ó
    ˆ*£h:Ž  ã                   ó"  • S SK Jr  SSKJr  SSKJrJrJr  \S+S j5       r\S,S j5       r	\S 5       r
\S	 5       r/ S
QrS r\S-S j5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S.S j5       rS rS rS rS r\S 5       r\S.S j5       r\S.S j5       r\S 5       r\S 5       r\S/S j5       r\S0S  j5       r S! r!S" r"\S /S4S# j5       r#\S/S 4S$ j5       r$S% r%S& r&S' r'S( r(\S1S) j5       r)\S* 5       r*g)2é    )Úprint_functioné   )Úxrangeé   )ÚdefunÚdefun_wrappedÚdefun_staticc                 ó:  ^ ^^^	• T R                  T5      mT R                  T5      mTS:  a  T R                  S5      $ [        T S5      (       a  T R                  nO	0 =nT l        TS:X  a3  TS:X  a  T R                  7$ TU;   a  UT   u  pEUT R
                  :¼  a  U7$ Sm	UU U	U4S jnT R
                  n TS:”  a$  ST l        T R                  UST R                  /SS	9m	US
-   [        TS-  5      -   T l        T R                  UST R                  /SS	9nT R                  T5      T-  ST-  -  T R                  T5      TS-   -  TS-   -  -
  SU-  T-  T	-  -   nUT l        TS:X  a'  T R                  T5      (       a  T R
                  U4UT'   U7$ ! UT l        f = f)Nr   z&Stieltjes constants defined for n >= 0Ústieltjes_cacher   c                 óú   >• U T-  nUTR                   -
  TR                  TTR                   U -  -
  5      T-  -  SUS-  -   -  TR                  STR                  -  U -  5      S-
  -  nTR	                  U5      T-  $ ©Nr   r   )ÚjÚlnÚexpÚpiÚ_re)ÚxÚxaÚvÚaÚctxÚmagÚns      €€€€ÚR/home/mande/repo/quber/.venv/lib/python3.13/site-packages/mpmath/functions/zeta.pyÚfÚstieltjes.<locals>.f   sw   ø€ Øˆq‰SˆØ�—‘‰X�s—v‘v˜a §¡ a¡™iÓ(¨!Ñ+Ñ+¨Q¨r°1©u©WÑ5°s·w±w¸qÀÇÁ¹xÈ¹zÓ7JÈ1Ñ7LÑMˆØ�w‰w�q‹z˜CÑÐó    é2   é   é   )Ú	maxdegreeé
   ç      à?r   )ÚconvertÚ
bad_domainÚhasattrr   ÚeulerÚprecÚquadÚinfÚintr   Úisint)
r   r   r   r   r(   Úsr   Úorigr   r   s
   ```      @r   Ú	stieltjesr/      s¢  û€ à�‰�A‹€AØ�‰�A‹€AØˆ1ƒuØ�~‰~ÐFÓGÐGÜˆsÐ%×&Ñ&Ø×-Ñ-‰à02Ð2ˆ˜#Ô-ØˆAƒvØ�‹6Ø—I‘I�:ÐØ�ÓØ% aÑ(‰GˆDØ�s—x‘xÓØ�r�	Ø
€C÷ ð  ð �8‰8€Dð
ð ˆr‹6ØˆCŒHØ—(‘(˜1˜q §¡˜k°Q�(Ð7ˆCØ˜"‘9œs 1 c¡6›{Ñ*ˆŒØ�H‰H�Q˜˜3Ÿ7™7˜¨rˆHÐ2ˆØ�F‰F�1‹I�q‰L˜!˜A™#Ñ §¡¨£¨Q¨q©SÑ!1°1°Q±3Ñ!7Ñ7¸!¸A¹#¸a¹%À¹)ÑCˆàˆŒØˆAƒv�#—)‘)˜A—,‘,Ø!Ÿh™h¨˜]ˆ˜ÑØˆ2€Iøð ˆ�ús   Â6B$F Æ	Fc                 óL  • [        U5      nXR                  :X  d  XR                  :X  a?  US:  a-  XR                  :X  a  US:X  a  U R                  $ U R                  $ U R                  $ US:X  aÎ  U R	                  U5      (       aX  U R                  SSU-  -   5      nU R                  SSU-  -
  5      nU R                  U R                  5      * S-  U-  SXE-
  -  -
  $ U R                  U5      (       a  U$ U R	                  U R                  SSU-  -   5      5      U R                  U R                  5      S-  U-  -
  $ US:”  aâ  SUS-
  -  U R                  US-
  SSU-  -
  5      -  nSUS-
  -  U R                  US-
  SSU-  -   5      -  nU R	                  U5      (       a3  US:X  a&  SU R                  U R                  5      -  SXE-   -  -   $ SXE-   -  $ US:X  a5  U R                  SU R                  U R                  5      -  SXE-   -  -   5      $ U R                  SXE-   -  5      $ g )Nr   r   ç      Ð?y              à?ù       €      à¿r   ç      à¿)r+   r*   ÚninfÚzeroÚ_imÚloggammar   r   ÚisinfÚ	polygammaÚlogr   )r   ÚtÚ
derivativeÚdr   Úbs         r   Úsiegelthetar?   ,   sê  € äˆJ‹€AØ	
�g‰g‹˜Ÿh™h›Øˆq‹5Ø—H‘H‹}  a£Ø—x‘x�Ø—7‘7ˆNà—8‘8ˆOØˆAƒvØ�7‰7�1�:‰:à—‘˜T $ q¡&™[Ó)ˆAØ—‘˜T $ q¡&™[Ó)ˆAØ—F‘F˜3Ÿ6™6“N�? 1Ñ$ QÑ&¨¨q©s©Ñ3Ð3à�y‰y˜�|‰|Ø�Ø—7‘7˜3Ÿ<™<¨¨T°!©V©Ó4Ó5¸¿¹¸s¿v¹v»ÀqÑ8HÈÑ8JÑJÐJØˆ1ƒuØ�a˜‘c‰N˜3Ÿ=™=¨¨1©¨d°4¸±6©kÓ:Ñ:ˆØ�Q�q‘S‰M˜#Ÿ-™-¨¨!©¨T°$°q±&©[Ó9Ñ9ˆØ�7‰7�1�:‰:Ø�A‹vØ˜CŸG™G C§F¡F›OÑ+¨D°!±#©JÑ6Ð6à˜Q™S‘zÐ!à�A‹vØ—w‘w˜t C§G¡G¨C¯F©F£OÑ3°D¸!¹#±JÑ>Ó?Ð?à—w‘w˜t Q¡S™zÓ*Ð*ð r   c           	      óÆ   ^ ^• ST R                   -  T R                  ST R                  ST-  S-   ST R                  -  -  5      -   5      -  nT R	                  U U4S jU5      $ )Nr   r   é   c                 óF   >• TR                  U 5      TR                  T-  -
  $ ©N)r?   r   )r;   r   r   s    €€r   Ú<lambda>Úgrampoint.<locals>.<lambda>S   s   ø€  #§/¡/°!Ó"4°S·V±V¸A±XÒ"=r   )r   r   ÚlambertwÚeÚfindroot)r   r   Úgs   `` r   Ú	grampointrJ   N   sU   ù€ ð 	
ˆ#�&‰&‰�—‘˜˜3Ÿ<™<¨¨1©¨Q©°°3·5±5±Ñ(9Ó:Ñ:Ó;Ñ;€AØ�<‰<Õ=¸qÓAÐAr   c           
      ó"  ^ ^^^^^^^^^^^^• [        UR                  SS5      5      nT R                  U5      nT R                  U5      nT R	                  U5      nT R
                  n [        U5      SU-  :”  aC  US-  U:  a:  T R                  X5      nT R                  U5      (       a  T R                  U5      $ U$ T =R
                  S-  sl        T R                  T R                  U5      5      nT R                  ST R                  U-  -   5      mUS:X  a6  UT-  nUT l        T R                  U5      (       a  T R                  U5      $ U7$ T R                  ST R                  U-  -   SS9mT R                  USS9mUS:X  aI  T R                  U-  TTT-  -   -  nUT l        T R                  U5      (       a  T R                  U5      $ U7$ T R                  ST R                  U-  -   SS9mT R                  USS9mTS-  T R                  T-  -
