�

¬�U�Usq`=@^RO�\�qIU�UsewCiUL^R\�^xUs=ZKPMaKªCFS(O@\FSTq`U@KT\�^ROLCFSP«�ChVIO�?! IKTOZU�#&UseR\FqRf�e¡UseR?BV¡CF=Z?y�qRKPU@?�OZvwCF=ZO@?F: S ?�klC�V EGST?GCF=@SNV \�=Z_R?G=�U@eR?lv"\�OZKPUsKNM�?�OZ\�STq`U@KT\�^`OpQxV KP^REG=Z?hCFO)"KT^Rf¡OZKT­[?�\FA�UseR?[KT=�~�E[\I\F=@_RKP^wCiUs? # \�=�?Gy�qRKNMiCFST?[^�U@SPV�Q�V�UseR?�OZKT­G?�\iA�U@eR?GKP=�{="EG\a\�=@_`KT^wCiU@?F:�c]e`?L�w=@O¤U�^R\�^xUs=ZKPMaKªCFS"O@\FSTq`U@KT\�^ �¸{ k Xs~ k ¢�«�KTSPS¨Q"?�EhCFSPST?[_�UseR?T� ������B�$�2 (��¢±�$ª]���/ª �Z¢L�¨�/�w:oN `CFklv`ST?$\ ' qRvRv"\�OZ?ds�� HI:gc�\��w^`_ �Dv"\�O@KNUsKPMF?�¢�OZ\�STqIUsKT\F^RO\FA U@eR?�EG\�=Z=@?!"

O@v"\�^R_RKP^Rf�;�?[STS&?[yaqRCiUsKP\�^�#R« ? =Z?[«�=@KNUs?�KNUjCFO9�| sx~ � � { �

CF^R_�U@eR?gvR=@\aEG?[?G_�Us\�OZ?hCF=ZE�erU@eR=@\�q`f�e�MFCiSTqR?[O]\FA(~®��9iX�HZX?W?W?W(X`U@\��w^R_�UseR\�OZ?A�\�=«�eRKPE�e 9�|es�~ � KTOLC�v"?G=¤A�?GEBUgO@y�qwCF=Z?F:�c]eRKPO�vR=@\aEG?[O@Ogy�qRKTE&%�SNV�VaKT?[ST_ROLU@eR?O@\�SPq`UsKP\�^RO���{KXs~`¢>� �Dm�X½H�¢!X��¤9��ZXh9�H�¢!X��.�F��X½�F�x¢!X��²�F���ZXstx�F�x¢(X?W*W?W@:�¬W^�Use`KTO�«�ChV « ?EhCF^�E[ST?hCi=@SPVl�w^R_�CFSTS0O@\�SPq`UsKP\�^ROYqRv�Us\�CF^xV�f�KNM�?[^�MiCFSPqR?�\FA�~¨:¬W^ c�CFQRSP?���«�? ?G^�qRk�?G=sC©Us? U@eR?�~B"WE[\I\F=@_RKP^wCiUs?\FA��w=ZOZUA�?B« v¨\FO@KPU@KPMF?pOZ\�STq�"

UsKP\�^RO��¸{���X@~��F¢ KT^ UseR?�EhCFOZ?vs�� HI: �L^R?�EGCF^ EGSP?hCF=ZSPV O@?[?�CF^ ?! `v"\�^R?[^xUsKªCiSKT^RE[=@?GCFO@?g\FA�~���«�KNUse *:³�?[=@? KPU,«YCiO*?hCiOZVgUs\j�w^R_ U@eR? �R=@OZU(A�?[« OZ\�STqIUsKT\F^RO,Q�V CjQR=@q`U@?("²A�\�=@E[? OZ?hCF=ZE�e&:

c]eRKTO�KPO]^R\FU�CFSP«�ChVaO�U@eR?pEhCFOZ?F: �7\F=]KT^RO¤U�CF^RE[?$#wA�\�=�s����`9�UseR?p?Gy�qwCiU@KT\�^�«�CFOCFST=Z?hCF_IV U@=@?GCiUs?[_ KT^�U@eR? 9�H�"²Use E[?G^xUsq`=ZV KP^ ¬W^R_RKTC�# CF^R_�KPU�«�CFO�A�\�q`^R_ UsewC©UUseR?]A�qR^`_wCFk�?G^xU�CFSwOZ\�STqIUsKT\F^�ewCFO ~ k � H�HF�ZXh9��Fm�Xs�����`:,¬�U�KTO>qR^`STK4%i?GSNV�UseRCiU UseRKPO«YCiO�A�\FqR^R_rQ�V�ewCF^R_rkl?[=@?[SPV�QxV�QR=Zq`Us?�A�\�=ZEG? �¯^`\FUseR?[= ?! `CFklv`ST?®\FA�CrSTCF=@f�?�A�q`^R_wCFk�?G^xU�CiS�OZ\�STqIUsKT\F^ A�\�=pCrOZk�CFSPS*MiCFSTq`?

\FANslKTO]fFKPM�?[^�QxVvs���9����Z#`«�eR?G^�~ k � 9���Xh9htx��X@t�Hit=Xstx���ZXh9��F�`:

Page 19: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ����

�I:N9�: �,�H�����/C�- �bOL:��GR ¯ =@?[k�Ci= %©CFQRSP?�A�?hCiU@qR=@?�\FA"Use`?�;(?GSPSw?Gy�qwCiU@KT\�^�KTO�U@eR??( IKTO¤Us?[^REG?�\FA�C�E[\�k�v¨\�OZKPU@KT\�^ SªCh« \�^ UseR?�O@?[U®\iA�O@\�SPq`UsKP\�^RO�#�UsqR=Z^RKT^Rf U@eR?GkKT^xUs\�C E[\�klk�q`U�CiU@KPMF?�f�=Z\�qRv&: ��^`?�«�C�Vl\FA,OZ?G?GKP^Rf�U@eRKTO�KTO Us\��R=@OZU�?G^R_`\�« U@eR?O@?BU�\FA �&  �$ªiO@\�SPq`UsKP\�^RO>«�KPU@e�Cpf�=@\FqRv�STC�«g: m ?[\�k�?[Us=ZKTEGCFSTSNVi#FU@eR?�=Z?hCFS7OZ\�STq`U@KT\�^`OA�\�=@k C e�Vav"?G=ZQ¨\�STC { � Y{s�~ � ��9«�KNUse�UW« \�O@e`?G?[U@O�#wCi^R_�« ?�_R\�Use`KTO �ZqROZUYA�\F=UseR?®=ZKTf�exUgO@e`?G?[U���{ �x¢!#5MIKTC�U@eR?�A�\�STSP\�«�KT^Rf�vwCi=sCFk�?[U@?G=@KP­hCiU@KT\�^�Q�V k�?hCi^RO\FA(UseR?gexVav¨?[=@Q"\�STKPELA�qR^RE[U@KT\�^`O�\

