� I�gh����dZddlZGd�de��ZddlmZd�Zd�Zd �Zd �Z dd �Z dgfd �Z dd �Z dd�Z dd�Zdd�Zedd���Zedd���ZdS)aI --------------------------------------------------------------------- .. sectionauthor:: Juan Arias de Reyna <arias@us.es> This module implements zeta-related functions using the Riemann-Siegel expansion: zeta_offline(s,k=0) * coef(J, eps): Need in the computation of Rzeta(s,k) * Rzeta_simul(s, der=0) computes Rzeta^(k)(s) and Rzeta^(k)(1-s) simultaneously for 0 <= k <= der. Used by zeta_offline and z_offline * Rzeta_set(s, derivatives) computes Rzeta^(k)(s) for given derivatives, used by z_half(t,k) and zeta_half * z_offline(w,k): Z(w) and its derivatives of order k <= 4 * z_half(t,k): Z(t) (Riemann Siegel function) and its derivatives of order k <= 4 * zeta_offline(s): zeta(s) and its derivatives of order k<= 4 * zeta_half(1/2+it,k): zeta(s) and its derivatives of order k<= 4 * rs_zeta(s,k=0) Computes zeta^(k)(s) Unifies zeta_half and zeta_offline * rs_z(w,k=0) Computes Z^(k)(w) Unifies z_offline and z_half ---------------------------------------------------------------------- This program uses Riemann-Siegel expansion even to compute zeta(s) on points s = sigma + i t with sigma arbitrary not necessarily equal to 1/2. It is founded on a new deduction of the formula, with rigorous and sharp bounds for the terms and rest of this expansion. More information on the papers: J. Arias de Reyna, High Precision Computation of Riemann's Zeta Function by the Riemann-Siegel Formula I, II We refer to them as I, II. In them we shall find detailed explanation of all the procedure. The program uses Riemann-Siegel expansion. This is useful when t is big, ( say t > 10000 ). The precision is limited, roughly it can compute zeta(sigma+it) with an error less than exp(-c t) for some constant c depending on sigma. The program gives an error when the Riemann-Siegel formula can not compute to the wanted precision. �Nc��eZdZd�ZdS)�RSCachec��ddiig|_dS)Nr� )� _rs_cache)�ctxs �g/home/asafur/pinokio/api/open-webui.git/app/env/lib/python3.11/site-packages/mpmath/functions/rszeta.py�__init__zRSCache.__init__6s���B��B��� � � �N)�__name__� __module__� __qualname__r �r r rr5s#������(�(�(�(�(r r�)�defunc�&�|dz}|dz }t|�d|dzz��d|zdz|�|��z ��}|�d|z��}t|�d|z��|�|��dz|z��}||_i}|j|d<|j|d <t dd|zd z��D]} || d z |jz|| <�|dz|_i} i} t d|d z��D]_} d | z|�d| z��z} |�| ��|�d| z��z } | |d| zz| | <�`t dd|zd z��D]5} |j|�| ��z } | d| zz } | || z| | <�6d |_d |�|��z }i}t d|d z��D]�} d |_t|�d| dzz��d| z|z��}||_d}t d| d z��D]%}|d |z| |z| d| zd|zz zz }�&d | d zz|j z|z|| <��i}t d|d z��D]�} d |_t|�d| dzz��d| z|z��}||_d}t d| d z��D]'}||j | |z z| |z| | |z zz }�(||| <��d |_d|�|��z }td d|z|z��}||_|� |�d����dz }|� d��dz }i}t d|��D]C} d |_td d| z|z��}||_||| z||| zz|d| z<�Dt d d|zd��D]} d|| <�||||gS)a� Computes the coefficients `c_n` for `0\le n\le 2J` with error less than eps **Definition** The coefficients c_n are defined by .. math :: \begin{equation} F(z)=\frac{e^{\pi i \bigl(\frac{z^2}{2}+\frac38\bigr)}-i\sqrt{2}\cos\frac{\pi}{2}z}{2\cos\pi z}=\sum_{n=0}^\infty c_{2n} z^{2n} \end{equation} they are computed applying the relation .. math :: \begin{multline} c_{2n}=-\frac{i}{\sqrt{2}}\Bigl(\frac{\pi}{2}\Bigr)^{2n} \sum_{k=0}^n\frac{(-1)^k}{(2k)!} 2^{2n-2k}\frac{(-1)^{n-k}E_{2n-2k}}{(2n-2k)!}+\\ +e^{3\pi i/8}\sum_{j=0}^n(-1)^j\frac{ E_{2j}}{(2j)!}\frac{i^{n-j}\pi^{n+j}}{(n-j)!2^{n-j+1}}. \end{multline} ��@r����(rr������� ��2g�?) �max�mag� _eulernum�prec�one�pi�range�mpf�fac�j�sqrt�expjpi)r�J�eps�newJ�neweps6�wpvw�E�wppi�pipower�n�v�w�va�wa�wpp1a�P1�wpp1�sump�k�P2�wpp2�wpc0�wpc�mu�nu�cs r �_coefrCNs���: �Q�3�D��"�f�G� �s�w�w�r�4��6�{�#�#�Q�t�V�A�X�c�g�g�g�.>�.>�%>� ?� ?�D� � � �a��f���A�" �s�w�w�r�$�w���������q��$�!6� 7� 7�D��C�H��G���G�A�J���G�A�J� �1�Q�t�V�A�X� � �)�)���Q�q�S�\�#�&�(��� � ��A�v�C�H��A��A� �1�T�!�V�_�_�����1�W�s�}�}�Q�q�S�)�)� )�� �W�W�R�[�[�����1���� %�� ���!�� �_��!��� �1�Q�t�V�A�X� � ���� �W�S�W�W�Q�Z�Z� �� �q�!�t�9�� ��� �]��!����C�H� ����� � � �E� �B� �1�T�!�V�_�_�'�'������3�7�7�2�q��s�8�$�$�Q�q�S��Y�/�/��������q��1���� 0� 0�A� �b�1�W��!��$�Q�q��s�1�Q�3�w�Z�/� /�D�D��a��c�{�C�E�!�$�&��1��� �B� �1�T�!�V�_�_��������3�7�7�2�q��s�8�$�$�Q�q�S��Y�/�/��������q��1���� 1� 1�A� �S�U�Q�q�S�\�Q�q�T�)�!�A�a�C�&�0� 0�D�D���1����C�H� �s�w�w�w��� �D� �a��$��t� � � �C��C�H� ���#�'�'�#�,�,� � �� !�B� ���D� � �!� �B��A� �1�T�]�]�#�#������!�A�a�C��H�o�o������B�q�E��"�R��U�(�"��!�A�#��� �1�Q�t�V�A� � ������!