  mUS:X  aS  UUUUU4S	 jn	T R                  U	S5      nU* U-  nUT l        T R                  U5      (       a  T R                  U5      $ U7$ T =R
                  S
-  sl        T R                  ST R                  U-  -   SS9mT R                  USS9mTS-  ST R                  -  T-  T-  -
  T-
  mUS:X  ab  UUUUUUU4S jn	T R                  U	S5      nT R                  * U-  U-  nUT l        T R                  U5      (       a  T R                  U5      $ U7$ T R                  ST R                  U-  -   SS9mT R                  USS9mU UUUU4S jn	T R                  U	S5      mUS:X  aV  UUU UUUUUUU4
S jn	T R                  U	S5      nX‡-  nUT l        T R                  U5      (       a  T R                  U5      $ U7$ US:”  a  U 4S jn
T R                  X¡US-
  S9$ g ! [         a     GNOf = f)Nr<   r   éô  r   é   r#   r   ©r<   c                  ó    >• ST-  T-  TTT -  /$ ©Nr   © )Úcomb1Útheta1ÚzÚz1Úz2s   €€€€€r   ÚtermsÚsiegelz.<locals>.termsz   s   ø€ Ø�b‘D˜‘K  Q u¡WÐ-Ð-r   r"   r    c                  ó2   >• ST-  T-  ST-  T -  TTT-  -   /$ )Nr    rQ   )rR   Úcomb2rS   rT   rU   rV   Úz3s   €€€€€€€r   rW   rX   ‡   s(   ø€ Ø�v‘X˜b‘[ ! B¡$ u¡*¨b°°5±©jÐ9Ð9r   é   c                  óv   >• TS-  ST R                   -  TS-  -  T-  STS-  -  ST-  T-  T R                   T-  /$ )Nr\   éúÿÿÿr   éýÿÿÿéüÿÿÿ©r   )r   rS   Útheta2Útheta3Útheta4s   €€€€€r   rW   rX   ‘   sL   ø€ Ø˜‘	˜2˜cŸe™e™8 F¨A¡IÑ-¨fÑ4°b¸À¹±lØˆv‰I�fÑ˜cŸe™e F™lð,ð 	,r   c                  ój   >
• STS-  -  T-  STR                   -  T-  T-  ST-  T-  ST-  T -  T	TT-  /$ )Né   r   r^   r\   ra   )
rZ   Úcomb3r   rS   rb   rT   rU   rV   r[   Úz4s
   €€€€€€€€€€r   rW   rX   –   sL   ø€ Ø�v˜q‘y‘[ ‘^ R¨¯©¡X¨b¡[°Ñ%7¸¸6¹À"¹Ø�2‘�e‘˜R  5¡ð*ð *r   c                 ó$   >• TR                  U SS9$ )Nr\   rN   )Úsiegelz)r   r   s    €r   rD   Úsiegelz.<locals>.<lambda>    s   ø€ �c—k‘k !°�kÑ2r   )r   )r+   Úgetr$   r   r6   r(   ÚabsÚrs_zÚ_is_real_typeÚNotImplementedErrorÚexpjr?   Úzetar   Úsum_accuratelyÚdiff)r   r;   Úkwargsr=   Út1Út2r(   r   Úe1rW   ÚhrR   rZ   rg   rS   rb   rc   rd   rT   rU   rV   r[   rh   s   `          @@@@@@@@@@@@r   rj   rj   V   sÝ  ÿü€ äˆF�J‰J�| QÓ'Ó(€AØ�‰�A‹€AØ	�‰�‹€BØ	�‰�‹€BØ�8‰8€DðÜˆr‹7�S˜‘XÓ " a¡%¨"£*Ø—‘˜“ˆAØ× Ñ  ×#Ñ#Ø—w‘w˜q“zÐ!ØˆHð ‡H‚H��N…HØ	�‰�#—/‘/ !Ó$Ó	%€BØ�‰��S—U‘U˜1‘W‘Ó€AØˆAƒvØˆq‰DˆØˆŒØ×Ñ˜Q×ÑØ—7‘7˜1“:ÐØˆrˆ	Ø	�‰�#�c—e‘e˜A‘g‘+¨!ˆÐ	,€BØ�_‰_˜Q¨1ˆ_Ð-€FØˆAƒvØ�U‰U�2‰X�r˜!˜F™(‘{Ñ#ˆØˆŒØ×Ñ˜Q×ÑØ—7‘7˜1“:ÐØˆrˆ	Ø	�‰�#�c—e‘e˜A‘g‘+¨!ˆÐ	,€BØ�_‰_˜Q¨1ˆ_Ð-€FØ�A‰I�c—e‘e˜F‘lÑ"€EØˆAƒv÷	.ñ 	.à×Ñ˜u aÓ(ˆØˆS�‰UˆØˆŒØ×Ñ˜Q×ÑØ—7‘7˜1“:ÐØˆrˆ	Ø‡H‚H��N…HØ	�‰�#�c—e‘e˜A‘g‘+¨!ˆÐ	,€BØ�_‰_˜Q¨1ˆ_Ð-€FØ�A‰I�a˜Ÿ™‘g˜f‘n VÑ+Ñ+¨FÑ2€EØˆAƒv÷	:ó 	:à×Ñ˜u aÓ(ˆØ�e‰eˆV�B‰Y�q‰[ˆØˆŒØ×Ñ˜Q×ÑØ—7‘7˜1“:ÐØˆrˆ	Ø	�‰�#�c—e‘e˜A‘g‘+¨!ˆÐ	,€BØ�_‰_˜Q¨1ˆ_Ð-€F÷,ñ ,ð ×Ñ˜u aÓ(€EØˆAƒv÷	*ö 	*ð ×Ñ˜u aÓ(ˆØ‰TˆØˆŒØ×Ñ˜Q×ÑØ—7‘7˜1“:ÐØˆrˆ	Øˆ1ƒuÜ2ˆØ�x‰x˜  !¡ˆxÐ$Ð$ð øôy ó Úðús   Á)AP  Â<P  Ð 
PÐP)dgŒ­�±úD,@g|¸Èc¤5@g‚÷�Ç9@glŠ®Äl>@gY°w@@gjDËâËB@gp`§•˜uD@g,¥Š‰Ý©E@ge¸¸È¨ H@gÜËPñãH@gæNƒ~3|J@gáC ¥9L@g7”ðk¬M@gï1ý·wjN@g?aóë3GP@g�ûž�ÅP@gMÎá>øbQ@gY`³OLR@g€§íR@gXàGEIS@gÊXe�—ÕS@giµ®CºT@gHõ&Q/U@gÜ©³7ÛU@gùK`zÈ3V@gÂwG{W@g‡¢êž¯©W@gr x¸÷W@gË»;I2µX@gSñ¾«WTY@gts7oîY@g´~Úx•\Z@gEA�†ÊÊZ@gKÐéãÁ[@g|jú÷[@gäZÝ~”\@gàU-î�]@gBí¼/œ²]@ggÁÆ °W^@gü'åÙ˜¼^@gè›·o_@g¾CYá_@gÚ¡«¾„2`@g#œ-XÎb`@gg’ví¯`@g®ñS5Ø`@gñåÓ�¶Ca@g™{�wa@gÎ:iõ£a@gÀñ¨=”ãa@g±l@b@g'B/K‡mb@gŸ;up¶Áb@gHýÙµ›Ýb@gÕ¬JÊ c@gŸ¤óóœƒc@gZUàx³c@gÒ�d3Ûc@gu…þ&d@g¼ªá’û`d@gÇÄ¸«/±d@g¸2«îæåd@g–1E#e@g× Lé.=e@gOÐ¢N+­e@g“ÆBV"Øe@gÄ'Ò: f@g£åx¹Lf@g�¦KÖS}f@g‡�c Æf@gÿ£ûg@goLc<)3g@gstUSgg@gô.õ+Q­g@g6×p^Ú h@g£™Î�"h@g%¢-!~hh@g@T¥#œh@go··j}Àh@g
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        S S & g s  snf )Nr   é   )ÚurllibÚurlopenÚ	readlinesÚfloatÚroundÚ_zeta_zeros)Úurlr|   r=   r   ÚLs        r   Ú_load_zeta_zerosr„   »   sU   € ÛØ�‰�sÓ€AØŸ;™;œ=Ó)š=�aŒˆqŽ™=€AÐ)ä��1‘‹;˜"ÓÐÐØ„K‘�Nùò 	*s   ¨Ac           	      óv  • [        U5      nUS:  a   U R                  U* 5      R                  5       $ US:X  a  [        S5      eU[	        [
        5      :”  a  US::  a  [        U5        U[	        [
        5      :”  a  [        S5      eU R                  SU R                  U R                  [
        US-
     5      5      $ )Nr   zn must be nonzeroi † zn too large for zetazerosr#   r   )r+   ÚzetazeroÚ	conjugateÚ
ValueErrorÚlenr�   r„   rp   ÚmpcrH   rj   )r   r   r‚   s      r   Úoldzetazeror‹   Ã   sœ   € äˆA‹€AØˆ1ƒuØ�|‰|˜Q˜BÓ×)Ñ)Ó+Ð+ØˆAƒvÜÐ,Ó-Ð-ØŒ3Œ{ÓÓ  V£Ü˜ÔØŒ3Œ{ÓÓÜ!Ð"=Ó>Ð>Ø�7‰7�3˜Ÿ™ S§[¡[´+¸aÀ¹cÑ2BÓCÓDÐDr   c           	      ó²  • US:X  a  U R                   $ [        U5      S:”  a[  U R                  U5      nSU R                  U R                  U5      5      -  n[        U5      [        U5      U R                  -  :  a  U$ [        U5      S:  a8  U =R
                  [        U R                  [        U5      S5      * 5      -  sl        U R                  =pEU R                  U5      nSn[        U5      [        U5      U R                  -  :”  aN  XV-  U-  nXEXpR                  US-   5      -  -  -  nUS-  n[        U5      [        U5      U R                  -  :”  a  MN  U$ )Nr   éè  r#   g{®Gáz„?r   r   )r5   rm   ÚliÚsqrtÚepsr(   r+   r:   Úoner   Ú	_zeta_int)r   r   r   r>   r-   r;   ÚuÚks           r   Úriemannrr•   Ð   s  € àˆAƒvØ�x‰xˆä