{���!Z¢T\b��EG\�OZe(��!Z¢!X ~0��!Z¢l\b� OZKT^Re,� !Z¢� s

¬�A�« ?p_R?G^R\iUs?gQxV����5� ��{�� !Z¢(Xs~0��!Z¢Z¢ UseR?gv"\�KT^xU]E[\�=@=Z?GOZv¨\�^`_RKT^Rf®Us\ ! UseR?[^rU@eR?f�=@\FqRv$STCh«�KTO�U@eR?r\�^R?rKT^ReR?[=@KNUs?G_ A�=Z\�k Use`?�CF_`_RKPU@KPM�?�f�=Z\�qRv \FAU@eR?r=@?hCiSTO�#^wCFk�?GSNV

�l��!Z¢������ ! � ¢l\b���l��!&| ! � ¢TW��\�=Z?�f�?[^R?G=@CFSTSNVi#7«�?�EGCF^�«�=ZKPUs? CF^xV�=@?hCiS�OZ\�STq`U@KT\�^�CFO�x�l��!Z¢�Ci^R_�Use`?G^ _R?!"EGSTCF=@?LU@eR?gf�=@\�q`vrSTCh«�U@\�Q"?���l��!Z¢��� � �l��! � ¢l\b��� � ��� !5| ! � ¢!# �X� � �}x�9�:¬W^ U@eRKTO5A�\�=Zk Use`KTO,_R\a?GO(STKPUZUsSP?�?( IEG?[v`U*U@\�_`?Gk�\�^ROZU@=sCiU@?>UseR?�?( IKTO¤Us?[^REG? \FA`U@eR?

f�=@\FqRv�SªCh«¥#xO@KP^REG?]=Z?GE[\�MF?G=ZKT^Rf !i|�! ��A�=@\�k !�Ci^R_ ! ��KT^xM�\�SNM�?[O CU@=sCF^`O@EG?[^R_R?[^�UsCFSKT^xM�?[=@OZKT\�^ v`=@\�QRSP?Gkr: ³�\�« ?[MF?G=�#�«�? klChV q`O@? Use`?¡CF_R_RKNUsKP\�^$A�\�=Zk�qRSTCF?rA�\F=UseR?�exVav¨?[=@Q"\�STKPE�A�qR^REBUsKT\F^RO�Us\ EG\�k�vRq`U@?�U@eR?�{ CF^R_ ~ E[\I\�=Z_RKT^RCiUs?[O�\FAU@eR?EG\�k�v"\�O@KNUsKT\F^&:]¬�A�« ?�O@?BU� � ��{KXs~`¢(#�� � � �¸{ � Xs~ � ¢YUseR?[^������ � � �¸{ � � Xs~ � � ¢«�KPU@e

{ � � � {w{ � |nsx~a~ � X ~ � � � {7~ � | { � ~2W�²�I:N9�¢¬W^�vRCF=ZU@KTEGq`SªCF=*#aU@eR?gKT^xM�?[=@O@?p\FA]��{KXs~`¢ KTOp�¸{KX?Yj~R¢½:��\iUs?®UsewCiUpA�=@\Fk �²�I:N9�¢�KPUpKTOg^R\FUpUs=@CF^ROZvwCF=@?[^xU�Us\�O@?[?lUseRCiU U@eR?lCF_R_RKNUsKT\F^

SªCh«�KTO�CFOZO@\aEGKªC©UsKPMF? �¬W^�U@eR?�A�\�=Zk �.�I:N9�¢5« ?YKPklk�?G_RKTCiUs?[SPVO@?G? UsewCiU�UseR? EG\�k�v"\�O@KNUsKT\F^p\iAK� �$¢L�¨�/�]�$ª

O@\�SPq`UsKP\�^RO�KTO�O¤UsKTSPS5=sCiU@KT\�^RCFS5CF^R_�_RKPUZUs\®A�\�=¥����¢)  ¤  (�]OZ\�STqIUsKT\F^RO�#R«�eRKPE�e¡CF=Z?p\FqR=f�\xCFS.:*c]e�qROj«�?gO@?[?pU@ewCiU]U@eR?gKT^xUs?[f�?G=�O@\FSTq`U@KT\�^RO A�\�=@k C�f�=Z\�qRv&:¯^�?GCFOZV «YChV Us\�=@?[EhCFSPSIU@eR?YEG\Fklv"\�O@KNUsKP\�^ STCh« �.�I:P9�¢(A�\�=*=@CiUsKP\�^wCFS�OZ\�STq`U@KT\�^`O

KTO�Us\lk�CFv�Use`?Gk Us\�y�qwCi_R=sCiU@KTEpKT=Z=sCiU@KT\�^wCiSTKPU@KT?[O�\j��{KX@~R¢��� ��� {®| � sx~]#"KP^«�eRKTE�erEhCiO@?gEG\�k�v"\�O@KNUsKT\F^�KTO]f�KNM�?[^�QxV�\�=Z_RKT^wCi=ZV�k�qRSPU@KTvRSPKTEhC©UsKT\F^&:� �$¢A�¨�/�]�$ªaO@\FSTq`U@KT\�^RO�Ci=@?�?hCiOZV�Us\��w^`_�QxV�Use`? �¤O@?GEGCF^xUjkl?BUseR\a_��R:

�����8�/@B>A<*� ��R ��R §��/¦ ¢�§��$¢r�$ª�ª�� �$¢L�¨�/�]�$ªp���/ª ��¢L�¨�/����)�®{ � Y}sx~ � � 9n�$�¡ ¤ ���$ (� ' ­��zYp9iX��x¢�¢)� ¤  (¢�§� (�¥¦«��¢J§

UI� 9 |ns"! �9�Y s"! � X H�!9�Ybs"! � ¢T\"!¥� �$¢A�¨�/�]�$ª [�W�²�I:bH�¢

³�KP^xU?\ ' U�CF=¤Uj«�KNUserUseR?gUs=ZKPMaKªCFS0O@\�SPq`UsKP\�^ �zYp9iX��x¢½: m KNM�?[^�C�v¨\�KP^xU��¸{KXs~`¢q^��zYp9iX��x¢,\�^�U@eR?]exVIv"?G=ZQ¨\FSªC�{ � Yds�~ � � 9i#�_`=sCh« UseR?YSPKT^R? EG\�^`^R?GEBUsKT^`fLUseR?�UW« \v¨\FKT^xUsO��zYp9iX��x¢]CF^`_ ��{KXs~`¢B:]c]e`KTOjSTKT^`?g«�KPSTS5KP^�U@?G=ZO@?GEBULUse`? ~B" C/ IKPO�C©ULC�v¨\�KP^xU

