403Webshell
Server IP : 35.80.110.71  /  Your IP : 216.73.216.221
Web Server : Apache/2.4.58 (Ubuntu)
System : Linux ip-172-31-21-44 6.17.0-1019-aws #19~24.04.1-Ubuntu SMP Tue Jun 23 18:53:06 UTC 2026 x86_64
User : ubuntu ( 1000)
PHP Version : 8.3.31
Disable Function : NONE
MySQL : OFF  |  cURL : ON  |  WGET : ON  |  Perl : ON  |  Python : OFF  |  Sudo : ON  |  Pkexec : OFF
Directory :  /lib/python3/dist-packages/mpmath/functions/__pycache__/

Upload File :
current_dir [ Writeable ] document_root [ Writeable ]

 

Command :


[ Back ]     

Current File : /lib/python3/dist-packages/mpmath/functions/__pycache__/zetazeros.cpython-312.pyc
�

�"`�x��$�dZddlmZmZd�Zd�Z		�d#d�Zd�Zd�Zd	�Z	d
Z
d�Zd�Zd
�Z
e�d$d��Zd�Zd�Zd�Zed��Zed��Z	gddg�d�ddg�d�ddg�d�ddg�d�ddg�d�d d!g�d�d"d#g�d�d$d%g�d�d&d'g�d�d(d)g�d�d*d+g�d�d,d-g�d�d.d/g�d�d0d1g�d2�d3d4g�d�d5d6g�d�d7d8g�d�d9d:g�d�d;d<g�d�d=d>g�d�d?d@g�d�dAdBg�d�dCdDg�d�dEdFg�d�dGdHg�d�dIdJg�d�dKdLg�d�dMdNg�d�dOdPg�d�dQdRg�d�dSdTg�d�dUdVg�d�dWdXg�d�dYdZg�d�d[d\g�d�d]d^g�d�d_d`g�d�dadbg�d�dcddg�d�dedfg�d�dgdhg�d�didjg�d�dkdlg�d�dmdng�d�dodpg�d�dqdrg�d�dsdtg�d�dudvg�d�dwdxg�d�dydzg�d�d{d|g�d�d}d~g�d�dd�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d2�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�dˑd�d�g�d�d�d�g�d�d�d�g�d�d�d�g�dԑd�d�g�d�d�d�g�dˑd�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�dԑd�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d�d�g�d�d��dg��d��d�dg�d��d�dg�d��d�dg�d��d�d	g�d��d
�dg�d��d�d
g�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d �d!g�d��d"�d#g�d��d$�d%g�d��d&�d'g�d��d(�d)g�d��d*�d+g�d��d,�d-g�d��d.�d/g�dˑ�d0�d1g�d��d2�d3g�d��d4�d5g�d��d6�d7g�d��d8�d9g�d��d:�d;g�d��d<�d=g�d��d>�d?g�d��d@�dAg�d��dB�dCg�d��dD�dEg�d��dF�dGg�d��dH�dIg�d��dJ�dKg�d��dL�dMg��dN��dO�dPg�d��dQ�dRg�d��dS�dTg��dU��dV�dWg�d��dX�dYg�d��dZ�d[g�d��d\�d]g�d��d^�d_g�d��d`�dag�d��db�dcg�d��dd�deg�d��df�dgg�d��dh�dig�d��dj�dkg�d��dl�dmg�d��dn�dog�d��dp�dqg�d��dr�dsg�d��dt�dug�d��dv�dwg�d��dx�dyg�d��dz�d{g�d��d|�d}g�dˑ�d~�dg�d��d��d�g�d��d��d�g�d��d��d�g��dU��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�dԑ�d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��dd�g�d��dĐd�g�d��dƐd�g�d��dȐd�g�d��dʐd�g�d��d̐d�g�d��dΐd�g�d��dАd�g�d��dҐd�g�d��dԐd�g�d��d֐d�g�d��dؐd�g�d��dڐd�g�dԑ�dܐd�g�d��dސd�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d��d�g�d��d��d�g�d��d��d�g��d��d��d�g�d��d��d�g�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d�d	g�d��d
�dg�d��d�d
g�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d2��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d �d!g�d��d"�d#g��dU��d$�d%g�d��d&�d'g�d��d(�d)g�d��d*�d+g�d��d,�d-g�d��d.�d/g�d��d0�d1g�d��d2�d3g�d��d4�d5g�d��d6�d7g�d��d8�d9g�d��d:�d;g�d��d<�d=g�d��d>�d?g�d��d@�dAg�d��dB�dCg�d��dD�dEg�d��dF�dGg�d��dH�dIg�d��dJ�dKg�d��dL�dMg�dԑ�dN�dOg�d��dP�dQg�d��dR�dSg�d��dT�dUg�d��dV�dWg�d��dX�dYg�d��dZ�d[g�d��d\�d]g�d��d^�d_g�d��d`�dag�d��db�dcg�d��dd�deg�d��df�dgg�d��dh�dig�d��dj�dkg�d��dl�dmg�d��dn�dog�d��dp�dqg�d��dr�dsg�d��dt�dug�d��dv�dwg�d��dx�dyg�d��dz�d{g�d��d|�d}g�d��d~�dg�d��d��d�g�d��d��d�g�d��d��d�g�d2��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g��dU��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g��d���d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g��d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��dÐd�g�d��dŐd�g�d��dǐd�g�d��dɐd�g�d��dːd�g�d��d͐d�g��dU��dϐd�g�d��dѐd�g�d��dӐd�g�d��dՐd�g�dˑ�dאd�g�d��dِd�g�d��dېd�g�d��dݐd�g�d��dߐd�g�dˑ�d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�dԑ�d�d�g�d��d�d�g�d��d�d�g�d��d�d�g�dԑ�d�d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��d�g�d��d��dg�d��d�dg�d��d�dg�d��d�dg�d��d�dg�d��d��d�g��d���d	�d
g��d���d�dg��d
��d�dg��d��d�dg��d
��d�dg��d���d�dg��d��d�dg��d���d�dg��d��d�dg��d
��d�d g��d��d!�d"g��d�Zy(%a�
The function zetazero(n) computes the n-th nontrivial zero of zeta(s).