ˆ1ƒv�ƒ}Ø�F‰F�1‹IˆØ�—‘�s—x‘x “{Ó#Ñ#ˆÜˆq‹6”C˜“F˜3Ÿ7™7‘NÓ"ØˆHÜ
ˆ1ƒv�ƒ}à�Š”C˜Ÿ™¤ Q£¨Ó*Ð*Ó+Ñ+�à�G‰G€O€AØ�‰ˆq‹	€AØ	€AÜ
ˆa‹&”3�q“6˜#Ÿ'™'‘>Ó
!Ø‰E�A‰IˆØ	�!—m‘m A a¡CÓ(Ñ(Ñ)Ñ)ˆØ	ˆQ‰ˆô ˆa‹&”3�q“6˜#Ÿ'™'‘>Õ
!ð €Hr   c                 óZ   • [        U5      nUS:  a  g[        U R                  U5      5      $ )Nr   r   )r+   r‰   Úlist_primes)r   r   s     r   Úprimepir˜   ç   s)   € äˆA‹€AØˆ1ƒuØÜˆs�‰˜qÓ!Ó"Ð"r   c                 óJ  • [        U5      nUS:  a  U R                  R                  $ US:  a*  U R                  R                  U R	                  U5      5      $ U R                  U5      nU R                  USS9U R                  USS9-  S-  U R                  SS9-  nU R                  U R                  R                  U5      U-
  R                  SS9nU R                  U R                  R                  U5      U-   R                  SS9nU R                  R                  XE/5      $ )Nr   ia
  r“   )ÚroundingrA   r=   )r+   Ú_ivr5   Úmpfr˜   rŽ   r�   r   r   Úfloorr   Úceilr>   )r   r   ÚmidÚerrr   r>   s         r   Úprimepi2r¡   ï   sñ   € äˆA‹€AØˆ1ƒuØ�w‰w�|‰|ÐØˆ4ƒxØ�w‰w�{‰{˜3Ÿ;™; q›>Ó*Ð*Ø
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;¸C¿F¹FÈC¸FÐ<PÑ
P€CØ�	‰	�3—7‘7—;‘;˜sÓ# CÑ'×*Ñ*°Sˆ	Ð9€AØ�‰�#—'‘'—+‘+˜cÓ" 3Ñ&×)Ñ)°CˆÐ8€AØ�7‰7�;‰;˜�uÓÐr   c                 ó¢  ^ ^^• T R                  T5      (       a  T$ T R                  T5      S::  a  [        S5      eTS:X  a  T R                  $ TS:X  a&  T R	                  T R
                  T R                  5      $ T R                  T5      nUT R                  :”  a  ST-  $ T R                  [        U5      -   mU UU4S jnT R                  U5      $ )Nr   z.prime zeta function defined only for re(s) > 0r   r#   c               3   óò   >#   • TR                   n Sn US-  nTR                  U5      nU(       d  M   TTl         UTR                  TR                  UT-  5      5      -  U-  nU(       d  g U Tl         Uv •  Md  7f)Nr   r   )r(   Úmoebiusr   rr   )r.   r”   r“   r;   r   r-   Úwps       €€€r   rW   Úprimezeta.<locals>.terms  sw   øé € Ø—8‘8ˆDð ˆAØØ�Q‘�Ø—K‘K “N�ÞÙØ�”Ø�c—f‘f˜SŸX™X a¨¡c›]Ó+Ñ+¨AÑ-�ÞØà�”Ø’ñ ùs   ƒA4A7)
ÚisnanÚrerˆ   r*   rŠ   r4   r   r(   r+   rs   )r   r-   ÚrrW   r¥   s   ``  @r   Ú	primezetarª   ý   s«   ú€ à
‡y�y�‡|�|ØˆØ
‡v�vˆaƒy�Aƒ~ÜÐIÓJÐJØˆAƒvØ�w‰wˆØˆCƒxØ�w‰w�s—x‘x §¡Ó(Ð(Ø�‰ˆq‹	€AØˆ3�8‰8ƒ|Ø�A‰vˆà�X‰Xœ˜A›Ñˆ÷	ð" ×Ñ˜eÓ$Ð$r   c                 óh  ^ ^^• [        T5      mTS:  a  [        S5      eTS:X  d  TS:X  a  TS:”  a  T R                  T5      $ TS:X  a*  T R                  SST-
  5      S-
  T R                  T5      -  $ TS::  aA  TS:X  a  TS-  $ TS:X  a  TS-
  $ TS:X  a  ST-  TS-
  -  S-   S-  $ TS:X  a  TTTS-
  -  S-   -  $ T R	                  T5      (       a  TT-  $ T R                  T5      (       a  T$ [        T5      S:”  a  U UU4S	 jnT R                  U5      TT-  -  $ U UU4S
 jnT R                  U5      $ )Nr   z-Bernoulli polynomials only defined for n >= 0r   r#   r    r   rf   g      ø?c               3   óæ   >#   • TR                   n U v •  TR                   T-  nSnUT::  aE  U TS-   U-
  -  U-  U-  n US:”  a
  US-  (       d  U TR                  U5      -  v •  US-  nUT::  a  MD  g g 7fr   )r‘   Ú	bernoulli)r;   r©   r”   r   r   rT   s      €€€r   rW   Úbernpoly.<locals>.terms6  sy   øé € Ø—‘ˆAØŠGØ—‘˜‘	ˆAØˆAØ�q“&Ø�q˜‘s˜1‘u‘I˜a‘K ‘M�Ø˜A› ! a§%Ø˜CŸM™M¨!Ó,Ñ,Ò,Ø�Q‘�ð	 �q—&ùs   ƒA*A1Á/A1c               3   óð   >#   • TR                  T5      v •  TR                  n SnUT::  aJ  U TS-   U-
  -  U-  T-  n TU-
  nUS:”  a
  US-  (       d  U TR                  U5      -  v •  US-  nUT::  a  MI  g g 7fr   )r­   r‘   )r;   r”   Úmr   r   rT   s      €€€r   rW   r®   B  s   øé € Ø—-‘- Ó"Ò"Ø—‘ˆAØˆAØ�q“&Ø�q˜‘s˜1‘u‘I˜a‘K !‘O�Ø�a‘C�Ø˜A› ! a§%Ø˜CŸM™M¨!Ó,Ñ,Ò,Ø�Q‘�ð �q—&ùs   ƒA/A6Á4A6)r+   rˆ   r­   Úldexpr8   r§   rm   rs   )r   r   rT   rW   s   ``` r   Úbernpolyr²   !  s@  ú€ ô 	ˆA‹€AØˆ1ƒuÜÐHÓIÐIØˆAƒv�!�q“&˜Q ›UØ�}‰}˜QÓÐØˆCƒxØ—	‘	˜!˜A˜a™CÓ  Ñ" C§M¡M°!Ó$4Ñ4Ð4ØˆAƒvØ�‹6˜!˜q™&�=Ø�‹6˜!˜c™'�>Ø�‹6˜1˜Q™3  !¡™9 Q™;¨™/Ð)Ø�‹6˜!˜Q  #¡™Y s™]Ñ+Ð+Ø
‡y�y�‡|�|Ø�A‰vˆØ
‡y�y�‡|�|ØˆÜ
ˆ1ƒv�ƒz÷		ð ×!Ñ! %Ó(¨1¨a©4Ñ/Ð/÷		ð ×!Ñ! %Ó(Ð(r   c                 óø  ^ ^^• [        T5      mTS:  a  [        S5      eTS::  a$  TS:X  a  TS-  $ TS:X  a  TS-
  $ TS:X  a  TTS-
  -  $ T R                  T5      (       a  TT-  $ T R                  T5      (       a  T$ TS-   nTS:X  a3  ST R	                  SU5      S-
  -  T R                  U5      -  U-  TS-  -  $ TS:X  a3  ST R	                  SU5      S-
  -  T R                  U5      -  U-  TS-  -  $ TS:X  af  TS-  (       a  T R                  $ TS:  d(  TT R                  ST-  5      -  T R                  S	-  :  a"  T R	                  T R                  T5      T* 5      $ U UU4S
 jnT R                  U5      U-  $ )Nr   z)Euler polynomials only defined for n >= 0r   r   r#   éþÿÿÿéd   g.eÏT>úÝ?r1   c               3   ó  >#   • TR                   n SnTR                  STS-   5      n TU-