Page 20: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� � ���������� ����������.��X !Z¢B: ' eR\�« UsewCiU�~ � ![�¸{�| 9�¢ CF^R_ O@q`QROZU@KPUsqIUs?�QwCFE&%�KT^xUs\�U@eR?�\�=ZKTf�KP^wCFS?Gy�qwCiU@KT\�^ { � Y sx~ � �d9�Us\ �w^`_ UseR?�?! `v`=@?GOZO@KP\�^��²�I:bH�¢�: ¯�=@f�q`?�UsewC©U®CFO !MiCF=@KP?GO�\�M�?[=lCiSTSY=@CiUsKP\�^wCFSPO�#�« ?rfF?[UlCFSTS�UseR?�=@CiUsKP\�^wCFS O@\�SPq`UsKP\�^RO�\FU@eR?G=�U@ewCF^�zYp9iX��x¢½:�I:bHI: �*E�P��8���8�v<�C]O -�P�>JC]E5<$R ¯O�« ?�O@C�« \F^�Use`?�QwCFOZKTO�\FA>?( `CFk�vRSP?GO�#7KP^�U@?Gf�?[=O@\�SPq`UsKP\�^RO(CF=Z?�ewCF=Z_R?G=(Us\�EG\�k�? QxVF:>�w=@\�k U@eR?�OZewCFv"?�\FAwU@eR?�E[\�klv"\�OZKPUsKP\�^gSªCh«�²�I:N9�¢Y«�? O@?[?�UsewC©ULKPA�UseR?[=@? KTO\F^R?�^R\F^�U@=@KNMIKTCFS5KT^xU@?Gf�?[=O@\FSTq`U@KT\�^�#RU@eR?G^ U@eR?G=Z?CF=@?�Ciq`Us\�klCiU@KTEhCiSTSPVb���4� ����¢£ (ª ­w�|�$��­�KP^�U@?Gf�?[=�O@\�SPq`UsKP\�^RO*\ S ?�klChV CFOZO@qRk�?UsewC©U�« ?¡ewChM�? C�v"\�O@KNUsKNM�? OZ\�STq`U@KT\�^ � � �¸{ k Xs~ k ¢(#]{ k Xs~ k �� CF^R_ Use`?G^EG\�k�v"\�O@KP^Rf®KPU]«�KPU@e�KNUsOZ?GSPA(« ? f�?BU�UseR?pO@\�SPq`UsKP\�^RO �p£ � ��� � �|#�W�W?W�# � £ � �� £ % � 1 k ( � � � �¸{���Xs~ �F¢�CF^R_ A�=Z\�k �²�I:N9�¢lKNUsO�KPO�E[ST?hCi=lUseRCiU�{���� � CFO �� � :���K�8C]����� ��R ��R ��$ %J�ls ���4��� �/¢ ���� (���? &�(¢H�������$�¡ �¢�§� (�u¢�§� (�¡ ¥�$�¡ ����4� �����¢) (ª ­|�|�$��­d����¢)  ¤  (�����/ª ��¢A�¨�/�����)� )  (ª�ª � ��  ���=�$¢L�¨�/��{ � Y�s�~ � � 98®' $�%���¦l G�� (� �/¢£  ' ­ � � ��{ k Xs~ k ¢�¢J§� ¥� ���]�B�$�2 (��¢±�$ª ���/ª �Z¢L�¨�/��¢�§� (� �$ª�ª

����¢)  ¤  (�����/ª ��¢L�¨�/�����$�&  ¤ ���$ (� ' ­p{�| � s�~��}x � � � w~���®�I:nm`:�� >LEHF5>LEH�wP��K� -HE5FK:��.��E�P�:=OD<�C]O�-KP�>LC]E R�S eRKPST?c]e`?G\�=Z?Gk �I:bm f�qwCF=)"CF^xUs?[?GO�U@eR?p?( IKTO¤Us?G^`EG?�\iA�O@\FSTq`U@KT\�^ROjCF^`_�UseRCiULCiSTS&OZ\�STq`U@KT\�^`O�Ci=@?gA�\�qR^R_rA�=@\�k%�^R\�«�ST?G_`f�?�\FA UseR?�A�q`^R_wCFk�?G^xU�CiS�O@\�SPq`UsKP\�^�#¨KPULU@?GSTSPOLqRO�^R\iUseRKP^Rf�CFQ"\�q`U�eR\�«Us\p�w^R_�U@eR?�A�qR^`_wCFk�?G^xU�CFS7OZ\�STq`U@KT\�^0: �L^R?k�?[U@eR\a_�\FA5E[\�qR=ZO@?KPO�U@\�OZ?hCF=ZE�e�CFO_R?GOZEG=ZKTQ"?G_®CFQ"\�M�?F:�³�\�«�?BM�?[= U@eR?G=Z?]KTO*CF^®CFSPU@?G=Z^wCiUsKNM�?�kl?BUseR\a_�«�eRKPE�e�KTO*k�\�=Z??/��EGKP?G^xU�CF^R_�KP^xM�\�SNM�?GO�U@eR?]EG\F^�U@KT^�qR?[_®A�=@CFE[U@KT\�^�?( IvwCF^ROZKT\�^�\iA � sw:*c]eR?]OZKPUsq�"CiUsKP\�^®KTO CFO A�\�STSP\�«�O*\�c]e`?EG\F^�U@KT^�qR?[_�A�=sCFEBUsKP\�^®?( IvwCF^ROZKT\�^�\FA � s�KTO�v¨?[=@KP\I_RKPE$#\FA(UseR?�A�\�=Zku\

� s�� +uz 3 z k X�W?W?W(X�z��?-«�eR?G=Z?���KPO]UseR?pklKP^RKTklCFS"v¨?[=@KP\I_�\FA�U@eR?g?( IvwCF^`O@KT\F^&:]§5?[U����41 k � _� 1 k Q"?pU@eR?��dY 9�¢£"�OZUjvwCF=¤UsKTCFS0E[\�^xM�?G=Zf�?G^xUG:�c]eR?[^

� ��41 k YbsB_��41 k � �zYp9�¢ � W

¬�A��rKTO� (�$ (�®UseR?[^rU@eR?�A�qR^R_wCFk�?G^xUsCFS5O@\�SPq`UsKP\�^�KTOp� ���41 k X¡_� 1 k ¢B:¬�A���KTOj���B� UseR?[^�UseR?�A�q`^R_wCFk�?G^xU�CiS`O@\FSTq`U@KT\�^ � KTO�f�KPMF?G^�QxV U@eR?g�.H �´Y�9�¢£"WO¤UvwCF=¤UsKªCiS0EG\F^�MF?G=Zf�?G^xU?#¨CF^R_�k�\�=@?[\�M�?G= � � �����41 k | � sB_����41 k � � ���41 k | � sB_��41 k ¢ � W

oN `CFklv`ST?GO*\ls®� ��\ c]eR?[^ � ��� +uH 9iXh9$Xh9iXst�-,ewCFOjv¨?[=@KP\I_���� tR: c]eR?�m�"=@_�vwCF=ZU@KªCFS0EG\�^xM�?[=@f�?[^xU�KTO�+nH 9iXh9iXG9(-&� �x�Fm�CF^R_�U@eR? A�q`^R_wCFk�?G^xU�CiS(OZ\�STqIUsKT\F^KTOp�D��X�mx¢½:s®� �`9$\�c]e`?G^ � �`9L�h+n� 9iXst=X�mZXh9iX½HZX�HZXh9iX�mZXst=Xh9iXG9ht*-�CF^R_�Use`KTOjCFSTSP\�«�O]qRO