��� �'�1�g� &�&r c����j}||dkr||dkr|d|dfS�jj} t�j||��}|�j_n#|�j_wxYw��jurlt �fd�|d���D����|d<t �fd�|d���D����|d<|�jdd�<�jd�jdfS)Nrrrrc3�L�K�|]\}}|��|��fV��dS�N��convert��.0r;r3rs �r � <genexpr>zcoef.<locals>.<genexpr>��6�����G�G�e�q���#�+�+�a�.�.�)�G�G�G�G�G�Gr c3�L�K�|]\}}|��|��fV��dSrFrGrIs �r rKzcoef.<locals>.<genexpr>�rLr )r�_mpr!rC�dict�items)rr*r+�_cache�orig�datas` r �coefrT�s��� �]�F��F�1�I�~�~�#����*�*��a�y�&��)�#�#� �7�<�D���S�W�a��%�%����� � ��t��� ����� �#�'����G�G�G�G�t�A�w�}�}���G�G�G�G�G��Q���G�G�G�G�t�A�w�}�}���G�G�G�G�G��Q���C�M�!�!�!�� �=�� �S�]�1�-� -�-s �A!�!A/c���d|z|z }|�|��}d|�|j��z|�|��z|�|��zdz |�d|jz��dz z }d|zdz }|�|dz��|�|dz dz��z dz |�|dzdz ��z } |||z| zkrp|dkrj|dz }|�|dz��|�|dz dz��z dz |�|dzdz ��z } |||z| zkr|dk�j|} | S)Ngb��k5�_@rrr�@r)�lnr#�loggamma) r�xA�xeps4�a�xB1�xL�aux1�aux2�m�aux3�xMs r �aux_M_Fprc�sh�� �r�>�%� �D� �6�6�$�<�<�D� �c�f�f�S�V�n�n� �S�V�V�C�[�[� (������� 2�A� 5�s�v�v�a���h�7G�7G��7I� I�D� �"��Q��A� �<�<��!�� � �S�\�\�!�C�%��'�2�2� 2�A� 5�s�|�|�Q�q�S�"�H�7M�7M� M�D� �!�D�&�$�,� � �A�a�C�C� ��E��� � �Q�q�S�!�!�#�,�,�q��u�Q�w�"7�"7�7��:�C�L�L�!�A�#�r��<R�<R�R�� �!�D�&�$�,� � �A�a�C�C� �B� �Ir c��|d|zz }||zdz}||�||dzz |dz dz ��z|z }t||��}|S)N�xgJ �R�_@rrr)�power�min) rrYrZr[r\rb�h1�h2�h3s r � aux_J_neededrk�s_�� ��B���B� �Q��&� &�B� �c�i�i��B��E��B�q�D�!�8�,�,� ,�r� 1�B� �R����B� �Ir c �"=���j}��|��}��|��}d|z }d�_��|d�jzz ��}||z}||z} ��d��|��dz ��} ��d��| ��dz ��} ��d| ��} | dz } | | zdz }| | zdz }d�_|dkr2d}tjd|��d z }tjd|��}d}n3d }tjd| ��d z }tjd| ��}d }|dkr2d}tjd|��d z }tjd|��}d}n3d }tjd| ��d z }tjd| ��}d }d�_d}d |z�� |d z��z��||z| ��z|krB|dz}d |z�� |d z��z��||z| ��z|k�Btd|��}d}d |z�� |d z��z��||z| ��z|krB|dz}d |z�� |d z��z��||z| ��z|k�Btd|��}d |zd|z|zdz ks\d |zdz|zdksMt|��|dz ks7d |zd|z|zdz ks%d |zdz|zdkst|��|dz kr|�_td���t||��}|d|zz }|d|zz }|d |zz }|d |zz }t�|||||��}t�|||||��} t|| ��}!t�|||||��}"t�||||| ��}#t|"|#��}"d}$d�jz|$z�� |$dz��z }%|%|"kr|$dz}$d�jz|%z|$z }%|%|"k�i}&tjtjz|z|z}'tjtjz|z|z}(t!dd��D]�})tj|'|)d z ��d|zz }*tj|(|)d z ��d|zz }+t|*|+��},�� |)dz���� |)dz d z��z }-tj|-��}-|,|-zt||��z|&|)<��td |zd z d��dz}.d|$z}/��d|$z��}0t!d|.��D]G})t|0��|/�� |)dz��z|&|)z ����}0�H|0��|��zdz�_��|d�jzz ��}��|��}1dd||1z zz }2��|2��d��|0zz��}3|2��d��|0zz|3z }4|4d kr|3}3n|3dz}3��|3��d��|0 zz��}2����d�jz��|$���� |$dz��d z|$zz }5i}6i}7t)�|$|5��\}7}8|7���}6i}9t!|!d |zdz ��D]}:d|9|:<�i}9|0�_t!d|!dz��D]d})d};t!d|$z|)z dz dd��D]}<|;|2z|6|<z};�|;|9|)<t!dd|$z|)z dz ��D]}<|<dz|6|<dzz|6|<<��ei}=td��d|z|z����}>d��dt|��z|z��z��|��z dz }?��d�j�jz|z|z|z|zz ��dz }@t!d|��D]o}:������ |:d z ��������|:|@z��z |?z}At|A|>��|=|:<�p|=ddz�_dd|zz }Bi}Cd|Cd<d|Cd <d|Cd!<d|Cd"<d|Cd#<d|Cd$<d|Cd%<d|Cd&<t!d|��D�]R}:|=|:dz�_t!dd |:zdzdz��D�]}<d |:zd|<zz })|)dkr_�j|)z }D|Ddz }E|B|Dzdz }F|)dz }G|G|Cd|:dz |<dz fz|E|Cd|:dz |<fzz|F|Cd|:dz |<dz fzz|Cd|:|<f<�sd|Cd|:|<f<t!d|<��D]w}H|Cd|:|Hf��d'����d|<zd|Hzz ��z��|<|Hz ��z z}I|Cd|:|<fxxd|<|Hz z|Izzcc<�x��d|Cd|:d(f<d|Cd|:df<d|Cd|:d |:zdzdzf<��Tt!d(|dz��D]e}Jt!d(|��D]R}:t!d)tdd |:zdzdz����D](}<|Jdks|:dks|<dks |<d |:zdzkrd|C|J|:|<f<�)�S�ft!d|dz��D]�}Jt!d|��D]�}:|=|:dz�_t!dd |:zdzdz��D]]}<d|Jzdz |C|Jdz |:dz |<d z fzd||:zdz z|C|Jdz |:dz |<d z fzz}/|/|C|Jdz |:dz |<dz fz |C|J|:|<f<�^����i}Ktd��d|z|z����}>d��dt|��z|z��z��|��z dz }L��d�j�jz|z|z|z|zz ��dz }Mt!d|��D]o}:������ |:d z ��������|:|Mz��z |Lz}Nt|N|>��|K|:<�p|Kddz�_dd|zz }Oi}Pd|Pd<d|Pd <d|Pd!<d|Pd"<d|Pd#<d|Pd$<d|Pd%<d|Pd&<t!d|��D�]R}:|K|:dz�_t!dd |:zdzdz��D�]}<d |:zd|<zz })|)dkr_�j|)z }D|Ddz }E|O|Dzdz }F|)dz }G|G|Pd|:dz |<dz fz|E|Pd|:dz |<fzz|F|Pd|:dz |<dz fzz|Pd|:|<f<�sd|Pd|:|<f<t!d|<��D]w}H|Pd|:|Hf��d'����d|<zd|Hzz ��z��|<|Hz ��z z}I|Pd|:|<fxxd|<|Hz z|Izzcc<�x��d|Pd|:d(f<d|Pd|:df<d|Pd|:d |:zdzdzf<��Tt!d(|dz��D]e}Jt!d(|��D]R}:t!d)tdd |:zdzdz����D](}<|Jdks|:dks|<dks |<d |:zdzkrd|P|J|:|<f<�)�S�ft!d|dz��D]�}Jt!d|��D]�}:|K|:dz�_t!dd |:zdzdz��D]]}<d|Jzdz |P|Jdz |:dz |<d z fzd||:zdz z|P|Jdz |:dz |<d z fzz}/|/|P|Jdz |:dz |<dz fz |P|J|:|<f<�^����i}Qi}Rd�_��d|dzz��}E��d*|dzz|z��}S��||ztj�j��z��dz }Tt!d|��D]�}<|S|<|Tzz ����|<d z����dz z}Ut|E|U��|��z dz��dzdzd+z|Q|<<t|E��|dz��|Uz��|��z dz��dzdz|R|<<��i}Vi}Wd�_��d|dzz��}E��d*|dzz|z��}X��||ztj�j��z��dz }Yt!d|��D]�}<|X|<|Yzz ����|<d z����dz z}Zt|E|Z��|��z dz��dzd+z|V|<<t|E��|dz��|Zz��|��z dz��dzdz|W|<<��i}[t!d|dz��D]9}Jt!d|��D]&}<t!d(d |<zdzdz��D] }\d|[|J|<|\f<� �'�:t!d|dz��D]�}Jt!d|��D]|}<|Q|<�_t!dd |<zdzdz��D]S}\|C|J|<|\f|9d |<zd|\zz z|8d|<z|\z z |[|J|<|\f<|[|J|<|\fd�jz|\zz |[|J|<|\f<�T�}���fd,�}]i}^t!d|dz��D]9}Jt!d|��D]&}<t!d(d |<zdzdz��D] }\d|^|J|<|\f<� �'�:t|Qd|Vd��d z�_|]|��}_��|_�� }`t!d|dz��D]�}at!d|��D]�}<|Q|<�_t!dd |<zdzdz��D]o}\d|^|a|<|\f<t!d|adz��D]Q}J��|a|J����|`|J��z|[|a|Jz |<|\fz}b|^|a|<|\fxx|bz cc<�R�p����i}ct!d|dz��D]9}Jt!d|��D]&}<t!d(d |<zdzdz��D] }\d|c|J|<|\f<� �'�:t!d|dz��D]�}Jt!d|��D]|}<|V|<�_t!dd |<zdzdz��D]S}\|P|J|<|\f|9d |<zd|\zz z|8d|<z|\z z |c|J|<|\f<|c|J|<|\fd�jz|\zz |c|J|<|\f<�T�}��i}dt!d|dz��D]9}at!d|��D]&}<t!d(d |<zdzdz��D] }\d|d|a|<|\f<� �'�:t!d|dz��D]�}at!d|��D]�}<|V|<�_t!dd |<zdzdz��D]o}\d|d|a|<|\f<t!d|adz��D]Q}J��|a|J����|`|J��z|c|a|Jz |<|\fz}b|d|a|<|\fxx|bz cc<�R�p����t|Qd|Vd��dz�_i}ed|ed<|ed|z |ed<t|Qd|Vd���_t!d|��D]}<|e|<dz |edz|e|<<�i}ft!d|dz��D]X}at!d|��D]E}<|Q|<�_t!dd |<zdzdz��D]}\|^|a|<|\f|e|<z|f|a|<|\f<��F�Yi}gt!d|dz��D]X}at!d|��D]E}<|V|<�_t!dd |<zdzdz��D]}\|d|a|<|\f|e|<z|g|a|<|\f<��F�Yi}ht!d|dz��D]U}at!d|��D]B}:|R|:�_d}it!dd |:zdzdz��D]}<|i|f|a|:|<fz }i�|i|h|a|:f<�C�Vi}jt!d|dz��D]U}at!d|��D]B}:|W|:�_d}it!dd |:zdzdz��D]}<|i|g|a|:|<fz }i�|i|j|a|:f<�C�Vi}kd�_tj�j��|z||zz }ld�_��d-|d zdzz|lz|z ��}mt|m��d|dzz����}m|m�_t!d|dz��D]8}ad|k|a<t!d|dz��D]}<|k|axx|h|a||<z fz cc<��9i}nd�_tj�j��|z||zz }od�_��d-|d zdzz|oz|z ��}pt|p��d|dzz����}p|p�_t!d|dz��D]8}ad|n|a<t!d|dz��D]}<|n|axx|j|a||<z fz cc<��9d�_dt��t|kd������t|nd������z}q| d |qzz }r|��|d�jzz ��z}sd.��dd|rz ��|| ��zz|sz��z}td.��dd|rz ��|| ��zz|sz��z}ut|t|u���_|d�jzz }v|dz ��|v��z|dz z �jdz z }w��|w ��}x|]|��}��|| ��}��|| ��} d|1dz z|z|xz}yd|1dz z| z|xz}zd�_d��|1��|1d|z ��z��|1��z| z ��z}{d��|1��|1d|z ��z��|1��z| z ��z}|t|{||��}}|}dz�_ ��|dt;|1��dz t!d|dz��d/��\}~}d�_t|~|��}�t|k||yz��}�td|��dd |�zd0|�zzz��z��}�|��_i}�t!d|dz��D]}a|~|a|k|a|yzz|�|a<�d�_t||��}�t|n||zz��}�td|��dd |�zd0|�zzz��z��}�|��_i}�t!d|dz��D]7}a||a|n|a|zzz|�|a<��|�|a��|�|a<�8|�_|�|�fS)1Nrrr�@rVrrr�5�U��@��`<@� Y���?r��?�9@�2Riemann-Siegel can not compute with such precisionr� ���s@���,�rrrr� �r�rr������rrr�rrr�rrr�rrr~�rrr�rrr�rrrz1.0r~������D��?c�x���j}|dz�_��|d�jzz ��}|�_|S�Nr�r!r(r#��t�wp�aars �r �trunc_azRzeta_simul.<locals>.trunc_a0�<��� �X����6��� �X�X�a��3�6��l� #� #������ r 皙����@rT�)r!�_im�_rer(r#rfr�math�pow�gammar�abs�NotImplementedErrorrcrkrgr$�floorr%rHrT�copyrWr"r&r'�binomial�expj�_zetasum�int�conj)�r�s�der� wpinitialr��xsigma�ysigmar[�xasigma�yasigma�xA1�yA1r+�eps1�xeps2�yeps2�xb�xcrYr\�yb�yc�yA�yB1r]�yL�L�xeps3�yeps3rZ�yeps4rb�yM�Mrj�h4r*�jvalue�eps5�xforeps5�yforeps5r`�xaux1�yaux1r^r_�twenty�aux�wpfp�N�p�num� difference�eps6�cc�cont�pipowers�Fpr2�sumPr;�xwpd�d1�xd2�xconst�xd3�xpsigma�xd�m1�c1�c2�c3�r�addr@�ywpd�yd2�yconst�yd3�ypsigma�yd�xwptcoef�xwpterm�xc2�xc4�xc3�ywptcoef�ywpterm�yc2�yc4�yc3� xfortcoef�ellr��xtcoefr��la�chi�tcoefter� yfortcoef�ytcoef�av�xtv�ytv�xterm�te�yterm�xrssum�xrsbound�xwprssum�yrssum�yrsbound�ywprssum�A2�eps8�T�xwps3�ywps3�tpi�arg�U�xS3�yS3�xwpsum�ywpsum�wpsum�xS1�yS1�xabsS1�xabsS2�xwpend�xrz�yabsS1�yabsS2�ywpend�yrzs�` r � Rzeta_simulr�sB#�����I� ���� � �A� �W�W�Q�Z�Z�F� ��Z�F��C�H� ����A�c�f�H����A��6�k�G��6�k�G� � � �!�S�W�W�W�%�%�a�'�(�(�C� � � �!�S�W�W�W�%�%�a�'�(�(�C� �)�)�A� �z� "� "�C� �r�6�D� �#�I�b�L�E� �#�I�b�L�E��C�H� ��z�z� �� �X�a�� � �� '�� �X�a�� � ����� �� �X�a��� � �� (�� �X�a��� � ���� ��z�z� �� �X�a�� � �� '�� �X�a�� � ����� �� �X�a��� � �� (�� �X�a��� � ���� �C�H� �B� �B�$�s�y�y��C�� � � �3�9�9�R��T�2�#�#6�#6� 6�%� ?� ?� ��T�� �B�$�s�y�y��C�� � � �3�9�9�R��T�2�#�#6�#6� 6�%� ?� ?� �Q�r���B� �B� �B�$�s�y�y��C�� � � �3�9�9�R��T�2�#�#6�#6� 6�%� ?� ?� ��T�� �B�$�s�y�y��C�� � � �3�9�9�R��T�2�#�#6�#6� 6�%� ?� ?� �Q�r���B� �2���1��Q��s�� � ��"��Q��v� �a���S��[�[�1�R�4�5G�5G� �2���1��Q��s�� � ��"��Q��v� �a���S��[�[�1�R�4�5G�5G����!�"V�W�W�W� �B�� � �A� �A�b�D�\�E� �A�b�D�\�E� �1�R�4�L�E� �1�R�4�L�E� �#�r�5�!�S�"� -� -�B� �#�r�5�!�S�"� -� -�B� �B�� � �A� �c�2�u�a��b� 1� 1�B� �c�2�u�a��b� 1� 1�B� �R����B� �A����h��]�S�Y�Y�q��s�^�^� +�F� �2�+�+� �a�C���C�F�(�F�"�1�$�� �2�+�+� �D��w�t�w��s�"�1�$�H��w�t�w��s�"�1�$�H� �1�R�[�[�/�/�����1�Q�3�'�'��b��1�����1�Q�3�'�'��b��1���5�%� � ���y�y��1��~�~�c�i�i��#��c� �2�2�2���y�������9�S��u�-�-�-��Q�����1��Q���^�^�A� �F� �q�&�C� �7�7�2�a�4�=�=�D� �1�V�_�_�>�>���4�����S�Y�Y�q��s�^�^�!3�D��G�!;�<�<�=�=����c�g�g�a�j�j� ��#�C�H� ����A�c�f�H����A� � � �!� � �A� �!�Q�q�S�'� �A� � � �!�S�W�W�S�\�\�4�'�(�)�)�C��c�g�g�c�l�l�D�(�)�#�-�J��S�������!�e�� � � �C�3�7�7�3�<�<�4�%�0�1�2�2�A� �9�9�S�[�[��3�6��*�*�A� .� .�� � �!�A�#���q�0@��0B� C�D� �B� �D��#�q�$�'�'�N�D�(� �y�y�{�{�B� �B� �1�a��c�!