  S-   nUS:”  a
  US-  (       d  SU-
  TR                  U5      -  U -  v •  US-  nUT:”  a  g U T-  TU-
  S-   -  U-  n US-  nMX  7f)Nr   r   r   r#   )r‘   r±   r­   )r;   r”   Úwr   r   r   rT   s       €€€r   rW   Úeulerpoly.<locals>.termsg  sœ   øé € Ø�G‰GˆØˆØ�I‰I�a˜˜!™ÓˆØØ�!‘�A‘ˆAØ˜“E˜a !ŸeØ˜‘s˜CŸM™M¨!Ó,Ñ,¨QÑ.Ò.Ø�‰FˆAØ�1‹uØØ�!‘�Q�q‘S˜‘U‘˜A‘ˆAØ�‰HˆAñ ùs   ƒA=B )r+   rˆ   r8   r§   r±   r­   r5   r   r(   Ú	_eulernumrs   )r   r   rT   r°   rW   s   ```  r   Ú	eulerpolyrº   N  sz  ú€ äˆA‹€AØˆ1ƒuÜÐDÓEÐEØˆAƒvØ�‹6˜!˜q™&�=Ø�‹6˜!˜c™'�>Ø�‹6˜!˜Q˜q™S™'�>Ø
‡y�y�‡|�|Ø�!‰tˆØ
‡y�y�‡|�|ØˆØ	ˆ!‰€AØˆAƒvØ�3—9‘9˜Q˜q“> !Ñ#Ñ$ S§]¡]°1Ó%5Ñ5°aÑ7¸!¸Q¹$Ñ>Ð>ØˆAƒvØ�#—)‘)˜A˜a“. Ñ"Ñ# C§M¡M°!Ó$4Ñ4°QÑ6¸¸A¹Ñ=Ð=ØˆCƒxØˆq�5Ø—8‘8ˆOàˆs‹7�a˜Ÿ™ 
¨1¡Ó-Ñ-°·±¸±Ó=Ø—9‘9˜SŸ]™]¨1Ó-°¨rÓ2Ð2÷ð ×Ñ˜eÓ$ qÑ(Ð(r   Fc                 ó  • [        U5      nU(       a  [        U R                  U5      5      $ US:  a   U R                  U R                  U5      5      $ US-  (       a  U R                  $ U R	                  U R                  US5      U5      $ )Nrµ   r   r#   )r+   r¹   rœ   r5   r±   rº   )r   r   Úexacts      r   Úeulernumr½   v  sm   € äˆA‹€AÞÜ�3—=‘= Ó#Ó$Ð$Øˆ3ƒwØ�w‰w�s—}‘} QÓ'Ó(Ð(Øˆ1‡uØ�x‰xˆØ�9‰9�S—]‘] 1 SÓ)¨1Ó-Ð-r   c                 óŽ   • U R                   7nU R                  nSnUn XeU-  -  nXG-  n[        U5      U:  a   U$ Xb-  nUS-  nM(  )Nr   )r�   r5   rm   )r   r-   rT   ÚtolÚlr”   ÚzkÚterms           r   Úpolylog_seriesrÃ   ‚  sa   € Ø�7‰7ˆ(€CØ�‰€AØ	€AØ	
€BØ
Ø�q‘D‰yˆØ	‰	ˆÜˆt‹9�s‹?Øð €Hð 	‰ˆØ	ˆQ‰ˆñ r   c                 óê  • US:  a  US-  $ SU R                   -  nX1-  * U R                  U5      -  U R                  XR                  U5      U-  5      -  nU R	                  U5      (       a  US:  a  U R                  U5      nU R                  U5      S:  d*  U R                  U5      S:X  aF  U R                  U5      S:¼  a1  XCU R                  U5      US-
  -  -  U R                  US-
  5      -  -  nU$ )Nr   y               @r   )r   Úfacr²   r   ro   r   r6   )r   r   rT   Útwopijr   s        r   Úpolylog_continuationrÇ   �  sÓ   € Øˆ1ƒuØ�‰sˆ
Ø�#—&‘&‰[€FØ	‰ˆ
�3—7‘7˜1“:Ñ §¡¨Q·±°q³	¸&Ñ0@Ó AÑA€AØ
×Ñ˜×Ñ  A£Ø�G‰G�A‹JˆØ
‡w�wˆqƒz�Aƒ~˜#Ÿ'™' !›*¨›/¨c¯g©g°a«j¸A«oØ	�C—F‘F˜1“I  !¡Ñ$Ñ$ S§W¡W¨Q¨q©S£\Ñ1Ñ1ˆØ€Hr   c                 ó¸  • U R                   7nUS:”  aè  U R                  nU R                  U5      nU R                  nSn X-
  S:w  aC  U R	                  X-
  5      U-  U R                  U5      -  nU(       a  [        U5      U:  a  OXH-  nXe-  nUS-  nMV  X@R                  U5      US-
  -  U R                  US-
  5      -  U R                  US-
  5      U R                  U R                  U5      * 5      -
  -  -  nO±US:  a¥  U R                  U* 5      U R                  U5      * US-
  -  -  nU R                  U5      nU R                  n	Sn
 U R                  X¡-
  S-   5      nU(       a2  X¹-  U R                  U
5      X¡-
  S-   -  -  n[        U5      U:  a  OXH-  nX•-  n	U
S-  n
MZ  [        eU R                  U5      (       a  US:  a  U R                  U5      nU$ )Nr   r   )r�   r5   r   r‘   rr   rÅ   rm   Úharmonicr­   rˆ   ro   r   )r   r   rT   r¿   rÀ   ÚlogzÚlogmzr°   rÂ   Úlogkzr”   r>   s               r   Úpolylog_unitcirclerÍ   ›  sÄ  € Ø�7‰7ˆ(€CØˆ1ƒuØ�H‰HˆØ�v‰v�a‹yˆØ—‘ˆØˆØØ‘˜‹zØ—x‘x ¡“} uÑ,¨s¯w©w°q«zÑ9�ÞœC ›I¨›OØØ‘	�Ø‰MˆEØ�‰FˆAñ ð 	
�V‰V�A‹Y˜˜1™Ñ˜cŸg™g a¨¡c›lÑ*¨C¯L©L¸¸1¹Ó,=¸c¿f¹fÀcÇfÁfÈQÃiÀZÓ>PÑ,PÑQÑQ‰Ø	
ˆQ‹Ø�G‰G�Q�B‹K˜#Ÿ&™& ›)˜ q¨¡sÑ+Ñ+ˆØ�v‰v�a‹yˆØ—‘ˆØˆØØ—‘˜a™c !™eÓ$ˆAÞØ‘w §¡¨£
¨A©C°©EÑ 2Ñ3�Ü�t“9˜s“?ØØ‘	�Ø‰MˆEØ�‰FˆAñ ô ÐØ
×Ñ˜×Ñ  A£Ø�G‰G�A‹JˆØ€Hr   c                 óN  • U R                   nU R                  U5      n[        U5      S:  d‘  U R                  nSU-
  nU R                  U* 5      SU R                  -  U-  -  nU R                  U5      XS-  U R                  USU-   5      -  XS* -  U R                  USU-
  5      -  -   -  SU R                  -  U-  -  $ SnSn U R                  X-
  5      U-  n	[        U	5      U R                  :  a  OX9-  nUS-  nXt-  nXx-  nMC  U R                  SU-
  5      U* US-
  -  -  U-   $ )Né   r   r   r#   r   )r5   r   rm   r   r   Úgammarr   r�   )
r   r-   rT   r   r“   r   Úyr;   r”   rÂ   s
             r   Úpolylog_generalrÒ   ¿  s1  € Ø�‰€AØ�‰ˆq‹	€AÜˆq‹6�A‹:Ø�E‰EˆØˆa‰CˆØ�F‰F�A�2‹J˜˜#Ÿ&™&™ ™
Ñ#ˆØ�y‰y˜‹|˜Q™T #§(¡(¨1¨S°©UÓ"3Ñ3°a¸±e¸C¿H¹HÀQÀsÈ1ÁuÓ<MÑ6MÑMÑNÐPQÐRU×RXÑRXÑPXÐ[\É}Ñ\Ð\Ø	€AØ	€AØ
Ø�x‰x˜™‹}˜qÑ ˆÜˆt‹9�s—w‘wÓØØ	‰	ˆØ	ˆQ‰ˆØ	‰ˆØ	‰ˆñ ð �9‰9�Q�q‘S‹>˜A˜2  1¡™+Ñ%¨Ñ)Ð)r   c           	      óì  • U R                  U5      nU R                  U5      nUS:X  a  U R                  U5      $ US:X  a  U R                  U5      * $ US:X  a  USU-
  -  $ US:X  a  U R                  SU-
  5      * $ US:X  a  USU-