Us\®EG\FklvRqIUs?LUse`?���SªCF=Zf�?�¢ A�q`^R_wCFk�?G^xU�CiS&O@\�SPq`UsKP\�^&:

Page 21: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ��� �Å

�`: � �,6�� 1 6 � �(3,3��x6w4Z��� �,3K�04 1 2*3c]eR? �jKP?GklCF^R^�­G?BU�C�A�qR^REBUsKT\F^rKPO�_R?[�w^`?G_�A�\F=jE[\�k�vRST?! r±p«�KNUser�j?��²±�¢ 9

QxV�UseR?gOZ?G=@KP?GO°"�.±©¢>� ��

��� k9 � W

S ?gf�KNM�?gCF^rKT^xU@=@\a_RqRE[U@KT\�^�U@\®KPUsO]QRCFO@KPE�vR=@\�v"?G=¤UsKP?GO �DOZ?G?2+nH�-¸¢½:¯ QwCFOZKTE�ADCFEBU,KTO,o>qRST?[=G¦uO5vR=@\a_RqREBU,A�\�=@k�q`SªC�#G«�e`KTE�ep_RKTOZvRSªChVaO&UseR? E[\�^R^R?[E[UsKP\�^

Q¨?BUW«�?[?G^�°¨�.±�¢YCF^R_�vR=@KPkl?[O�\���K�8C]����� ��RA��R �H�/�j�j?��²±�¢� 9 �*°"�²±�¢;� �$� '  ,�¡ £�]�¡ &�� (��¢)  � ' ­�¢J§� d�&�/���$ (�!�¤  (��¢��]�&���(���(¢p�/�$ (���$ª�ª��]�(���2 &���

°"�²±�¢�� -'

99pY � 1 � W

) �&���)�*®�c]eR?�KP_R?hC KPO�Us\ ?! `vRCF^R_ ?hCFE�e ADCFEBUs\�=�� 9,Y � 1 � ¢ 1 k CiO®C fF?G\�k�?[U@=@KTEO@?[=@KT?[O

99�Y � 1 � � ��

���39� � �

CF^R_�Us\�k�qRSNUsKPvRSPV®Us\Ff�?[U@eR?G=YU@eR?g=@?GOZqRSPU@KT^Rf®OZ?G=@KP?GO-'

99�Y � 1 � � � 9

� � � k � � �� * * *�W�W?W�* * * � ����¢ � W

S ?gEGCF^�«�=@KPU@?�UseRKPO�CFOjC®OZqRk��� � k

z"� 5¢ �

«�eR?G=Z? z¨� 5¢�KPO�UseR? ^�qRk�Q"?G=�\FAp«�C�VaOr\FA ?( IvR=@?[O@OZKT^Rf UseR? KP^�U@?Gf�?[= CFOrCvR=@\a_RqREBUY\FA&vR=@KPkl?jv¨\�« ?G=ZOG:�X V®UseR?L�7q`^R_wCFk�?G^xU�CiS"c]e`?G\�=Z?Gk \iA5¯=ZKPUse`kl?BUsKTEm`:bHZ#�UseRKPO�EhCi^ Q¨?�_R\F^R?�KT^ \F^R?�CF^R_ \�^`SPV \�^R?�«�ChVi#�K²:u?F:�z¨� 5¢®� 9i#�«�e`KTE�evR=@\�MF?GO*U@eR?�vR=Z\I_`qRE[U(A�\�=@k�qRSªC�#�\�^`EG? « ?YE�eR?[E&% U@ewCiU�?[M�?[=ZV�Use`KT^RfLKPO*CiQRO@\�SPq`Us?[SPVEG\�^xMF?G=@fF?G^xUKNA���?x�.±�¢ 9�:¯O�U@eR?�CFQ"\�M�?�Ci=@f�qRk�?G^xU�O@eR\�«�O�# Use`?�vR=@\a_RqREBU�A�\�=Zk�qRSªC�KTO QRq`U®CrA�\�=@k�\FAUseR?p�wqR^R_RCFkl?[^xU�CFS&c]e`?G\�=Z?Gk \FA�¯=ZKPUse`kl?BUsKTEi:�`:N9�: ��EK:�O�Q�P�>L@�@iC]E�P�>LE�-H:ZP�>LC]E :�EHF -HE5@$P�>JC]EH:=O´� ��-K:�P�>JC]EnC� °"�²±�¢ R c(\A�qR=ZU@eR?G=�?! `v`ST\�=Z?LUseR?gEG\F^R^R?GEBUsKP\�^�Q"?[UW« ?G?[^�Use`?gU@eR?G\F=ZV�\FA*vR=@KPkl?[O�CF^R_r°"�²±�¢(#«�?Y«�KPSTS`CF^wCFSNVaU@KTEGCFSTSNV�EG\�^xU@KT^�qR?j°"�²±�¢,U@\gCFSTSaMFCiSTqR?[O�\FA&±x: S ?]q`O@?]UseR? m CFk�k�CA�qR^REBUsKT\F^rfFKPM�?[^�A�\�=��j?x�²±�¢ ��QxV�UseR?gKP^�U@?Gf�=@CFS&=Z?GvR=Z?GO@?[^xU�CiU@KT\�^

� �²±�¢>��� �3 �1 � ! � s"!!

Page 22: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� � ���������� ���������Us\®_R?B�w^R?gUseR?��&�/���]ª4 (¢£  ���/ (¢¨��� �Z� �(¢L�¨�/�lQxV

° £ �²±�¢T\b� � 1 � � � � � ±H ¢¤°¨�.±�¢c]eR?gQwCFOZKTE�ADCFE[UjCiQ¨\�qIU]UseRKPOYMFCi=@KªCi^�U�\iA�°"�²±�¢ KPO���K�8C]����� ��R4��R 9F: ! §� o�&�/���]ª4 (¢£  ��/ (¢±��� �Z� �(¢L�¨�/� ° £ �.±©¢u§��/�w� �2 (�¡�/�

�2�/�±��§��¨�,�&�/��¢A��� ���$¢A�¨�/�w¢£�|¢�§� r (��¢A���¡ �±i�4�]ª �$�  �®HI:>° £ �²±�¢q�4�,�$�]�$ª ­/¢L�¨�r  ©Z�& £�]¢]�?�/���(�����]ª4 T���/ª4 &�q�$¢�±L� ��Xh98®m`: %!¢ ���$¢A�4� �T &�q¢�§� «� ��� �(¢A�¨�/�]�$ªN  �����$¢A�¨�/�

° £ �²±�¢ ��° £ �¤9�Y ±�¢¯OYCF^�KTk�k�?G_RKTCiUs?]E[\�^RO@?[y�qR?G^RE[? \iA&UseRKPO ADCFEBU?#�«�?�\FQRO@?[=ZM�?LU@ewCiU�°�£��²±�¢�eRCFO

^R\ ­G?G=Z\�O�\Fq`UsOZKT_R?�Use`?�E[=@KPU@KTEGCFSYOZU@=@KPv �ec ��?��.±©¢Gc 9�: c]e`KTO�eR\�ST_`O�OZKT^RE[?� �²±�¢pKPO�^R?[MF?G=®­G?[=@\=# Ci^R_ °¨�.±�¢pKTO�CF^wCFSNV�UsKTE�CF^R_ ^R\F^R­G?[=@\ KT^�U@eR?�=@?[f�KT\F^ \FAEG\�^xMF?G=@fF?G^RE[?®�j?��²±�¢ 9i#"O@\�U@ewCiUjUseR? EG\�k�vRST?BUs?[_�­G?BU�ClA�q`^RE[U@KT\�^�°a£G�²±�¢�^� �KT^ ��?x�.±�¢� 9 �QxV$UseR? A�qR^`E[UsKP\�^wCFS�?Gy�qwCiU@KT\�^ #jU@eR?¡OsCFk�? KPO�Us=ZqR? A�\�=�U@eR?OZVak�kl?BUs=@KPE�=@?[f�KT\�^ ��?x�²±�¢;f �I:���\�=Z?G\�M�?[=�#*OZKT^RE[? � �.±©¢LKPO�CF^RCFSPV�UsKPE�?( IEG?[v`UA�\�=�O@KPklv`ST?�v"\�ST?[OlC©U�± �<��X?Yp9iX?YHZX?W?W?W(#Y°"�.±©¢�KPO�^R\�^R­[?G=@\ KT^$�j?��²±�¢uf �?( IEG?[v`U�A�\F= OZKTk�vRST?®­G?[=@\�O�CiUgUseR?l^R?Gf�CiUsKNM�?�?[M�?[^ KT^xUs?[f�?G=ZO�±�� YHZX?Yjt=X?W�W?W��U@\rklC$%i?�q`v A�\�=LU@eR?lOZKTk�vRST?�v¨\�SP?GOg\FA � � �� ¢�CiUgUseR?[O@?�v¨\FKT^xUsOs¢B:�c]eR?GOZ?�CF=Z?EhCFSPST?[_®U@eR?¥¢A�!�����L�$ªZ�/ (�¡����\iA5°"�²±�¢��U@eR?j^R\�^xUs=ZKPMaKªCFS`\�^R?GO CF=Z?�U@eR?�­G?[=@\�O�\FA5°�£��.±�¢CF^R_rCFO]« ? ewChMF? OZ?G?G^�Use`?[VrCFSPS0STKP?�KT^�U@eR?gEG=@KNUsKPEhCFS¨OZUs=ZKTv&:) �&���)�*®�� ' %F?BUsE�e7¢ S ?gO¤U�CF=¤Uj«�KNUse�Use`?pKP^xUs?GfF=sCFS&=Z?GvR=Z?GOZ?G^xU�CiU@KT\�^

� 1 � � � � � ±H ¢ 9 � � � �3 �1���� � � ! � � � s"!!

«�eRKTE�egOZeR\�«�O,U@ewCiU5« ? ewChM�?�CF^�KT^xU@?Gf�=@CFSF=Z?GvR=Z?GO@?[^xU�CiU@KT\�^p\FAI° £ �²±�¢"A�\�=&�j?��.±�¢ 9pCFO

° £ �.±©¢>� � �3 � ��!Z¢NY 9H ! � � � s"!!�.�`:N9�¢

«�eR?G=Z?�UseR?�Use`?[U�C�"�A�qR^REBUsKP\�^�KTO]f�KNM�?G^�A�\�= ! �®Q�V� � !Z¢ � ��

���+1��1���� � �

X V�;(\�KTOZO@\�^rO@qRk�klCiUsKP\�^�# � ��!Z¢YewCiO�C�U@=sCF^RO¤A�\�=@klCiU@KT\�^�A�\�=Zk�qRSTC� � 9 ! ¢ � � ! � � !Z¢�.�`:bH�¢

X�=@?GC$%�KT^Rf q`v U@eR?¡=@?[f�KT\F^ \FApKT^xUs?[f�=sC©UsKT\F^ KT^ U@eR?¡KT^xUs?[f�=sCiS=@?[vR=@?[O@?[^�UsCiUsKP\�^�`:N9pU@\�CF^ KP^xUs?GfF=sCFS5\�M�?[=®�.�ZXh9�¢]CF^R_�\�^R?�\�MF?G=�� 9iX � ¢!#7« ?�E�eRCF^Rf�?�MiCF=ZKªCFQ`ST?GO! �� 9�� !�Us\�U@\�Us=@CF^RO¤A�\�=@k Use`? KT^xUs?[f�=sCiSj\�MF?G= �D��WP9�¢®Us\�\�^R?�\�MF?G= � 9iX � ¢B:

Page 23: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ��� � �S ?�UseR?[^rq`O@?�UseR?�Us=sCi^ROZA�\�=Zk�C©UsKT\F^�A�\�=@k�qRSªC �`: H A�\F= � � !Z¢�Us\��w^`_�C©A¸Us?G=�O@\�k�?k�Ci^RKTvRq`SªCiU@KT\�^�UsewCiU

° £ �²±�¢>� Y 9± Y9

9pY ± | � �k

� ��!Z¢NY 9H � ! � � � | ! % k 1 � (�� � � s"!!�.�`:nmx¢

' KT^`EG? � ��!Z¢/Y�9� ��� �1�� � ¢,CFO ! � ��#©UseR? KT^xUs?[f�=sCiSaKPO*CiQRO@\�SPq`Us?[SPVgE[\�^xM�?G=Zf�?G^xU

A�\�=>CFSPS7±�Ci^R_lKPO�UseR?[=@?BA�\�=@?jCF^®?G^xU@KT=@?YA�q`^RE[U@KT\�^®\FA&±�:�c]e�qRO�A�=Z\�k �D�`:nmx¢�«�?�f�?BUUseR?�k�?G=Z\�k�\�=@vRe`KTE�E[\�^xUsKP^aqRCiUsKP\�^�#�«�KPU@e UseR?�\�^`SPV v¨\�SP?GO�Q"?GKP^Rf UseR?�OZKTk�vRST?\�^R?[O�CiU�±�� ��Xh9�:*�w=@\�k U@eR?jOZVaklk�?[U@=ZV®\FA��.�`:nmx¢*«�KPU@el=Z?GO@v"?GEBU�U@\�± �� 9NY�±«�?gf�?BU�UseR?�A�qR^REBUsKP\�^wCFS&?[y�qwCiUsKP\�^&:

�`:bHI: �,C]EHEH��@$P�>LEH�bP��K� �H�/>A�.��<2:=E5F`P��H���B����C]<�C� °¨�.±�¢ R ��KT?[k�Ci^R^�#0KP^eRKTO�OZ?Gk�KT^wCiS*vwCFv"?G=g\FA�9��x�i�o+u��-A#,qRO@?[_ °"�²±�¢jU@\�f�KPMF?lC�A�\�=Zk�qRSTC�A�\�=����¸{¨¢�KP^Us?[=@k�O�\FA>U@eR?�­[?G=Z\�O\FA�°"�.±©¢½:j³�KTO�A�\�=Zk�qRSªC®f�KNM�?GOC�EGSP?hCF=jqR^R_`?G=@O¤U�CF^`_RKT^Rf�CFOUs\«�exV®§5K²��{¨¢*KPO�Use`?]EG\�=Z=@?GEBU�CFvRvR=Z\? `KPk�C©UsKT\F^ Us\ � ��{¨¢B:�¬W^ROZU@?hCF_�\FA"CA�\�=Zk�qRSTCA�\�= ����{¨¢!#xKPU�KTO*k�\�=Z? E[\�^xM�?G^`KT?G^xU Us\f�KPMF?YC�A�\�=Zk�qRSTCjA�\�=,UseR? «�?[KTf�exUs?[_®O@qRk�\FAvR=@KPkl?����/¦l (�¡��� � c {K#¨?GCFE�e¡vR=ZKTk�?pv"\�«�?[= � � «�?[KTf�exU@?G_¡QxV�UseR?�SP\�fxCF=ZKPU@eRkST\�f ��\FA(UseR?gE[\�=@=Z?GO@v"\�^R_`KT^Rf®vR=ZKTk�?F:��L^R?g_R?[�R^R?GO

� ��{¨¢T\b� �'

���� '���� �

SP\�f0�

c]eR?=Z?Gv"?[U@KPUsKP\�^RO0� � A�\�= � 0 Hpf�KNM�?�C EG\F^�U@=@KPQRq`UsKP\�^�\FA0U@eR?\�=Z_R?G=�\FA&CiU�k�\�O¤U� {,: c]eR?�vR=@KPkl?[O � � � 9�¢�f�KPMF?�C EG\�^xUs=ZKTQRqIUsKT\F^ «�eRKPE�e�#YKNA�\�^R?rQ¨?[STKP?[M�?[Om CiqRO@O[¦>CFO@OZ?G=¤UsKT\F^ U@ewCiU�U@eR?�_R?G^`O@KPUWV \FAjvR=@KPkl?[O�^R?hCi=�{ KPO®CFQ¨\Fq`U�9���ST\Ff�{K#KTOpCFQ"\�q`Ug{,:�c]e�qROp«�?l?( Iv"?GE[U��.CF^`_ U@eR?�CiQ¨\�M�?�CF=@fFqRkl?[^xU KPOg?hCFOZKTSPV k�CF_`?=@KPf�\�=@\FqRO�¢"U@ewCiU5U@eR? ;>=@KTk�?>�jqRk�Q"?G=,c]eR?G\F=@?Gk KTO5?Gy�qRKNMiCFST?[^�U5U@\]UseR? CFOZO@?[=ZUsKP\�^UsewC©U � �¸{¨¢ � {,:pc]eRKTOLKPOLklCF_R?�U@=sCF^ROZvwCF=Z?G^xUgQxV UseR? A�\�=@k�q`SªC¡��_RqR?�Us\�MF\�^��CF^RfF\�ST_`U�¢

� ��{¨¢ � {2Y � {�Y ° �° �.�x¢�.�`:ut�¢

«�eR?G=Z?>UseR? O@q`k KTO5\�M�?[=*CiSTSF­[?G=Z\�O � \FA`°"�²±�¢B:(��\iUs?>UsewC©U5«�?�EhCF^`^R\FU,?( Iv"?GEBU�U@eR?A�\�=@k�qRSªCLUs\ EG\�^xMF?G=@fF?�CFQROZ\�STq`U@?GSNVi#aO@KT^`EG?jKPU�«�\�q`ST_lU@eR?G^�_R?[�w^`?�Cv�&�/��¢L��� ���2�B�A�qR^REBUsKT\F^�\FA,{K#`«�eRKPST? � ��{¨¢ KPO]C�OZU@?Gv�A�qR^REBUsKP\�^�«�KPU@e%�ZqRklv`O�«�eR?G^�{�� � � KPOC�vR=@KPkl?Lv"\�«�?[=G:c]eR?�EG\�^xUs=ZKTQRqIUsKT\F^$\FAUseR?�Us=ZKPMaKªCFS�­G?G=Z\�O � � YHZX?Yjt=X?Y���X?W?W�WlKPO�?GCFO@KPSPV

O@qRk�k�?G_ Us\�?[y�qwCFS k� ST\�fw�¤9�Y { 1� ¢CF^`_ KPOL^R?[f�STKPf�KTQRSP?F:�c]eR?®EG\�^RO¤U�CF^xU�U@?G=@kKTO�°�� �i°¨�D�x¢>��SP\�f H��>:�c]eR? KPklv"\�=¤U�CF^xU]vwCi=ZUjKTOYU@eR? OZqRk \�M�?[=jU@eR? ^`\�^xUs=@KNMaKªCFS­G?[=@\�O*#`«�eRKTE�e�« ?�?( Iv"?GEBU]Us\�Q"?L\FA,O@klCFSPST?G=�\�=Z_R?G=�U@ewCF^�{,:>¬�U]KPO�Use�qRO]E[=@qRE[KªCFSUs\®qR^R_`?G=@O¤U�CF^`_�Use`?p_`KTOZU@=@KPQRq`UsKP\�^�\FA�UseR?g­[?G=@\FOG:

Page 24: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� � ���������� ����������`:nm`: �,�H� �>J���.:�EHE��vQ �«C�P��K�Z<�>A<$R ¯O�^`\FUs?[_�KT^ O@?[E[UsKP\�^ �I:P9i#,Use`?�^R\�^�"Us=ZKPMaKªCFS0­G?G=Z\�O\iA °"�²±�¢jCiSTS5STKP?gKT^�UseR? EG=@KNUsKPEhCFS5O¤Us=@KPv �2c �j?��²±�¢jc 9�:�¬�A � KTO�C­G?[=@\�U@eR?G^�QxV�U@eR?A�qR^`E[UsKP\�^wCFS7?[yaqRCiUsKP\�^�° £ �²±�¢>� ° £ � 9 Y ±©¢!#`OZ\�KTOj9DY � #`CF^R_O@KP^REG? °"� �±�¢Y� °"�²±�¢g� ���_R?[^R\FU@KT^RflEG\�k�vRST?! rEG\�^ �ZqRfxCiU@KT\�^7¢!#w«�? f�?[U­G?G=Z\�OLC©U ��CF^R_ 9�Y �� ��Use`?�UW«�\�OZVak�kl?BUs=ZKT?GOp±��� �±lCF^R_ ± �� 9�Y ±�EG\�KP^REGKP_R?�\�^¡U@eR?�ZEG=ZKPU@KTEhCiS0STKP^R? �®��?x�.±�¢>� 9���H�¢½:c]eR?®�w=@O¤UgA�?[« ­[?G=@\FOp« ?G=Z?lE[\�klv`q`Us?[_ QxV ��KT?GklCF^R^¡e`KTk�O@?GSNA)#,CF^R_ CFSTS*STKT?