�e�_�_������1��� �B��C�H� �1�Q�q�S�\�\�#�#�����q��s�1�u�Q�w�r�"�%�%� %� %�A��1�H�r�!�u�$�D�D���1���q��1��Q��q��!�!� #� #�A��q�S�2�a��c�7�N�B�q�E�E� #� �D� �Q�s�w�w�r�!�t�A�v��� � �B� �S�W�W�a��F� � �m�R�'� (� (� (������� 7�� 9�C� �V�V�A�s�v�c�f�}�Q��q�(��,�S�0�1� 2� 2�A� 5�F� �1�Q�Z�Z�����g�g�c�h�h�s�y�y��3��/�/�0�0�1�1�#�)�)�A�f�H�2E�2E�E�c�I���C�� � ��Q����A�w�r�z�C�H���6��l�G� �B��B�v�J�Q�"�V�*�!��5� �q�R��Y��B�w�K�q�2�g�;�Q�"�V�*�1��6� � �1�Q�Z�Z�7�7����7�2�:����q��1��a����"�"� 3� 3�A��!��A�a�C��A��!�t�t��W�Q�Y���q�D���B�J��>���q�S�6���R��!�A�#�a��c� �]�*�2�b��1�Q�3�q��k�>�9�"�R��!�A�#�a��c� �]�:J�J��1�Q�q�5� � ���1�Q�q�5� ��q����3�3�A��1�Q�q�5� �3�7�7�5�>�>�#�'�'�!�A�#�a��c�'�2B�2B�#B�3�7�7�1�Q�3�<�<�#O�P�C��q��1�u�I�I�I�2��1��+�s�!2�2�I�I�I�I�3���1�Q�r�6� ��b��1�R��j�Q�B�q��1�Q�3��6�!�8�|�$4�$4��B�s�1�u�o�o�#�#���r�!��� #� #�A��2�c�!�A�a�C��F�1�H�o�o�.�.� #� #����T�T�a��c�c�a��c�c�a��!��Q��h�h�!"�B�r�!�A�v�J�� #� #��A�c�!�e�n�n�4�4���q���� 4� 4�A��A�w�r�z�C�H��1�Q�q�S�!�V�A�X�&�&� 4� 4���r�T�!�V�R��1��Q�q�S��1�� �-�-�a����!��n�R��1��Q�q�S��1�� �=M�.M�M�� �2�b��d�1�Q�3�q��s�l�#3�3��2�a��6� � � 4� 4� �D� �Q�s�w�w�r�!�t�A�v��� � �B� �S�W�W�a��F� � �m�R�'� (� (� (������� 7�� 9�C� �V�V�A�s�v�c�f�}�Q��q�(��,�S�0�1� 2� 2�A� 5�F� �1�Q�Z�Z�����g�g�c�h�h�s�y�y��3��/�/�0�0�1�1�#�)�)�A�f�H�2E�2E�E�c�I���C�� � ��Q����A�w�r�z�C�H���6��l�G� �B��B�v�J�Q�"�V�*�!��5� �q�R��Y��B�w�K�q�2�g�;�Q�"�V�*�1��6� � �1�Q�Z�Z�7�7����7�2�:����q��1��a����"�"� 3� 3�A��!��A�a�C��A��!�t�t��W�Q�Y���q�D���B�J��>���q�S�6���R��!�A�#�a��c� �]�*�2�b��1�Q�3�q��k�>�9�"�R��!�A�#�a��c� �]�:J�J��1�Q�q�5� � ���1�Q�q�5� ��q����3�3�A��1�Q�q�5� �3�7�7�5�>�>�#�'�'�!�A�#�a��c�'�2B�2B�#B�3�7�7�1�Q�3�<�<�#O�P�C��q��1�u�I�I�I�2��1��+�s�!2�2�I�I�I�I�3���1�Q�r�6� ��b��1�R��j�Q�B�q��1�Q�3��6�!�8�|�$4�$4��B�s�1�u�o�o�#�#���r�!��� #� #�A��2�c�!�A�a�C��F�1�H�o�o�.�.� #� #����T�T�a��c�c�a��c�c�a��!��Q��h�h�!"�B�r�!�A�v�J�� #� #��A�c�!�e�n�n�4�4���q���� 4� 4�A��A�w�r�z�C�H��1�Q�q�S�!�V�A�X�&�&� 4� 4���r�T�!�V�R��1��Q�q�S��1�� �-�-�a����!��n�R��1��Q�q�S��1�� �=M�.M�M�� �2�b��d�1�Q�3�q��s�l�#3�3��2�a��6� � � 4� 4��H� �G��C�H� ����Q�q�S�� � �B� �'�'�"�a��c�(�2�+� � �C� �'�'�#�a�%�� �#�&�)�)�)� *� *�1� ,�C� �1�Q�Z�Z�G�G���A�c�E�k�#�'�'�#�'�'�!�C�%�.�.�1�1�"�4�4���2�c�#�'�'�%�.�.�0��2�3�3�A�5�r�9�3�>��� ��"�S�W�W�Q�q�S�\�\�#�-�c�g�g�e�n�n�<�Q�>�?�?��A�2�E��� � � �H� �G��C�H� ����Q�q�S�� � �B� �'�'�"�a��c�(�2�+� � �C� �'�'�#�a�%�� �#�&�)�)�)� *� *�1� ,�C� �1�Q�Z�Z�F�F���A�c�E�k�#�'�'�#�'�'�!�C�%�.�.�1�1�"�4�4���B�s�3�7�7�5�>�>�1�!�3�4�4�b�8�#�=��� ��"�S�W�W�Q�q�S�\�\�#�-�c�g�g�e�n�n�<�Q�>�?�?��A�2�E��� � ��I��A�c�!�e�n�n�&�&���q���� &� &�A��R��!��Q��q��)�)� &� &��$%� �"�Q�s�(�#�#� &� &��A�c�!�e�n�n�I�I���q���� I� I�A���{�C�H��Q�q��s�A�v�a�x�(�(� I� I��$&�r�!�C�x�L��A�a�C��#��I��$>�x��!��C��?P�$P� �"�Q�s�(�#�$-�b��3�h�$7�!�C�E�'�C��$H� �"�Q�s�(�#�#� I� I� ����� �F��A�c�!�e�n�n�#�#���q���� #� #�A��R��!��Q��q��)�)� #� #��!"��r�!�C�x� � � #� #��8�A�;�x��{�+�+�A�-�C�H���� � �B� �&�&��*�*��B��Q�s�1�u�~�~�2�2���q���� 2� 2�A���{�C�H��Q�q��s�A�v�a�x�(�(� 2� 2��#$��s�1�S�y�!���3�q�5�/�/�2�2�B� �\�\�#�b�1�1�#�)�)�B�r�2B�2B�B�9�S�QS�V�TU�VY�\�CZ�Z�H��3�q��9�%�%�%��1�%�%�%�%�2� 2� 2��I��A�c�!�e�n�n�&�&���q���� &� &�A��R��!��Q��q��)�)� &� &��$%� �"�Q�s�(�#�#� &� &��A�c�!�e�n�n�I�I���q���� I� I�A���{�C�H��Q�q��s�A�v�a�x�(�(� I� I��$&�r�!�C�x�L��A�a�C��#��I��$>�x��!��C��?P�$P� �"�Q�s�(�#�$-�b��3�h�$7�!�C�E�'�C��$H� �"�Q�s�(�#�#� I� I� �F��Q�s�1�u�~�~�$�$���q���� $� $�A��R��!��Q��q��)�)� $� $��"#��s�1�S�y�!�!� $� $��Q�s�1�u�~�~�2�2���q���� 2� 2�A���{�C�H��Q�q��s�A�v�a�x�(�(� 2� 2��#$��s�1�S�y�!���3�q�5�/�/�2�2�B� �\�\�#�b�1�1�#�)�)�B�r�2B�2B�B�9�S�QS�V�TU�VY�\�CZ�Z�H��3�q��9�%�%�%��1�%�%�%�%�2� 2� 2��8�A�;��� �,�,�Q�.�C�H� �B� �B�q�E� �q�E�!�G�B�q�E��8�A�;�x��{�+�+�C�H� �1�Q�Z�Z� � ���1�Q�3��"�Q�%���1��� �C��Q�s�1�u�~�~�:�:���q���� :� :�A���{�C�H��Q�q��s�A�v�a�x�(�(� :� :��!'��A�c� �!2�B�q�E�!9��C��#�I��� :� :� �C��Q�s�1�u�~�~�:�:���q���� :� :�A���{�C�H��Q�q��s�A�v�a�x�(�(� :� :��!'��A�c� �!2�B�q�E�!9��C��#�I��� :� :� �E��Q�s�1�u�~�~�����q���� � �A��q�z�C�H��B��1�a��c�1�f�Q�h�'�'� #� #���c�#�a��'�l�"����E�#�a�%�L�L�  � �E��Q�s�1�u�~�~�����q���� � �A��q�z�C�H��B��1�a��c�1�f�Q�h�'�'� #� #���c�#�a��'�l�"����E�#�a�%�L�L�  � �F� �C�H��y��� � �2�%��1��-�H� �C�H��w�w�s�Q�q�S�1�H�~�h�.��6�7�7�H��8�S�W�W�R��1��X�.�.�/�/�H��C�H��Q�s�1�u�~�~�*�*����s� ��q��1���� *� *�A� �3�K�K�K�5��Q�q�S��>� )�K�K�K�K� *� �F� �C�H��y��� � �2�%��1��-�H� �C�H��w�w�s�Q�q�S�1�H�~�h�.