  S-  -  $ [	        U5      S::  d%  U R                  U5      (       d  [	        U5      S:  a  [        XU5      $ [	        U5      S:¼  aT  U R                  U5      (       a>  SUS-   -  [        XSU-  5      -  [        U [        U R                  U5      5      U5      -   $ U R                  U5      (       a%  [        U [        U R                  U5      5      U5      $ [        XU5      $ )Nr   éÿÿÿÿr   r   ç      è?gÍÌÌÌÌÌì?gffffffö?)r$   rr   Úaltzetar   rm   r,   rÃ   rÇ   r+   r¨   rÍ   rÒ   )r   r-   rT   s      r   Úpolylogr×   Ó  sK  € à�‰�A‹€AØ�‰�A‹€AØˆAƒvØ�x‰x˜‹{ÐØˆBƒwØ—‘˜A“ˆÐØˆAƒvØ�!�A‘#‰wˆØˆAƒvØ—‘�q˜‘s“ˆ|ÐØˆBƒwØ�!�A‘#˜‘‰zÐÜ
ˆ1ƒv�ƒ~˜cŸi™i¨Ÿl™l¬s°1«v¸«|Ü˜c aÓ(Ð(Ü
ˆ1ƒv�ƒ}˜Ÿ™ 1Ÿ™Ø�a˜‘c‰{œ>¨#°!°A±#Ó6Ñ6Ô9MÈcÔSVÐWZ×W]ÑW]Ð^_ÓW`ÓSaÐcdÓ9eÑeÐeØ
‡y�y�‡|�|Ü! #¤s¨3¯6©6°!«9£~°qÓ9Ð9Ü˜3 1Ó%Ð%r   c                 óª  • U R                  U5      (       a  US:  a  [        U5      S-  S:X  a  US-  $ U(       a  U R                  U5      nOU R                  U5      nU R	                  U5      (       a6  U R	                  U5      (       a   U R                  U R                  X5      5      $ SU-  nSU R                  X5      U R                  X5      -
  -  $ )Nr   r   r   r2   )r,   r+   Úexpjpirq   ro   Úimr×   ©r   r-   rT   r   r   r>   s         r   ÚclsinrÜ   é  s­   € à
‡y�y�‡|�|˜˜A›¤# a£&¨1¡*°£/Ø�‰sˆ
Þ	Ø�J‰J�q‹M‰à�H‰H�Q‹KˆØ
×Ñ˜×Ñ × 1Ñ 1°!× 4Ñ 4Ø�v‰v�c—k‘k !Ó&Ó'Ð'Ø	ˆ!‰€AØ�C—K‘K Ó$ s§{¡{°1Ó'7Ñ7Ñ8Ð8r   c                 óª  • U R                  U5      (       a  US:  a  [        U5      S-  S:X  a  US-  $ U(       a  U R                  U5      nOU R                  U5      nU R	                  U5      (       a6  U R	                  U5      (       a   U R                  U R                  X5      5      $ SU-  nSU R                  X5      U R                  X5      -   -  $ )Nr   r   r   r#   )r,   r+   rÙ   rq   ro   r¨   r×   rÛ   s         r   ÚclcosrÞ   ö  s­   € à
‡y�y�‡|�|˜˜A›¤# a£&¨1¡*°£/Ø�‰sˆ
Þ	Ø�J‰J�q‹M‰à�H‰H�Q‹KˆØ
×Ñ˜×Ñ × 1Ñ 1°!× 4Ñ 4Ø�v‰v�c—k‘k !Ó&Ó'Ð'Ø	ˆ!‰€AØ�—‘˜AÓ  3§;¡;¨qÓ#3Ñ3Ñ4Ð4r   c                 ól   •  U R                   " U40 UD6$ ! [         a    U R                  U5      s $ f = frC   )Ú_altzetarp   Ú_altzeta_generic)r   r-   ru   s      r   rÖ   rÖ     s;   € ð'Ø�|Š|˜AÑ( Ñ(Ð(øÜó 'Ø×#Ñ# AÓ&Ò&ð'ús   ‚ •3²3c                 ó‚   • US:X  a  U R                   SU-  -   $ U R                  SSU-
  5      * U R                  U5      -  $ )Nr   r   r   )Úln2Úpowm1rr   )r   r-   s     r   rá   rá   
  s@   € àˆAƒvØ�w‰w˜˜1™‰}ÐØ�I‰I�a˜˜1™ÓÐ §¡¨£Ñ+Ð+r   Nc                 óÞ  • [        U5      nUS:X  a"  U(       d  U(       d   U R                  " U40 UD6$ U R                  U5      nU R                  nUR                  S5      nUR                  S5      nU(       d-  U(       d&  U R                  S5      U R                  U5      S   -
  $ US:X  aŽ  US:w  aˆ  [        U R                  U5      5      n	[        U R                  U5      5      n
[        U	5      SU-  :”  a  SU
-  U:  a  US	::  d  US
:X  a-    U(       a  [        S5        U R                  " X40 UD6Xpl        $ US:X  a  U R                  $ [        U5      nX°R                  :X  aB  U R                  U5      U R                  :X  a  US:X  a  U R                  $ U R                   $ US-  $ U R#                  U5      (       a  SU-  $ U R                  U5      SU R                  -  :”  a-  US:X  a'  U(       d   U R                  U R%                  SU* 5      -   $ U R&                  " XU40 UD67$ ! [         a     GNf = f! [         a    U(       a  [        S5         Of = f Xpl        GN)! Xpl        f = f)Nr   ÚmethodÚverboser#   r   zeuler-maclaurinrL   r"   r\   zriemann-siegelz4zeta: Attempting to use the Riemann-Siegel algorithmz0zeta: Could not use the Riemann-Siegel algorithmr   )r+   Ú_zetarp   r$   r(   rl   rœ   Ú_convert_paramrm   r6   r   ÚprintÚrs_zetar*   r¨   r‘   r5   r§   ÚpowerÚ_hurwitz)r   r-   r   r<   ræ   ru   r=   r(   rç   rÚ   r¨   Úabsss               r   rr   rr     s  € äˆJ‹€AØˆAƒv–qžFð	Ø—9’9˜QÑ) &Ñ)Ð)ð 	�‰�A‹€AØ�8‰8€DØ�Z‰Z˜Ó!€FØ�j‰j˜Ó#€GÞž
Ø�w‰w�s‹|˜c×0Ñ0°Ó3°AÑ6Ñ6Ð6ØˆAƒv�&Ð-Ó-Ü�—‘˜“‹_ˆÜ�—‘˜“‹_ˆô ˆr‹7�S˜‘XÓ " R¡%¨$£,°:À³?ØÐ&Ó&ð
 ðÞÜÐTÔUØŸ;š; qÑ?¸Ñ?ð  •ØˆAƒvØ�w‰wˆÜˆq‹6€DØ�w‰wƒØ�6‰6�!‹9˜Ÿ™ÓØ�A‹vØ—w‘w�Ø—8‘8ˆOØ�‰sˆ
Ø	�‰�4�‰Ø�‰sˆ
Ø
‡v�vˆaƒy�1�S—X‘X‘:Ó ! q£&¶Ø�w‰w˜Ÿ™ 1 q bÓ)Ñ)Ð)Ø�LŠL˜˜qÑ+ FÑ+Ð+Ð+øô] #ó 	Úð	ûô6 +ó ÞÜÐPÔQÙðúð à–ø˜4•ús5   ¡H( Ä$H9 È(
H6È5H6È9IÉI$ ÉIÉI$ É$I,c                 ó  • U R                   nUR                  S5      n SnU =R                   U-  sl         U R                  U5      u  p(U R                  U5      S:  a&  U(       a  [	        S5         [        XX#U5      XPl         $ U(       a  [	        S5         XW-   U l         [        XX#US-   U5      u  pšU R                  U	5      U R                  Xš-   5      -
  nU(       a%  [	        SU	5        [	        SU
5        [	        S	US
5        X·:  a
  Xš-   XPl         $ [        SU-  [        US-   SU-  5      5      nXtR                  SSU-  5      :”  a  U R                  S5      eMÆ  ! [         a     Of = fU(       d  Mò  [	        S5        Nþ! XPl         f = f)Nrç   r"   r   z#zeta: Attempting reflection formulazzeta: Reflection formula failedz)zeta: Using the Euler-Maclaurin algorithmzTerm 1:zTerm 2:zCancellation:Úbitsr   rÏ   rµ   Úmaxpreczzeta: too much cancellation)r(   rl   ré   r¨   rê   Ú_hurwitz_reflectionrp   Ú_hurwitz_emr   ÚmaxÚminÚNoConvergence)r   r-   r   r=   ru   r(   rç   Ú	extraprecÚatypeÚT1ÚT2Úcancellations               r   rí   rí   F  sq  € à�8‰8€DØ�j‰j˜Ó#€GðØˆ	Ø�Š�IÑ�à×%Ñ% aÓ(‰ˆØ�6‰6�!‹9�q‹=ÞÜÐ;Ô<ðÜ*¨3°1¸Ó?ð, �ö# ÜÐ=Ô>ØØÑ'ˆCŒHÜ  ¨¨t°B©w¸Ó@‰FˆBØŸ7™7 2›;¨¯©°±«Ñ7ˆLÞÜ�i Ô$Ü�i Ô$Ü�o |°VÔ<ØÓ'Ø‘wð �ô	    )¡¬S°ÀÑ1AÀ3ÀtÁ8Ó-LÓM�	ØŸz™z¨)°S¸±XÓ>Ó>Ø×+Ñ+Ð,IÓJÐJñ øô 'ó ÙðúçˆwÜÐ7Õ8øð$ �ús>   ŸAF Á1E ÂBF ÄA