\�^ UseR?lEG=@KNUsKPEhCFS�SPKT^R?���?x�.±©¢g� 9��FHI:�c]eR?BV�Ci=@? � � � 9���H�| � ! �r«�KPUse ! k �9ht=WN9�mpW?W?W(X ! ��� HI9iWn�xHTW?W?W(X ! ��� H��ZWn�`9 W?W?W(?[U@EF:7��QxV�OZVak�kl?BUs=¤Vi#�« ?]\�^RSPVp^R?[?G_Us\®EG\F^RO@KP_R?G=�v"\�O@KNUsKPMF? !Z¢B:�>J�Z�o:�E5E � <g�GQ �«C�P��K��<?>A<��Z������=� ª�ª]� �/��¢L�!�����L�$ª��/ (�¡�����)��°"�.±©¢�ª �¨ ¥�/��¢J§� �(�!��¢A�¨� �$ª�ª ���  ��j?��.±�¢>� 9���H=®c]eR?j��KT?GklCF^R^®³�Vav¨\FU@eR?GOZKTO ewCFO>Q"?G?[^lE�e`?GE&%F?[_�?( aU@?G^ROZKPM�?[SPV�CF^`_�KPO�«�KT_R?[SPV

Q¨?[STKP?[M�?[_ U@\Q"?�U@=@qR? #iUse`\�qRf�e�Ci^�?( IvRSTCF^wCiU@KT\�^ CF^R_ vR=@\a\FARCF=@?�OZU@KTSTSxk�KTOZO@KP^RfjU@\_wCiU@?F:�¬�U@O�O@KPf�^RKN�wEhCF^`EG?�Us\ U@eR?�U@eR?G\F=ZV�\FA5vR=ZKTk�?GO KTO�KTk�k�?G^ROZ?F:��w\�=�KT^`OZU�Ci^REG? #«�?EhCF^�q`O@?��j³ U@\�?( IvRSTCFKT^�UseR?LOZk�CiSTSwO@KP­G?L\iA&UseR?=@?GklCFKP^R_R?G= Us?[=@k §5KW�¸{¨¢ Y����{¨¢�KP^�c*CiQRST?�9F:>c�\®O@?[?�UseRKPO�#RKNU]O@q��E[?GO�Us\®O@eR\�«�UsewCiU � �¸{¨¢NY {�KTO]O@klCFSPS¨#CF^R_ KT^ ADCiE[U�« ?�OZewCFSPS>CF=@fFqR?�UsewC©U�KPU�KPO�\FA�\�=@_`?G=pCiU�k�\�OZU � {SP\�f � {,:�c]eRKPOKTOY=Z?hCFOZ\�^wCFQ`ST?gKPA,« ?pSP\I\$%�CiU]U@eR?�A�\�=@k�qRSªC A�\F= � �¸{¨¢�KT^ �.�`:ut�¢&#R«�eRKTE�e�« ?p«�KPSTS«�=@KNUs?gCFO

� ��{¨¢ � {2Y � {�k � � � � ���9���H]| ��! � | W?W�W

«�eR?G=Z?�U@eR?�OZqRk KTO�^R\�« \�^`SPV�\�M�?[= U@eR?�^`\�^xUs=@KNMaKªCFSw­[?G=@\FO�#�UseR?j\�k�KPU@U@?G_�Us?[=@k�OQ¨?[KT^Rfl^R?GfFSTKTfFKTQRSP?F:Y¬�A>« ?�CFOZO@qRk�? U@eR?%! ��CF=@?��¡  �$ª #¨OZ\.g {�k � � � � � � gI� � {H#¨KNUKPOUs?[klv`U@KT^Rf�U@\�UseR?[^�qROZ?pU@eR?�Us=ZKªCF^RfFST?LKP^R?Gy�qwCFSPKPUWV�Us\®_R?[_RqREG?

g � ��{¨¢NY {«g8c � { � 9g 9��FH]| � ! ��g

CF^R_®O@\gO@C�V�UsewCiU � �¸{¨¢ZYl{lKTO>\FA"\�=Z_R?G= � {(:*c]eR?jCF=Zf�qRk�?G^xU KTO�^`\FU y�qRKPU@?]EG\�=)"=@?[E[U?#©CFO5KNU&Us=@CF^ROZvRKT=Z?GO&UseRCiU&Use`?�OZqRk \iA`CiQRO@\�SPq`Us?�MiCFSPqR?GO5_`KPM�?[=@f�?[O�\� 9��]gb9���H�|� ! �]g5� � :��j?[MF?G=ZU@eR?GSP?GOZO�#>UseRKPO fFKPM�?[O U@eR?l?[O@OZ?G^RE[?�\FAY«�ewCiU�KTOpewCFvRv"?G^`KT^Rf=#CF^R_lKT^lADCiE[U�UsC$%�KT^Rf kl\F=@?jEhCF=Z?LCF^R_�qRO@KP^Rf�k�\�=@?jKT^`A�\F=@klCiUsKP\�^�\�^�U@eR?_RKPOZU@=@K "QRq`U@KT\�^ \iA�­[?G=@\FO�#5\�^R?�EGCF^ O@eR\�«�UsewC©U � �¸{¨¢´Y {� � {�ST\�f � {,: c]eRKTO�fFKPM�?[O����{¨¢ Y §5K²��{¨¢� � {SP\�f�{�CF^R_�OZ\�?( IvRSTCFKT^RO�UseR?\�QRO@?[=ZMiCiU@KT\�^�=Z?GfxCi=@_RKP^Rf U@eR?O@KP­G?�\FA�UseR?�Use`KT=@_�E[\�STq`kl^�KT^�c�CFQRSP?�9�:c]eR?���KT?[k�Ci^R^�³�Vav¨\FU@eR?GOZKTO�CF^R_�KNUsO�f�?G^R?[=sCFSPKT­GCiUsKP\�^�� m ��³¢>U@\�\FU@eR?G= �� N"