��6�7�7�H��8�S�W�W�R��1��X�.�.�/�/�H��C�H��Q�s�1�u�~�~�*�*����s� ��q��1���� *� *�A� �3�K�K�K�5��Q�q�S��>� )�K�K�K�K� *� �C�H� �S�����V�A�Y���(�(�#�'�'�#�f�Q�i�.�.�*A�*A� B� B� C�B� ��"��:�D� �3�6�6�!�Q�s�v�X�,� � ��A� ����!�Q�t�V�S�Y�Y�q�&��%9�%9�9�9�1�<�=�=� =�E� ����!�Q�t�V�S�Y�Y�q�&��%9�%9�9�9�1�<�=�=� =�E��5�%� � �C�H� �Q�s�v�X�,�C� �Q�3����s� � � �Q�q�S� !�#�&��(� *�C� ���#����A���� � �A��i�i��F�7�#�#�G��i�i��F�7�#�#�G� �!�A�#�;�'� !�A� %�C� �!�A�#�;�'� !�A� %�C��C�H�����!�C�I�I�a��&��1�1�1�3�6�6�!�9�9�<�d�B�C�C�C�F�����!�C�I�I�a��&��1�1�1�3�6�6�!�9�9�<�d�B�C�C�C�F� ��� � �E��b�y�C�H��$�|�|�A�q�#�a�&�&��(�E�!�C��E�N�N�D�A�A�H�C���C�H� ��S��]�]�F� ����s�"� #� #�F� ��I�c�g�g�a��6��!�F�(�):�&;�=�=�=� ?� ?�F��C�H� �C��Q�s�1�u�~�~�,�,���s�8�F�3�K��O�+��C����C�H� ��S��]�]�F� ����s�"� #� #�F� ��I�c�g�g�a��6��!�F�(�):�&;�=�=�=� ?� ?�F��C�H� �C��Q�s�1�u�~�~�&�&���s�8�F�3�K��O�+��C���8�8�C��H�%�%��C����C�H� ��8�Or c ��"��t|��}�j}��|��}��|��}d�_��|d�jzz ��}��||��}��d��|��dz ��} ��d| ��} | dz } | | zdz } d�_|dkr2d} tj d|��d z }tj d|��}d}n3d } tj d| ��d z }tj d| ��}d }d�_d}d |z�� |d z��z��| |z| ��z| krB|dz}d |z�� |d z��z��| |z| ��z| k�Btd|��}d |zd|z|zdz ks%d |zdz|zdkst|��|dz kr|�_td���| d|zz }|d |zz }t�|||||��}i}t|d |zdz ��D]}d||<�|d|zz }�j�jz|z|z��d ��ztjztjz}|��||dzz |dz d z ��z|z }t!||��}d}d�jz|z�� |dz��z }||kr|dz}d�jz|z|z }||k�i}tjtjz|z|z}tdd��D]s}tj ||d z ��d|zz }�� |dz���� |dz d z��z } tj| ��} || z|z||<�tt!d |zd z d��dz}!d|z}"��d|z��}#td|!��D]G}t|#��|"�� |dz��z||z ����}#�H|#��|��zdz�_��|d�jzz ��}��|��}$dd||$z zz }%��|%��d��|#zz��}&|%��d��|#zz|&z }'|'d kr|&}&n|&dz}&��|&��d��|# zz��}%��d�jz|���� |dz��d z|zz }(i})i}*t)�||(��\}*}+|*���})i}t|d |zdz ��D]}d||<�|#�_td|dz��D]d}d},td|z|z dz dd��D]}-|,|%z|)|-z},�|,||<tdd|z|z dz ��D]}-|-dz|)|-dzz|)|-<��ei}.td��d|z|z����}/d��dt|��z|z��z��|��z dz }0��d�j�jz|z|z|z|zz ��dz }1td|��D]o}������ |d z ��������||1z��z |0z}2t|2|/��|.|<�p|.ddz�_dd|zz }3i}4d|4d<d|4d <d|4d!<d|4d"<d|4d#<d|4d$<d|4d%<d|4d&<td|��D�]B}|.|dz�_tdd |zdzdz��D]�}-d |zd|-zz }|dkr_�j|z }5|5dz }6|3|5zdz }7|dz }8|8|4d|dz |-dz fz|6|4d|dz |-fzz|7|4d|dz |-dz fzz|4d||-f<�rd|4d||-f<td|-��D]i}9|4d||9f�j��d|-zd|9zz ��z��|-|9z ��z z}:|4d||-fxxd|-|9z z|:zzcc<�j��d|4d|d'f<d|4d|df<d|4d|d |zdzdzf<��Dtd'|dz��D]e};td'|��D]R}td(tdd |zdzdz����D](}-|;dks|dks|-dks |-d |zdzkrd|4|;||-f<�)�S�ftd|dz��D]�};td|��D]�}|.|dz�_tdd |zdzdz��D]]}-d|;zdz |4|;dz |dz |-d z fzd||zdz z|4|;dz |dz |-d z fzz}"|"|4|;dz |dz |-dz fz |4|;||-f<�^����i}<i}=d�_��d|dzz��}6��d)|dzz|z��}7��||ztj�j��z��dz }>td|��D]�}-|7|-|>zz ����|-d z����dz z}8t|6|8��|��z dz��dzdz|<|-<t|6��|dz��|8z��|��z dz��dzdz|=|-<��i}?|D]9};td|��D]&}-td'd |-zdzdz��D] }@d|?|;|-|@f<� �'�:|D]�};td|��D]|}-|<|-�_tdd |-zdzdz��D]S}@|4|;|-|@f|d |-zd|@zz z|+d|-z|@z z |?|;|-|@f<|?|;|-|@fd�jz|@zz |?|;|-|@f<�T�}���fd*�}Ai}B|D]9}Ctd|��D]&}-td'd |-zdzdz��D] }@d|B|C|-|@f<� �'�:|<dd z�_|A|��}D��|D�� }E|D]�}Ctd|��D]�}-|<|-�_tdd |-zdzdz��D]^}@d|B|C|-|@f<td|Cdz��D]@};��|C|;��|E|;zz|?|C|;z |-|@fz}F|B|C|-|@fxx|Fz cc<�A�_����|<ddz�_i}Gd|Gd<|Gd|z |Gd<|<d�_td|��D]}-|G|-dz |Gdz|G|-<�i}H|D]X}Ctd|��D]E}-|<|-�_tdd |-zdzdz��D]}@|B|C|-|@f|G|-z|H|C|-|@f<��F�Yi}I|D]U}Ctd|��D]B}|=|�_d}Jtdd |zdzdz��D]}-|J|H|C||-fz }J�|J|I|C|f<�C�Vi}Kd�_tj�j��|z| |zz }Ld�_��d+|d zdzz|Lz| z ��}Mt|M��d|dzz����}M|M�_|D]8}Cd|K|C<td|dz��D]}-|K|Cxx|I|C||-z fz cc<��9d�_d��|Kd��z}N| d |Nzz }O|��|d�jzz ��z}Pd,��dd|Oz ��|| ��zz|Pz��z}Q|Q�_|d�jzz }R|dz ��|R��z|dz z �jdz z }S��|S ��}T|A|��}��|| ��}d|$dz z|z|Tz}Ud�_d��|$��|$d|z ��z��|$��z| z ��z}V|Vdz�_ ��|dt;|$��dz |��d}Wd�_t|W|��}Xt|K||Uz��}Ytd|��dd |Xzd-|Yzzz��z��}Z|Z�_i}[|D]}C|W|C|K|C|Uzz|[|C<�|�_|[S).a� Computes several derivatives of the auxiliary function of Riemann `R(s)`. **Definition** The function is defined by .. math :: \begin{equation} {\mathop{\mathcal R }\nolimits}(s)= \int_{0\swarrow1}\frac{x^{-s} e^{\pi i x^2}}{e^{\pi i x}- e^{-\pi i x}}\,dx \end{equation} To this function we apply the Riemann-Siegel expansion. rrrrmrVrrrrnrorprrqrrrsrrertrurvrwrxryrzrrrr{r|rr}rr�r�r�r�r�r�r~r�r�c�x���j}|dz�_��|d�jzz ��}|�_|Sr�r�r�s �r r�zRzeta_set.