F Å
E*Å'F Å)E*Å*
F Å6F ÆF
c                 óX  ^ ^^^• US:w  a  [         eT R                  U5      nU* nT R                  U5      (       a,  [        U5      nUS::  a  T R	                  SU-
  U5      US-
  -  $ US:X  d  US:X  d  [         eSU-
  mSnSn	UmT R                  T5      S:”  a)  TS-  mUTU-  -  nU	S-  n	T R                  T5      S:”  a  M)  T R                  T5      S::  a)  UTU-  -  nTS-  mU	S-  n	T R                  T5      S::  a  M)   UR
                  u  n
mX©T-  -  n
SU
s=::  a  T::  d   e   eT R                  UU UU4S j[        STS-   5       5       5      nUST R                  T5      -  ST R                  -  T-  T-  -  -  nX‹-  nU$ !   U[        U5      :X  d   e[        U5      n
Sm N™= f)Nr   r   ÚQÚZc              3   óˆ   >#   • U  H7  nTR                  TS -  S U-  T-  -
  5      TR                  TUT45      -  v •  M9     g7f)r   N)Úcospirí   )Ú.0r”   r>   r   Úqr;   s     €€€€r   Ú	<genexpr>Ú&_hurwitz_reflection.<locals>.<genexpr>Ž  sE   øé € ð ÚˆAð —‘˜1˜Q™3˜q ™s 1™u™9Ó% c§l¡l°1°a¸°UÓ&;Ö;Úùs   ƒ?Ar   )
rp   r¨   Úisnpintr+   r²   Ú_mpq_ÚfsumÚrangerÐ   r   )r   r-   r   r=   rø   ÚresÚnegsr   r   ÚshiftÚprI   r>   r  r;   s   `           @@@r   rò   rò   k  s¿  û€ àˆAƒvÜ!Ð!Ø
�&‰&�‹)€CØˆ2€Dà
‡{�{�1‡~�~Ü�‹HˆØ�‹6Ø—<‘<  !¡ QÓ'¨1¨Q©3Ñ/Ð/Ø�S‹L˜E S›LÜ!Ð!Ø	ˆ!‰€Aà	€AØ€EØ	€AØ
�&‰&�‹)�a‹-Ø	ˆQ‰ˆØ	ˆQ�‰W‰ˆØ�‰
ˆð �&‰&�‹)�a�-ð �&‰&�‹)�q‹.Ø	ˆQ�‰W‰ˆØ	ˆQ‰ˆØ�‰
ˆð �&‰&�‹)�q�.ð
Ø�w‰w‰ˆˆ1ð
 ˆq‰�L€AØ��;�Q‹;Ð‰;Ðˆ;Ø�‰÷ Ü�q˜˜1™”óó 	€Aàˆˆ3�9‰9�Q‹<‰˜˜3Ÿ6™6™ !™ a™Ñ	'Ñ'€AØ�F€AØ€HøðØ”C˜“F‹{Ðˆ{Ü�‹FˆØŠús   ÄF Æ F)c           
      ó¼  • U R                  U5      nU* nSnUS-  nUn	Sn
U R                  U5      (       a  [        U R                  U5      5      nUS-
  n U R	                  XU-   X‡-
  S-
  U/5      S   S   nX¬-  n
X‚-   nU R                  U5      nXã-  nU/nSU-  nSU-  nXÑ* -  nU(       a   U R                  US-   X¾-  5      X³S-   -  -  nO
SX½U-  -  -  nUSU-  U-  -  nS/nUnSn[        SU	S-   5       GH  nSU-  nUS:X  a  S/nO
US-
  US-
  /nU H”  n[        UUS-   5      nUU::  a  UR                  US   U-  5        S/US-   -  n[        U5       H  nSU-
  U-
  UU   -  UU'   M     [        SUS-   5       H  nUU==   UUS-
  -
  UUS-
     -  -  ss'   M!     UnUU-  nM–     U R                  UU5      U-  U R                  U5      -  U* -  nUU-  nU R                  U5      U:  a  U
SU-  U-  4s  $ UUS-   US-   -  -  nGM     U(       a  [        SXxSU R                  W5      S	U5        XˆS-  p‡U R                  U5      S:  a  X™S-  -  n	GM  )
Nr   r    r   r#   r   rÔ   z
Sum range:zterm magnitudeÚ	tolerance)r$   r,   r+   r   Ú_zetasumr   Úgammaincr  rõ   Úappendr   Úfdotr­   r   rê   r¨   ) r   r-   r   r=   r(   rç   r¿   ÚM1ÚM2ÚNÚlsumÚs1rÀ   ÚM2aÚlogM2aÚlogM2adÚlogsÚlogrÚrM2aÚM2asÚtailsumÚUr©   Úfactr   Új2Úupdsr°   ÚDÚUnÚir;   s                                    r   ró   ró   ”  sº  € à�‰�A‹€AØˆ%€Cà	
€BØ	�‰€BØ
€AØ€Dà
‡y�y�‡|�|Ü�—‘˜“
‹OˆØ	
ˆ1‰€BØ
à�L‰L˜˜q™D "¡%¨¡'¨A¨3Ó/°Ñ2°1Ñ5ˆð
 	‰	ˆØ‰dˆØ—‘˜“ˆØ‘)ˆØˆyˆØ�‰xˆØ�‰uˆØ�R‰yˆÞØ—l‘l 1 Q¡3¨©	Ó2°R¸A¹#±YÑ>‰Gà˜" R™i™Ñ(ˆGØ�3˜‘= 4Ñ'Ñ'ˆØˆCˆØˆØˆÜ�q˜!˜A™#—ˆAà�1‘ˆBØ�A‹vØ�s‘à˜1™˜b ™d�|�Û�Ü˜˜!˜A™#“J�Ø˜“6Ø—K‘K  R¡¨4¡Ô0Ø�S˜!˜A™#‘Y�Ü ž�A¨Q¨q©S°©U°A°a±D©L B q£E™Ü  ! A¡#ž�A¨¨1«°!°Q°q±S±'¸1¸Q¸q¹S¹6Ñ1AÑ(A­™Ø�Ø�T‘	’ñ ð —‘˜˜DÓ! AÑ%¨¯©°bÓ(9Ñ9¸D¸5ÑAˆAØ�q‰LˆGØ�w‰w�q‹z˜CÓØ˜b 1™W wÑ.Ð.Ò.Ø�R˜‘T˜B˜q™D‘MÑ!‹Dñ) ö* Ü�, Ð(8¸#¿'¹'À!»*ÀkÐSVÔWØ˜‘TˆBØ�6‰6�!‹9�q‹=Ø�A‘‰IˆAòa r   c                 ó¾  ^ ^^
^• [        T R                  U5      5      ST R                  -  :  a   T R                  UTX4U5      $ T R                  USS9mUS/:g  n[        U5      S:H  nU(       ds  U(       d+  T R                  UU4S j[        US-   5       5       5      // 4$ U(       a:  US   m
T R                  UU U
U4S j[        US-   5       5       5      nST
-  U-  // 4$ [        U5      n	U(       d  [        U	S-   5      nU V
s/ s H  n
T R                  PM     nn
U(       a  U V
s/ s H  n
T R                  PM     nn
O/ n[        US-   5       Hê  nTU-   nUT-  nU(       a   T R                  T R                  Xï-  -  5      nU(       aŒ  T R                  U5      * nU(       a0  UU	-  nUS==   UU-  -  ss'   U(       a  US==   WU-  -  ss'   M‚  M„  T R                  nU H/  m
UT
==   UU-  -  ss'   U(       a  UT
==   WU-  -  ss'   UU-  nM1     MÇ  US==   U-  ss'   U(       d  MÝ  US==   W-  ss'   Mì     X¼4$ ! [         a     GNf = fs  sn
f s  sn
f )	a‹  
Returns [xd0,xd1,...,xdr], [yd0,yd1,...ydr] where

xdk = D^k     ( 1/a^s     +  1/(a+1)^s      +  ...  +  1/(a+n)^s     )
ydk = D^k conj( 1/a^(1-s) +  1/(a+1)^(1-s)  +  ...  +  1/(a+n)^(1-s) )

D^k = kth derivative with respect to s, k ranges over the given list of
derivatives (which should consist of either a single element
or a range 0,1,...r). If reflect=False, the ydks are not computed.