A�qR^REBUsKT\F^RO���KPO�\�^R?�\iA(U@eR?Lk�\�OZUYKTk�v¨\�=¤U�CF^xU qR^ROZ\�SPMF?G_�vR=@\FQRST?[klOYKP^���q`k�Q¨?[=c]eR?G\F=ZVi#aCF^`_�KNUsO*MiCFSPKT_RKNUWVpewCFO>^�qRk�?G=@\FqRO KTk�vRSTKPEhCiU@KT\�^RO[:,�w\�=�KT^`OZU�Ci^REG? #�U@eR?G=Z?CF=@? CiSTf�\�=ZKPU@eRklO®A�\�=2�]�!���|�$ª ��¢L­ ¢) &�(¢L��� ¤ \iA�KP^�U@?Gf�?[=@O�«�eRKPE�e CF=Z?¡vR=@\�MF?G_�U@\=@?[yaq`KT=@?v¨\�SNVa^R\�k�KªCFSRU@KTk�?��¸UsewCiUYKTO�#aUs?[OZUsKP^Rf�KNA KTO vR=@KPkl?�\�=�^R\iU]=@?[yaq`KT=@?[O�C^�qRk�Q"?G=�\FA�\�v¨?[=sCiU@KT\�^`OYv¨\FSPVa^R\�k�KªCFS"KT^�SP\�f� 5¢(#wv`=@\�MaKT_R?[_r« ?pCFO@OZqRkl? m ��³p:

Page 25: empslocal.ex.ac.ukempslocal.ex.ac.uk/people/staff/mrwatkin/zeta/rudnick_intro_NT.pdf · KT^`O@v"?GE[U@KT\ ^ O@e`\ « OgUseRCiUgUseR? f =@?GCiUs?[OZUpEG\Fklk \ ^ _RKPMaKPO@\ =ZO \FAYtCF^R_

� ����=������� �� ����=����������� ��� ��� � �¯O�U@\ «�ewCiU�«�CFO�CFE[U@qwCFSPSPV vR=@\�MF?G_ O@\ ADCF=*# U@eR?�OZKTf�^RKN�wEhCi^�U�ADCFE[U®KPO�UsewC©U

UseR?[=@? CF=@?g^R\�­G?[=@\�O�\�^�UseR?pQ¨\FqR^R_wCF=¤V�\FA(UseR?pEG=ZKPUsKPEhCFS0OZUs=ZKTv�\ °"�¤9 | ��!Z¢,^� ��#O@\���f��j?x� � ¢�f�9�A�\�=LCiSTS,^R\�^xUs=ZKPMaKªCiS(­[?G=Z\�OG:Lc]e`KTOKPO�?[^R\�qRfFe Us\�vR=Z\�M�?�U@eR?;>=@KTk�?���qRk�Q¨?[= c]eR?[\�=@?[ku#*CFO «�CFO _R\�^R?��DKP^R_R?Gv"?G^`_R?G^xUsSNV`¢pQxV ³CF_wCik�CF=Z_CF^R_ _`?�SªC ��CFSTS��?G?l;�\�qROZO@KP^ KT^ 9��F���`: ��^`?�ewCFO�KP^ ADCFEBU®C�­[?G=@\/"�A�=@?[?�=Z?Gf�KP\�^^R?hCi=YUseR?�Q"\�qR^R_RCF=ZV�\iA�U@eR?�EG=ZKPUsKPEhCFS¨OZU@=@KTv #`«�eR\�OZ?g«�KP_`Use�OZeR=@KP^=%�O�U@\®­G?G=Z\�CFO«�?�f�\�qRv&:g³j\�« ?[MF?G=*#(U@\�_wCiU@?�«�?�_`\�^R\FULeRC�MF?�C�vR=@\a\FA>U@ewCiULU@eR?G=Z?�KTO�Ci^�VOZU@=@KTv \FAYU@eR?lA�\�=Zk��j?��²±�¢� 9¥Ye¬�KT^ «�eRKTE�e U@eR?G=Z?�CF=Z?�^R\�­G?[=@\�O*#�A�\�=�Ci^�V¬ �`:

��6��`6 � 673K�&678������� ÄB'�»����� ° ¼��ZÁ����©·��������©·���µ���� ¶�� �]· ¶!�#"¤Á $ %�&%��²¿'�(�)���) ° ��ÁG-0»&*,+�¼-).(�/&&%0��)1��Ä&27¼-%3-3ZÁ-0»&*,+�¼-)1(�/&BÁ ��4&44 Ä�  ��� Ä º�Äi�5(�6&»½¼7(�398:�N·��]µ�;<;�= >@?`·¤¶¸µ,ACB�;�" ¶!�#D�;iÁFE(¾Z»&(�%*G)b¾@27¼-%3-3 �%4 H�I Ä� J��,K Ä � Ä � »B¼-(�L�»&�F(g�,Ä º�ÄNM�¼-)�/[¿�@Á�O:;P�Q;i¶R�-D%STBF" ¶!�#D�;�¶UD�¶Q�©·Y¶Q�©·7D���VWDNX';FB��ZY²·��7>�[#\&¿�-^] »½¼-��F( ° �_27¼-%3-3¤ÁF`ba c ° ¼7(j%b��)��¤¼73d)��NLe2"¼��3�3ZÁ�)0�6gf ° ¼-h�Á �%4 H�4i Ä� IT� º�Ä�ºkj�¼d�NLeO:�W¶!�Q;C= >@"7D�;%l¤·-"¤¶!B��.·mX�D��^n����Q�'� ¶!�Qos·,��D�D�¶R>¨ºg»T�.¿�Äh+U�p��%]�]1)�/��©¾�¤¼rq st[ �%4&uui Á

� ° Ä I Á©À 4�v�wpH Ä� À ��x Ä"#9)1�*]»&���zy{|Y�·��}ST�N·~O:;��@µ%� �bS[·��������Q���@µ%���n·�;gB�;i¶ª·��g·��Q;�·����G·!�G·�Y²·�;F·�;���� yDd�7· Áº ° �p�²»&3�+�Ä�(�Z¼ x Z¼�]1)1��¤¼|E0h�»&(�Ä�[ �%u À u ��w à iCwpH���v�w&u éÁ%)1�t�5·�>7>¤µ��]·�� ¶¸·0�gµ½¶Q�©·��]µ�¶!�.>�"-�©·

��·����[·,ÂT�F(~%(�)���) ° ��Á<\�%j�+���Z¼ZÁ����)1 ���)1/ �%4&u Â�� ° Ä�!(+.+WÄ��� ���r�������������r�������C�����<�� ¢¡��p���£�� ¢¤������¥��¦¨§©��ªC¤����b��ª�«� <���g�C <«Q�F�� ¢�F¬&­

®����_¯5��«1�±°²��«1���F��¬�«.ª¢�^­³®����W¯5��«1�¥´�µFµ<¶ ·C­�¸7¬%�������º¹¼»p½�¾¿CÀÁ�Â�Ã&ÄFÅ�Æ�ÇÉÈUÇ�ÊT½�ÈNÊFÁ�È�À&Ë Ì