<locals>.trunc_a�r�r r�rr�)rr!r�r�r(r#rfrr�r�r�r�r�rcr$�ergr�r%rHrTr�rWr"r&r'r�r�r�r�)\rr�� derivativesr�r�r��sigmar[�asigma�A1r+r��eps2�brB�A�B1r��eps3�eps4r�r�r2rhrirjr*r�r��foreps5r`r^r_r�r�r�r�r�r�r�r�r�r�r�r�r;�wpdr��d2�const�d3�psigma�dr�r�r�r�r�r�r@�wptcoef�wpterm�c4�fortcoefr�r��tcoefr�r�r�r�r��tv�termr��rssum�rsbound�wprssumrrr�wps3rrr�S3r �S1�absS1�absS2�wpend�rzs\` r � Rzeta_setr=�s5���$ �k� � �C���I� ���� � �A� �G�G�A�J�J�E��C�H� ����A�c�f�H����A� �Y�Y�q�%� � �F� ���1�c�g�g�f�o�o�a�'� (� (�B� �)�)�A� �z� "� "�C� �r�6�D� ��8�B�;�D� �C�H� �q�y�y� �� �H�Q�u� � �g� %�� �H�Q�u� � �� ��� �� �H�Q��v� � �w� &�� �H�Q��v� � �� ���C�H� �A� �A�#�c�i�i��#���� ����1�Q�3��r�!2�!2� 2�d� :� :� �a�C�� �A�#�c�i�i��#���� ����1�Q�3��r�!2�!2� 2�d� :� :� �A�a���A� �1���!��A��c� � � �q��s�1�u�U�{�1�}�}�#�e�*�*�a��d�2B�2B����!�"V�W�W�W� �!�A�#�J�D� ��1��:�D� ��a��q�"�a�(�(�A� �B� �1�a��c�!�e�_�_������1��� �s�1�u��B� �����r� �!� �S�X�X�a�[�[� (��� /��� 6�B� �c�i�i��A�q�D��1�Q�3��'�*�*� *�Q� .�B� �R����B��A����h��]�S�Y�Y�q��s�^�^� +�F� �2�+�+� �a�C���C�F�(�F�"�1�$�� �2�+�+� �D��g�d�g�o�b� ��"�G� �1�R�[�[�!�!���x���1��%�%�t�A�v�.���y�y��1��~�~�c�i�i��#��c� �2�2�2���y������t�)�D�.��Q�����1��Q���^�^�A� �F� �q�&�C� �7�7�2�a�4�=�=�D� �1�f� � �>�>���4�����S�Y�Y�q��s�^�^�!3�D��G�!;�<�<�=�=����c�g�g�a�j�j� �2�%�C�H� ����A�c�f�H����A� � � �!� � �A� �!�Q�q�S�'� �A� �)�)�A�s�w�w�q�z�z�4�'�(� )� )�C��c�g�g�a�j�j�$�&�'��+�J��C�������!�e�� � � �C�3�7�7�1�:�:���.�/�0�0�A� �9�9�Q�s�v�X�q� !� !�3�9�9�Q�q�S�>�>�!�#3�A�#5� 6�D� �B� �D��#�q�$�'�'�N�D�(� �����B� �B� �1�a��c�!�e�_�_������1����C�H� �1�Q�q�S�\�\�$�$�����q��s�1�u�Q�w�r�"�%�%� &� &�A��1�H��1��%�D�D���1���q�!�A�#�a�%��'�"�"� $� $�A��q�S�B�q��s�G�O�B�q�E�E� $� �C� �Q�����1��Q���� � �B� �C�G�G�Q�s�5�z�z�\�1�$� %� %� %�c�g�g�d�m�m� 3�A� 5�B� �F�F�1�c�f�S�V�m�A�o�a�'��*�2�-�.� /� /�� 2�E� �1�Q�Z�Z���� �W�W�S�X�X�c�i�i��#��.�.�/�/� 0� 0����1�U�7�1C�1C� C�B� F���R�����A����1�v�b�y�C�H� ��%��[�F� �A��A�f�I�1��6��q�a��h�1�!�E�(��A�g�J�Q�!�G�*�!��&� �q�Q�v�Y� �1�Q�Z�Z�4�4���q�6�"�9����q��1��a����"�"� 2� 2�A��!��A�a�C��A��1����W�Q�Y����T���R�i��]����s�V���a��!�A�#�a��c� �l�?�2�a��!�A�#�a��j�=�8��A�a��!��A�a�C�i�L��H��!�A�a�%�����!�A�a�%���q����2�2�A��A�a��E�(�C�G�C�G�G�A�a�C��!��G�,<�,<�$<�S�W�W�Q�q�S�\�\�$I�J�C��a��!�e�H�H�H�"��!���c� 1�1�H�H�H�H�2���!�A�b�&� �q�Q�q��2�v�Y�!�!�A�a��!��Q��q��L�/�/��B�s�1�u�o�o�"�"���r�!��� "� "�A��2�c�!�A�a�C��F�1�H�o�o�.�.� "� "����T�T�a��c�c�a��c�c�a��!��Q��h�h� !�A�b��1�f�I�� "� "� �A�c�!�e�n�n�2�2���q���� 2� 2�A��1�v�b�y�C�H��1�Q�q�S�!�V�A�X�&�&� 2� 2���r�T�!�V�Q�r�!�t�A�a�C��!��|�_�,�Q��a��� �]�1�R��T�!�A�#�a��c�\�?�-J�J���!�B�q�D��1��Q�q�S�L�/�1��"�Q�q�&� � � 2� 2��G� �F��C�H� ����Q�q�S�� � �B� ����Q�q�S��!�� � �B� ����A��d�i���'�'�'� (� (�� *�B� �1�Q�Z�Z�B�B�� �!�B�$�Y�s�w�w�s�w�w�q��u�~�~�.�.�r�1� 1����B�s�w�w�t�}�}�,�Q�.�/�/��1�2�5��� ���3�7�7�1�Q�3�<�<��?�3�7�7�4�=�=�8��:�;�;�A�=�r�A��q� � � �H��%�%���q���� %� %�A��R��!��Q��q��)�)� %� %��#$���A�c��"�"� %� %��G�G���q���� G� G�A��q�z�C�H��Q�q��s�A�v�a�x�(�(� G� G��#$�R��#�X�;�r�!�A�#�a��e�)�}�#<�X�a��c�#�g�=N�#N���A�c��"�#+�B�q��H�#5��#�%��#�~�#F���A�c��"�"� G� G� ����� �E��#�#���q���� #� #�A��R��!��Q��q��)�)� #� #��!"��c�!�C�i� � � #� #��q�z�!�|�C�H� �����B� �&�&��*�*��B��1�1���q���� 1� 1�A��q�z�C�H��Q�q��s�A�v�a�x�(�(� 1� 1��#$��c�!�C�i� ���3�q�5�/�/�1�1�B�"�|�|�C��3�3�b�"�f�<� ��R���#��.� /�H��#�a��)�$�$�$��0�$�$�$�$�1� 1� 1��q�z�A�~�C�H� �B� �B�q�E� �q�E�!�G�B�q�E��q�z�C�H� �1�Q�Z�Z� � ���1�Q�3��"�Q�%���1��� �B��8�8���q���� 8� 8�A��q�z�C�H��Q�q��s�A�v�a�x�(�(� 8� 8�� %�c�!�C�i� 0�"�Q�%� 7��3�q��9� � � 8� 8� �D������q���� � �A��a�y�C�H��B��1�a��c�1�f�Q�h�'�'� "� "���b��Q�q��k�!����D��Q��K�K�  � �E� �C�H��i�����!�#�a��c�*�G� �C�H��g�g�c�A�a�C�!�8�n�W�,�t�3�4�4�G��'�3�7�7�2�q��s�8�,�,�-�-�G��C�H��(�(����c� ��q��1���� (� (�A� �#�J�J�J�$�s�1�Q�3�w�-� '�J�J�J�J� (� �C�H� �S�W�W�U�1�X� � � �B� ��2��;�D� �C�F�F�1�a���h�<� � � �A� �s�w�w��1�T�6�3�9�9�Q��v�#6�#6�6�6��9�:�:� :�D��C�H� �Q�s�v�X�,�C� �Q�3����s� � � �Q�q�S� !�#�&��(� *�C� ���#����A���� � �A� �Y�Y�q�5�&� !� !�F� ��1��+�� �!� #�B��C�H� �����3�9�9�Q�q��w�/�/�/�������:�4�?�@�@� @�E��r�z�C�H� � ���a��C��F�F�1�H�k� 2� 2�1� 5�B��C�H� ��3��L�L�E� ��c� �R�� � �E� ��9�s�w�w�q�!�E�'�!�E�'�/�':�;�;�;� <� <�E��C�H� �B��(�(���S�'�%��*�R�-�'��3����C�H� �Ir c���|�d��|j|zz}|j}d|_|d|jzz }|dz|�d|dzz|�|��z��z}|dz|�d|z|�|��z��z}||_|�|��}||_t||t|dz����} |dkrM|� |� d|dz ��dz |�|j��dz z ��} |dkr7|� |j|� d|dz ��zd z ��} |dkr0|� |� d|dz �� d z ��} |dkr8|� |j |� d|dz ��zd z ��} |� |��}|dkrd|z| dz}|dkrd |z}|| | dz| dzz}|dkrCd|z}| d| dz| z| d| dzzz| dz|j| dz| zz z}|dkrod |z}d| dz| dzz| d| dzzzd| z| dzz}||d| dz| zz d| dz| z| zz | dz| d| zz z}|d kr�d|z}d | dz| dzz| d| d zzzd| dzz| dzz}|d| dz| z| zz d| dz| dzz| zz d| dz| zz d| dz| z| zz }|d | z| dzzd | dz| zz d | dz| z| zz | d z|j| dz| zz}||z}||_|� |��S)z- z_half(t,der=0) Computes Z^(der)(t) �0.5rrrrr�rtrrr|��@y���@r�(@�@) r%r'r!r#rrW� siegelthetar=r$r��psir�)rr�r�r�r��tt�wptheta�wpz�thetar<�ps1�ps2�ps3�ps4�exptheta�z�zfs r �z_halfrRrs�� �g�g�e�n�n�S�U�1�W��A���I��C�H� �A�c�f�H��B���l�S�W�W�Q��C��[�������%;�<�<�<�G� �a�-�#�'�'�"�R�%����r� � �"2�3�3� 3�C��C�H� �O�O�A� � �E��C�H� �3�q�%��A��,�,� '� '�B� �Q�w�w�c�g�g�c�g�g�a��!