r#   T©r¼   r   r   c              3   ó4   >#   • U  H  nTU-   T-  v •  M     g 7frC   rQ   )r  r”   r   r
  s     €€r   r  Ú_zetasum.<locals>.<genexpr>í  s   øé € Ð>²+¨Q˜a ™c Dž[²+ùs   ƒc              3   ód   >#   • U  H%  nTR                  TU-   5      T-  TU-   T-  -  v •  M'     g 7frC   )r   )r  r”   r   r   r=   r
  s     €€€€r   r  r*  ð  s.   øé € ÐKº{¸!˜Ÿ™  !¡› a™¨1¨Q©3°©+Ö5º{ùs   ƒ-0rÔ   )rm   r¨   r(   Ú_zetasum_fastrp   Úfnegr‰   r  r   rô   r  r5   Úconjr‘   r   )r   r-   r   r   ÚderivativesÚreflectÚhave_derivativesÚhave_one_derivativer   Úmaxdr=   ÚxsÚysr”   r·   ÚxtermÚytermÚlogwr;   r
  s   ` `       `        @r   r  r  Õ  s?  û€ ô ˆ3�6‰6�!‹9ƒ~˜˜cŸh™h™Ó&ð	Ø×$Ñ$ Q¨¨1¸7ÓCÐCð �8‰8�A˜Tˆ8Ð"€DØ" q cÑ)ÐÜ˜kÓ*¨aÑ/ÐÞÞØ—H‘HÕ>´&¸¸1¹´+Ó>Ó>Ð?ÀÐCÐCÞØ˜A‘ˆAØ—‘×K¼vÀaÈÁc¼{ÓKÓKˆAØ˜!‘G˜a‘K�= "Ð$Ð$Üˆ{Ó€DÞÜ˜D ™F“mˆÙ'Ó	(šK�qˆ#�(Œ(™K€BÐ	(ÞÙ +Ó,¢˜1ˆc�hŒh¡ˆÐ,ˆàˆÜ�A�a‘CŽ[ˆØ�‰EˆØ�T‘	ˆÞØ—H‘H˜SŸW™W¨©	Ñ2Ó3ˆEÞØ—F‘F˜1“I�:ˆDÞ"Ø˜t‘|�Ø�1“˜ ™Ñ%“ÞØ�q“E˜U T™\Ñ)•Eñ ð —G‘G�Û$�AØ�q“E˜U Q™YÑ&“EÞØ˜1› ¨¡Ñ*›Ø˜‘I’Aó	 %ð ˆq‹E�U‰N‹EßˆwØ�1“˜‘•ñ- ð. ˆ6€MøôW #ó 	Úð	üò 
)ùâ,s   ±I ÄIÄ*IÉ
IÉIc           	      óò  • U R                  U5      n[        U5      n[        U5      nUS:”  a  [        S5      eU R                  n U =R                  S-  sl        US:X  ad  SnU HB  nU(       d  M  US:w  d  M  SnU R
                  7n	U =R                  SUS-   -  -  sl        X-  nMD     U(       a  U R                  7X`l        $ U R                  n
[        SUS-   5       Hz  nX+U-     (       d  M  US:X  aF  X¢X´-     U R                  XU4S5      U R                  XU45      U R                  U5      -  -
  -  -  n
M]  X¢X´-     U R                  XU45      -  -  n
M|     X¤U-  -  n
X`l        U
7$ ! X`l        f = f)Nr   zarbitrary order derivativesr"   r   TF)r$   r‰   r+   rp   r(   r�   r*   r5   r  rr   r:   )r   r-   Úchir<   r  r=   r(   Ú	have_poler   ry   rT   r  s               r   Ú	dirichletr<    so  € à�‰�A‹€AÜˆC‹€AÜˆJ‹€AØˆ1ƒuÜ!Ð"?Ó@Ð@Ø�8‰8€DðØ�Š�B‰�Ø�‹6ØˆIÛ�ß�1˜˜a�Ø %�IØŸ™˜�AØ—H’H  1 Q¡3¡Ñ'•HØ‘F’Añ ö ØŸ™�xð �ð �H‰HˆÜ�q˜˜1™–ˆAØ�Q‘3�x‰xØ˜“6Ø˜Q™S™ S§X¡X¨a°A°¸Ó%:ØŸ™  q EÓ*¨3¯7©7°1«:Ñ5ñ&6ñ 7ñ 7’Að ˜Q™S™ C§H¡H¨Q°1°Ó$6Ñ6Ñ6’Añ ð 	
�‰T‰	ˆàŒØˆ2€Iøð �ús&   Á'E. Á1E. Á9AE. Ã(E. Ã1A4E. Å.E6c                 ón  ^ ^^^• T R                   nUU UU4S jnT R                  =pgT R                  nSn	X„:”  aC  Xg-  nU	S-  n	T R                  T R	                  U	5      5      mU" U	5      n[        U5      nX„:”  a  MC  Sn
UR                  S5      (       a�  T R                  T5      nST R                  S-  -  [        SU5      -  TUS-
  -  -  T R                  TST R                  -  -  5      -  T R                  STTS-  -  5      -  [        T R                  TS-  5      5      -  n
[        U
5      n
U7X©4$ )	Nc                 óF   >• TR                  ST-  TTS-  -  SS9TT* -  -  $ )Nr#   r   T)Úregularized)r  )r   r   r   Úgammr-   s    €€€€r   rD   Ú&secondzeta_main_term.<locals>.<lambda>7  s+   ø€ �#—,‘,˜s 1™u a¨¨a©¡i¸T�,ÐBÀ4È1È"Á:ÒMr   r   r   Úerrorr#   rÔ   r   r3   )r�   r5   r*   rÚ   Úzetazero_memoizedrm   rl   r¨   r   rô   r:   r  rÐ   )r   r-   r   ru   r¿   r   ÚtotsumrÂ   Úmgr   r    Úsgr@  s   ```         @r   Úsecondzeta_main_termrG  5  s"  û€ Ø
�'‰'€CßM€AØ—H‘HÐ€FØ	�‰€BØ	€AØ
‹(Ø‰ˆØ	ˆQ‰ˆØ�v‰v�c×+Ñ+¨AÓ.Ó/ˆÙ�‹tˆÜ�‹Yˆð �(ð €CØ‡z�z�'×ÑØ�V‰V�A‹YˆØ�#—&‘&˜2‘,Ñœs 1 R›yÑ(¨¨R°©V©Ñ4°S·W±W¸TÀ1ÀSÇVÁVÁ8¹_Ó5MÑMØ�\‰\˜$  $¨¡'¡	Ó*ñ+Ü+.¨s¯y©y¸¸1¹«~Ó+>ñ?ˆä�#‹hˆØˆ7�Cˆ?Ðr   c                 ó4  ^ ^^• T R                   nUU U4S jnT R                  =pgT R                  nSn	X„:”  d  U	S:  a>  Xg-  nU	S-  n	U" U	5      nUS:X  a  T R                  nO[        U5      nX„:”  a  M6  U	S:  a  M>  UR	                  S5      (       a  Un
U7W
U	4$ )Nc                 óP  >• TR                  SST-
  -  STR                  U 5      S-  -  TS-  -  5      STR                  U 5      -  TS-
  -  -  TR                  U 5      -  TR                  U 5      -  STR	                  ST-  5      -  TR                  TR
                  5      -  -  $ )Nr#   r   r1   r   rÔ   )r  r:   Úmangoldtr�   rÐ   r   ©r   r   r   r-   s    €€€r   rD   Ú'secondzeta_prime_term.<locals>.<lambda>K  s›   ø€ �#—,‘,˜s A a¡C™y¨¨c¯g©g°a«j¸!©mÑ);¸aÀ"¹gÑ)EÓFØ
ˆc�g‰g�a‹j‰.˜A˜a™CÑ	 ñ"Ø"%§,¡,¨q£/ñ2Ø25·(±(¸1³+ñ>à	
ˆ3�9‰9�S˜‘UÓÑ	˜CŸH™H S§V¡VÓ,Ñ	,ò.r   r   é	   r   rB  )r�   r5   r*   rm   rl   )r   r-   r   ru   r¿   r   rD  rÂ   rE  r   r    s   ```        r   Úsecondzeta_prime_termrN  I  s�   ú€ Ø
�'‰'€Cö	.€Að —H‘HÐ€FØ	�‰€BØ	€AØ
‹(�a˜!“eØ‰ˆØ	ˆQ‰ˆÙ�‹tˆØ�1‹9Ø—‘‰Bä�T“ˆBð �(�a˜!•eð ‡z�z�'×ÑØˆØˆ7�C˜ˆ?Ðr   c                 óè  ^ ^^• T R                  T5      (       aZ  T R                  T5      S::  aE  [        [        T R                  T5      5      5      nUS-  (       d  T R	                  S5      U* S-  -  $ T R
                  nUU U4S jnT R                  nU" S5      nT R                  nSn	X„:”  a#  Xg-  nU	S-  n	U" U	5      n[        U5      nX„:”  a  M#  TST-  -  U-  T R                  ST-  5      -  n
U
$ )Nr   r   z-0.25r   c                 óJ   >• ST-  U -  U ST-  -   TR                  U 5      -  -  $ )Nr1   r#   )rÅ   rK  s    €€€r   rD   Ú%secondzeta_exp_term.<locals>.<lambda>c  s'   ø€ �4˜‘6˜A‘+  # a¡%¡¨¯©°«Ñ3Ò4r   r#   )
r,   r¨   r+   r€   rœ   r�   r5   r*   rm   rÐ   )r   r-   r   r°   r¿   r   rD  rÂ   rE  r   r   s   ```        r   Úsecondzeta_exp_termrR  ]  sÝ   ú€ Ø
‡y�y�‡|�|˜Ÿ™˜q›	 Q›Ü”�c—f‘f˜Q“iÓ Ó!ˆØ�1�uØ—7‘7˜7Ó# q b¨!¡eÑ,Ð,Ø
�'‰'€CÞ4€AØ�X‰X€FÙˆQ‹4€DØ	�‰€BØ	€AØ
‹(Ø‰ˆØ	ˆQ‰ˆÙ�‹tˆÜ�‹Yˆð	 �(ð
 	
ˆC�‰E‰
�6Ñ˜#Ÿ)™) C¨¡EÓ*Ñ*€AØ€Hr   c           	      óœ  ^ ^^• TSTS-
  -  -  ST R                  T R                  5      -  T R                  ST-  5      -  -  nT R                  U5      nT =R                  U-  sl        TSTS-
  -  -  ST R                  T R                  5      -  T R                  ST-  5      -  -  nT R
                  nUU U4S jnT R                  nT R                  n	Sn
U" U
5      n[        U5      nXÆ:”  a@  XÉ::  a;  X‹-  nU
S-  n
U" U
5      nX‹-  nU
S-  n
U" U
5      nUn	[        U5      nXÆ:”  a  XÉ::  a  M;  X‹-  nSTS-
  S-  -  T R                  T R                  ST R                  S-  -  T-  5      -   TS-
  S-  -  -   nXMU-   -  nS	nUR                  S
5      (       a�  XÆ:”  a  XÉ::  dz  XÆ::  a8  T R                  S5      [        T R                  [        XF-  5      S5      5      -  nXÉ:”  a8  T R                  S5      [        T R                  [        XI-  5      S5      5      -  n[        UT R
                  S-  5      nT =R                  U-  sl        U7U4$ )Nr#   r   r\   c                 ó¸   >• TR                  U S5      STR                  T5      -  U -  -  TR                  SU -  5      -  TU -   S-
  TR                  U 5      -  -  $ )NrÕ   r\   r#   r   )r²   r�   rÐ   rÅ   rK  s    €€€r   rD   Ú*secondzeta_singular_term.<locals>.<lambda>v  sX   ø€ �#—,‘,˜q Ó&¨¨#¯(©(°1«+©¸Ñ'9Ñ9Ø
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        f = f)a—  
Evaluates the secondary zeta function `Z(s)`, defined for
`\mathrm{Re}(s)>1` by

.. math ::

    Z(s) = \sum_{n=1}^{\infty} \frac{1}{\tau_n^s}

where `\frac12+i\tau_n` runs through the zeros of `\zeta(s)` with
imaginary part positive.