��n�n�Q�.���������1A�A�B�B�� �Q�w�w�c�g�g�c�e�C�G�G�A�a��c�N�N�2�1�4�5�5�� �Q�w�w�c�g�g�s�w�w�q��1��~�~�o�a�/�0�0�� �Q�w�w�c�g�g�s�u�f�S�W�W�Q�q��s�^�^�3�B�6�7�7���x�x����H� �a�x�x� �h�J�r�!�u� �� �a�x�x� ��[�� ��B�q�E� �"�Q�%�� �� �a�x�x� ��\�� �C��2�a�5����R��U�3��6�\�)�"�Q�%�/���b��e� �C��?� @�� �a�x�x� ��\�� �b��e�G�C��F�N�2�a�5��a��<� '��#��b��e� � 3�� ��"�R��U�(�3�,��r�"�Q�%�x��|�C�/�/��1��5�b��e�C�i�?� @�� �a�x�x� �x�Z�� �b��e�G�C��F�N�2�a�5��a��<� '��#�q�&���A��� 6�� �c�"�Q�%�i��m�C�� ��2�a�5���a���� 3� 3�B�r�!�u�H�S�L� @��2�a�5����S�� P�� ��#��b��e� �O�A�b��e�G�C�K� '��"�Q�%��� �C�� 7��1�� =�c�e�B�q�E�k�#�o� M�� �q�D���C�H� �7�7�1�:�:�r c� �|j}|�|��}|�|��}d|_|dkr#|�t |����}n)d|jz|dz zt d|z ��d|z zz}|dkr$d|�|d|jzz ��z}nd|z|z}t d|z ��}d|zt j|��z} tdd|� |��z��} d|� d |zd |z|zzd |z|z| zz��z|zdz} t| | ��} tdd|� d |z|z��z|zdz��} d|� d d|zz��z|zdz} | |_|� ||j ||� d��z zz ��}|dkrM|�|� d|dz ����dz |�|j��dz z }|dkr0|�|� d|dz ���� dz }|dkr0|�|� d|dz ���� dz }|dkr/|�|� d|dz ����dz }| |_t||t!|dz����}i}t!d|dz��D] }|�||��||<�!| |_|�d|z��}|dkr|d||dzz}|dkr&|d d|dz|zz }|d||zz}|dkrFd|dz|zd|dz|dzzz|dzd|dz|zz}|d||zz}|dkr{d|dz|dzzd|dz|dzzz d|dz|zz d|dz|zz }|d|dz|z|zz |dz d|dz|zz}|d||zz}|dkr�d|dz|dzzd|dz|dzzzd|dz|dzzz}|d|dz|z|zzd|dz|dzz|zz}|d|dz|zzd|dz|dzzz d|dz|zzd|dz|zz }|d|dz|z|zz |dzd|dz|zz }|d||zz}||_|S)z? zeta_half(s,k=0) Computes zeta^(k)(s) when Re s = 0.5 �5rrrrqrrr��(\���@�3333335@��������?皙����@皙�����?r?r|r@r~rA�����rDrC� ��H@rt)r!r�r�r(r�r#r��logrrrEr'r%rFrWr=r$r�r�)rr�r;r�rr��X�M1�abstr�wpbasic�wpbasic2rH�wpRrJrKrLrMrNrrr�rO�zv�zv1s r � zeta_halfrg�s�����I� �G�G�A�J�J�E� ���� � �A��C�H� �q�y�y� �H�H�S��V�V� � ��� �s�v�X��q�� !�C��!��H�H�s�5�y�$9� 9�� �q�y�y� �s�x�x��1�S�V�8� �%�%� %��� ��U�Q�Y�� �s�1�u�:�:�D� �4��������A��!�A�c�g�g�a�j�j�L�!�!�G������b���b����*�3�r�6�!�8�A�:�5�6�6�6�y�@��B�H��'�8�$�$�G��!�Q�s�w�w�s�2�v�a�x�(�(�(��2�1�4�5�5�G� �C�G�G�C��!��G� � � �Y� &�q� (�C��C�H� �O�O�A�c�e�U�3�7�7�5�>�>�%9�:�:� ;� ;�E��1�u�u�S�W�W�S�W�W�Q�q��s�^�^�,�,�a�/�#�&�&���.�.��2B�B�c��1�u�u�c�g�g�c�g�g�a��!��n�n�-�-�.�q�0�c��1�u�u�c�g�g�c�g�g�a��!��n�n�-�-�.�q�0�c��1�u�u�S�W�W�S�W�W�Q�q��s�^�^�,�,�b�0�c��C�H� �C��%��!��*�*� %� %�C� �C��Q�q��s�|�|�&�&���8�8�C��H�%�%��C����C�H��x�x��5��!�!�H��!�t�t� ��V�H�S��V�O� #���!�t�t��1�v�g��#�a�&��� �$�� ��V�h�s�l� "���!�t�t���A��h�s�l�1�S��V�8�S�!�V�,�,�S��V�3�B�s�1�v�I�c�M�A�� ��V�H�S�L� ���!�t�t��#�a�&�j��a����#�a�&���a���/��#�a�&��� �<�R��A��Y�s�]�J���C��A��J�s�N�3�&�&�s�1�v�-�a��A��h�s�l�:�� ��V�H�S�L� ���A�v�v���Q��i��Q����3�q�6� �#�q�&� 0�0��C��F��3��6�1A�A���3�s�1�v�:�c�>�#�%�%�c�#�a�&�j�#�q�&�&9�#�&=�=���#�c�!�f�*�S�.� ��C��F��3��6�!1�1�!�C��F�(�3�,�>�q��Q��x��|�K���"�S��V�)�C�-��#�#�C��F�*�2�c�!�f�9�S�=�8�� ��V�H�S�L� ���C�H� �Ir c�� �|j}|�|��}|�|��}d|_|dkr$|�t |��d��}nK|�d|jz|dz ��|�t d|z ��d|z ��z}|dkr$d|�|d|jzz ��z}nd|z|z}d|z dkr$d|�|d|jzz ��z}nLd|z|�d|jz| ��z|�t |��|dz ��z}t d|z ��} d| ztj| ��z} tdd|� |��z��} d|� d |zd |z|zzd |z|z| zz��z|zdz} t| | ��} tdd|� d |z|z��z|zdz��} d|� d d|zz��z|zdz}| |_|� ||j ||� d��z zz ��}|}|�d|z ��}||_t|||��\}}|dkrW|�d|dz ��|�dd|z dz ��zdz |�|j��dz z }|dkrA|j |�d|dz ��|�dd|z dz ��z zdz }|dkr:|�d|dz ��|�dd|z dz ��z dz }|dkrB|j |�d|dz ��|�dd|z dz ��z zdz }| |_|�d|z��}|dkr|d||dzz}|dkr&|d d|dz|zz }|d||zz}|dkrFd|dz|zd|dz|dzzz|dzd|dz|zz}|d||zz}|dkr{d|dz|dzzd|dz|dzzz d|dz|zz d|dz|zz }|d|dz|z|zz |dz d|dz|zz}|d||zz}|dkr�d|dz|dzzd|dz|dzzzd|dz|dzzz}|d|dz|z|zzd|dz|dzz|zz}|d|dz|zzd|dz|dzzz d|dz|zzd|dz|zz }|d|dz|z|zz |dzd|dz|zz }|d||zz}||_|S)z+ Computes zeta^(k)(s) off the line rTrrqrrrrrrUrVrWrXrYr?r|r@r[r~rArZrDrCr\r]rt)r!r�r�rfr�r#r(r�r^rrrEr'r%r�rrFrWr�)rr�r;r�rr�r_r`�M2rarrbrcrHrdrJ�s1�s2rrrKrLrMrNrOrerfs r � zeta_offlinerl�s8����I� �G�G�A�J�J�E� ���� � �A��C�H� �q�y�y� �I�I�c�!�f�f�c� "� "��� �I�I�a���h��a�� (� (����3�q��s�8�8�C��I�)F�)F� F�� �� � � �s�x�x��1�S�V�8� �%�%� %��� ��U�Q�Y�� �%��!� � � �s�x�x��1�S�V�8� �%�%� %��� �q�S����1�S�V�8�e�V�,�,� ,�S�Y�Y�s�1�v�v�e�C�i�-H�-H� H�� �s�1�u�:�:�D� �4��������A��!�A�c�g�g�a�j�j�L�!�!�G������b���b����*�3�r�6�!�8�A�:�5�6�6�6�y�@��B�H��'�8�$�$�G��!�Q�s�w�w�s�2�v�a�x�(�(�(��2�1�4�5�5�G� �C�G�G�C��!��G� � � �Y� &�q� (�C��C�H� �O�O�A�c�e�U�3�7�7�5�>�>�%9�:�:� ;� ;�E� �B� ���!�B�$���B��C�H��3��1�%�%�H�C���1�u�u�S�W�W�Q�r�!�t�_�_�S�W�W�Q��"��a�x�%8�%8�8�!�;�c�f�f�S�V�n�n�Q�>N�N�c��1�u�u�C�E�3�7�7�1�R��T�?�?�3�7�7�1�a��d�A�X�+>�+>�>�?��A�c��1�u�u�c�g�g�a��1��o�o�c�g�g�a��2��q��&9�&9�9�:�2�=�c��1�u�u�S�U�F�C�G�G�A�b��d�O�O�C�G�G�A�q��t�Q�h�,?�,?�?�@��C�c��C�H��x�x��5��!�!�H��A�v�v� ��V�H�S��V�O� #���A�v�v��1�v�g�a��A��h�s�l�"�� ��V�H�S�L� ���A�v�v���A��h�s�l�1�S��V�8�S�!�V�,�,�c�!�f�4�R��A��Y�s�]�B�� ��V�H�S�L� ���A�v�v��#�a�&�j��a����3�q�6��#�q�&��0��3�q�6��#��=�b��Q��i��m�K���C��A��J�s�N�3�&�&�s�1�v�-�a��A��h�s�l�:�� ��V�H�S�L� ���A�v�v���Q��i��Q����3�q�6� �#�q�&� 0�0��C��F��3��6�1A�A���3�s�1�v�:�c�>�#�%�%�c�#�a�&�j�#�q�&�&9�#�&=�=���#�c�!�f�*�S�.� ��C��F��3��6�!1�1�!�C��F�(�3�,�>�q��Q��x��|�K���"�S��V�)�C�-��#�#�C��F�*�2�c�!�f�9�S�=�8�� ��V�H�S�L� ���C�H� �Ir c ���|�d��|j|zz}|}|�d|z ��}|j}d|_|�|��dkrYd|�|�|��d|jzz ��z}|�t|����}njd|jz|�|��dz ztd|z ��d|�|��z zz}d|�|��z|z}|�|��dkr7d|�|�|��d|jzz ��z} nhd|�|��zd|jz|�|��dz zztd|z ��d|�|��z zz} dt|� |����z} |�|��} | || zz} d|z} td d|� | ��z|� | d d| zzz��| z��}td|� d | z��| z��}|� d| z��| z}||_|� |��}||_t|||��\}}d d |zz}d d |zz }|dkrNd |� d|��|� d|��zz|�|j��dz z }|dkr0d|� d|��|� d|��z z}|dkr0d|� d|��|� d|��zz}|dkr0d|� d|��|� d|��z z}||_|�|��}|dkr||dz|d|z z}|j}|dkr5||z|d|d|zzz||d|d|zzz|z z }|dkrs|d|dz|z|d|dzzz |dz ||dz|zzz}|d|dz|z|d|dzzz |dz ||dz|zz |z z}|dkr�d|dz|dzz|d|dzzz d|dz|zz |dz|dz|zz}|d|dz|z|zz|dz |d|zz|z|z}d|dz|dzz|d|dzzzd|dz|zz|dz|dz|zz}||d|dz|z|zz|dz|d|zz z|z }||z}|dk�r�d|dz|dzz|d|dzzzd |dz|dzzz}|d|dz|z|zz d|dz|dzz|zz d|dz|zz }|d|dz|z|zz d|dz|zzd|dz|zz d|dz|z|zz }||dz||dz|zz}d|dz|dzz|d|dzzzd |dz|dzzz}|d|dz|z|zzd|dz|dzz|zzd|dz|zz}|d|dz|z|zz d|dz|zzd|dz|zz d|dz|z|zz }||dz||dz|zz }||z||z z}||_|S)z8 Computes Z(w) and its derivatives off the line r?r�#rrrqrrr�gR���Q@g�?y�?y�?g��y���r~r�rBrCrD)r%r'r�r!r�r(r�r#r�rErrrrFrWr�)rr4r;r�rjrkr�r`r_rirr^r_rarbrHrdrJrr�pta�ptbrKrLrMrNrOrer'rf�zv2s r � z_offlinerss��� ������s�u�Q�w��A� �B� ���!�B�$���B���I��C�H� ���� � �q��� �s�x�x����� � �Q���Z�0�1�1� 1�� �H�H�S��W�W� � ��� �s�v�X�������Q�� '�#�a��d�)�)�c�#�'�'�"�+�+�o�*F� F�� �������_�Q� �� ���� � �q��� �s�x�x����� � �Q���Z�0�1�1� 1��� �������_�a���h�#�'�'�"�+�+�a�-�8� 8��Q�r�T���S����QS���_�9U� U�� �#�c�o�o�a� � � !� !�!�A� �8�8�A�;�;�D� ��B��<�D� �i�<�D��!�Q�s�w�w�q�z�z�\�3�7�7�4��A�a�C��=�#9�#9�$�#>�?�?�G��!�C�G�G�D��I�&�&�t�+�,�,�G� �'�'�!�D�&�/�/�$� �C��C�H� �O�O�A� � �E��C�H��3�q��#�#�H�C�� ��a��-�C� ��a��-�C��1�u�u�D�#�'�'�!�C�.�.�����3���7�8�3�6�6�#�&�>�>�!�;K�K�c��1�u�u�T�C�G�G�A�c�N�N�3�7�7�1�S�>�>�9�:�c��1�u�u�V�c�g�g�a��n�n�S�W�W�Q�s�^�^�;�<�c��1�u�u�V�c�g�g�a��n�n�S�W�W�Q�s�^�^�;�<�c��C�H��x�x����H��A�v�v� �c�!�f�_�S��V�H�_� ,�� ��A��A�v�v� �x�Z��Q���A��s� �*� +�A�s�1�v�c�!�f�S�j�/@�,A�(�,J� J���A�v�v� �r�#�a�&�y��}�S��V�C��F�]�2�3�q�6�9�!�C��F�(�3�,�F� G�� �"�S��V�)�C�-��A��s�A�v� �-�c�!�f�4�Q�s�1�v�X�c�\�A�8�K� K���A�v�v���Q��i��Q���s�1�v�c�1�f�}�,�Q�s�1�v�X�c�\�9�!�A�#�c�!�f�*�S�.�H���B�s�1�v�I�c�M�#�%�%�c�!�f�,�S��V�C�Z�7��:�8�C����A��h�s�A�v�o�c�!�f�S�!�V�m�+�A�c�!�f�H�S�L�8��1��S��V��C��G����r�#�a�&�y��}�S�(�(�#�a�&�0��Q����;�<�X�E�� ��W���A�v�v���A��h�s�A�v�o�c�!�f�S�!�V�m�+�a��A��h�s�A�v�o�=���#�c�!�f�*�S�.��$�$�R��A��Y�s�A�v�%5�c�%9�9�"�S��V�)�C�-�G���!�C��F�(�3�,�s�"�"�1�S��V�8�C�<�/��#�a�&��� �<�Q�s�1�v�X�c�\�#�=M�M���#�a�&�j��3�q�6��#��%����A��h�s�A�v�o�c�!�f�S�!�V�m�+�a��A��h�s�A�v�o�=���#�c�!�f�*�S�.��$�$�R��A��Y�s�A�v�%5�c�%9�9�"�S��V�)�C�-�G���!�C��F�(�3�,�s�"�"�1�S��V�8�C�<�/��#�a�&��� �<�Q�s�1�v�X�c�\�#�=M�M���#�a�&�j��3�q�6��#��%�� �c�\�#�h�,� &���C�H� �Ir c �v�|dkrt�|�|��}|�|��}|�|��}|dkr>|�|�|�|��|����}|S|dk}|rt |||��St|||��S)Nrrrq)r�rHr�r�r��rs_zetargrl)rr�� derivative�kwargs�re�imrP� critical_lines r rurubs����A�~�~�!�!� � � �A���A� ������B�#�'�'�!�*�*�R� �A�v�v� �H�H�S�[�[����!���j�9�9� :� :�����3�Y�M��0���a��,�,�,��C��J�/�/�/r c��|�|��}|�|��}|�|��}|dkrt|| |��S|dk}|rt |||��St |||��S)Nr)rHr�r��rs_zrRrs)rr4rvrxryrzs r r|r|qs��� � � �A���A� ������B�#�'�'�!�*�*�R� �A�v�v��C�!��Z�(�(�(��1�W�M��-��c�1�j�)�)�)���a��,�,�,r )r)�__doc__r��objectr� functionsrrCrTrcrkrr=rRrgrlrsrur|rr r �<module>r�sx��0�0�d � � � �(�(�(�(�(�f�(�(�(�������*t'�t'�t'�l .� .� .�6 � � � � � �D�D�D�D�L$%�#�q�q�q�q�h &�&�&�&�P<�<�<�<�|@�@�@�@�DF�F�F�F�P� 0� 0� 0��� 0�� -� -� -��� -� -� -r
Memory