`Z(s)` extends to a meromorphic function on `\mathbb{C}`  with a
double pole at `s=1` and  simple poles at the points `-2n` for
`n=0`,  1, 2, ...

**Examples**

    >>> from mpmath import *
    >>> mp.pretty = True; mp.dps = 15
    >>> secondzeta(2)
    0.023104993115419
    >>> xi = lambda s: 0.5*s*(s-1)*pi**(-0.5*s)*gamma(0.5*s)*zeta(s)
    >>> Xi = lambda t: xi(0.5+t*j)
    >>> chop(-0.5*diff(Xi,0,n=2)/Xi(0))
    0.023104993115419

We may ask for an approximate error value::

    >>> secondzeta(0.5+100j, error=True)
    ((-0.216272011276718 - 0.844952708937228j), 2.22044604925031e-16)

The function has poles at the negative odd integers,
and dyadic rational values at the negative even integers::

    >>> mp.dps = 30
    >>> secondzeta(-8)
    -0.67236328125
    >>> secondzeta(-7)
    +inf

**Implementation notes**

The function is computed as sum of four terms `Z(s)=A(s)-P(s)+E(s)-S(s)`
respectively main, prime, exponential and singular terms.
The main term `A(s)` is computed from the zeros of zeta.
The prime term depends on the von Mangoldt function.
The singular term is responsible for the poles of the function.

The four terms depends on a small parameter `a`. We may change the
value of `a`. Theoretically this has no effect on the sum of the four
terms, but in practice may be important.

A smaller value of the parameter `a` makes `A(s)` depend on
a smaller number of zeros of zeta, but `P(s)`  uses more values of
von Mangoldt function.

We may also add a verbose option to obtain data about the
values of the four terms.

    >>> mp.dps = 10
    >>> secondzeta(0.5 + 40j, error=True, verbose=True)
    main term = (-30190318549.138656312556 - 13964804384.624622876523j)
        computed using 19 zeros of zeta
    prime term = (132717176.89212754625045 + 188980555.17563978290601j)
        computed using 9 values of the von Mangoldt function
    exponential term = (542447428666.07179812536 + 362434922978.80192435203j)
    singular term = (512124392939.98154322355 + 348281138038.65531023921j)
    ((0.059471043 + 0.3463514534j), 1.455191523e-11)

    >>> secondzeta(0.5 + 40j, a=0.04, error=True, verbose=True)
    main term = (-151962888.19606243907725 - 217930683.90210294051982j)
        computed using 9 zeros of zeta
    prime term = (2476659342.3038722372461 + 28711581821.921627163136j)
        computed using 37 values of the von Mangoldt function
    exponential term = (178506047114.7838188264 + 819674143244.45677330576j)
    singular term = (175877424884.22441310708 + 790744630738.28669174871j)
    ((0.059471043 + 0.3463514534j), 1.455191523e-11)

Notice the great cancellation between the four terms. Changing `a`, the
four terms are very different numbers but the cancellation gives
the good value of Z(s).

**References**

A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier,
53, (2003) 665--699.

A. Voros, Zeta functions over Zeros of Zeta Functions, Lecture Notes
of the Unione Matematica Italiana, Springer, 2009.
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ST R                  -  mUU U	U
UU4S jnUST R                  UST R                  /5      -  -  nT R!                  U5      (       dR  T R!                  T5      (       d<  T R!                  T5      (       d&  T R                  U5      S:  a  T R#                  U5      nU$ )a®	  
Gives the Lerch transcendent, defined for `|z| < 1` and
`\Re{a} > 0` by

.. math ::

    \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(a+k)^s}

and generally by the recurrence `\Phi(z,s,a) = z \Phi(z,s,a+1) + a^{-s}`
along with the integral representation valid for `\Re{a} > 0`

.. math ::

    \Phi(z,s,a) = \frac{1}{2 a^s} +
            \int_0^{\infty} \frac{z^t}{(a+t)^s} dt -
            2 \int_0^{\infty} \frac{\sin(t \log z - s
                \operatorname{arctan}(t/a)}{(a^2 + t^2)^{s/2}
                (e^{2 \pi t}-1)} dt.

The Lerch transcendent generalizes the Hurwitz zeta function :func:`zeta`
(`z = 1`) and the polylogarithm :func:`polylog` (`a = 1`).

**Examples**

Several evaluations in terms of simpler functions::

    >>> from mpmath import *
    >>> mp.dps = 25; mp.pretty = True
    >>> lerchphi(-1,2,0.5); 4*catalan
    3.663862376708876060218414
    3.663862376708876060218414
    >>> diff(lerchphi, (-1,-2,1), (0,1,0)); 7*zeta(3)/(4*pi**2)
    0.2131391994087528954617607
    0.2131391994087528954617607
    >>> lerchphi(-4,1,1); log(5)/4
    0.4023594781085250936501898
    0.4023594781085250936501898
    >>> lerchphi(-3+2j,1,0.5); 2*atanh(sqrt(-3+2j))/sqrt(-3+2j)
    (1.142423447120257137774002 + 0.2118232380980201350495795j)
    (1.142423447120257137774002 + 0.2118232380980201350495795j)

Evaluation works for complex arguments and `|z| \ge 1`::

    >>> lerchphi(1+2j, 3-j, 4+2j)
    (0.002025009957009908600539469 + 0.003327897536813558807438089j)
    >>> lerchphi(-2,2,-2.5)
    -12.28676272353094275265944
    >>> lerchphi(10,10,10)
    (-4.462130727102185701817349e-11 - 1.575172198981096218823481e-12j)
    >>> lerchphi(10,10,-10.5)
    (112658784011940.5605789002 - 498113185.5756221777743631j)

Some degenerate cases::

    >>> lerchphi(0,1,2)
    0.5
    >>> lerchphi(0,1,-2)
    -0.5

Reduction to simpler functions::

    >>> lerchphi(1, 4.25+1j, 1)
    (1.044674457556746668033975 - 0.04674508654012658932271226j)
    >>> zeta(4.25+1j)
    (1.044674457556746668033975 - 0.04674508654012658932271226j)
    >>> lerchphi(1 - 0.5**10, 4.25+1j, 1)
    (1.044629338021507546737197 - 0.04667768813963388181708101j)
    >>> lerchphi(3, 4, 1)
    (1.249503297023366545192592 - 0.2314252413375664776474462j)
    >>> polylog(4, 3) / 3
    (1.249503297023366545192592 - 0.2314252413375664776474462j)
    >>> lerchphi(3, 4, 1 - 0.5**10)
    (1.253978063946663945672674 - 0.2316736622836535468765376j)

**References**

1. [DLMF]_ section 25.14

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