The general strategy is to locate a block of Gram intervals B where we
know exactly the number of zeros contained and which of those zeros
is that which we search.

If n <= 400 000 000  we know exactly the Rosser exceptions, contained
in a list in this file. Hence for n<=400 000 000 we simply
look at these list of exceptions. If our zero is implicated in one of
these exceptions we have our block B.  In other case we simply locate
the good Rosser block containing our zero.

For n > 400 000 000 we apply the method of Turing, as complemented by
Lehman, Brent and Trudgian  to find a suitable B.
�)�defun�
defun_wrappedc�t�ttt�dz�D]�}td|zd}td|zd}||dz
ks�*|dz
|ks�3|j|�}|j|�}|jj|�}|jj|�}||z
dz
}	||z
}
td|zdz}|	||g||g||gfcS|dz
}t
||�\}}
}|g}|
g}|dkr?|dz}t
||�\}}
}|jd|�|jd|
�|dkr�?||z
dz
}	|dz
}t
||�\}}
}|j|�|j|
�|dkr=|dz
}t
||�\}}
}|j|�|j|
�|dkr�=|	||g||fS)z;for n<400 000 000 determines a block were one find our zero��r)	�range�len�_ROSSER_EXCEPTIONS�	grampoint�_fp�siegelz�compute_triple_tvb�insert�append)�ctx�n�k�a�b�t0�t1�v0�v1�my_zero_number�zero_number_block�pattern�t�v�T�V�ms                 �</usr/lib/python3/dist-packages/mpmath/functions/zetazeros.py�find_rosser_block_zeror#s���
�3�)�*�A�-�
.�=��
�Q�q�S�
!�!�
$��
�Q�q�S�
!�!�
$��
��1��W�1�Q�3�!�8����q�!�B����q�!�B�������$�B�������$�B��q�S��U�N� !�!���(��1��Q��/�G�"�Q�q�E�B�r�7�R��G�<�<�=�	
�!��A��s�A�&�E�A�a��	
��A�	
��A�
�a�%�	�Q���"�3��*���!�A�	����1�
�	����1�
�	�a�%�
�q�S��U�N�	�!��A��s�A�&�E�A�a���H�H�Q�K��H�H�Q�K�
�a�%�	�Q���"�3��*���!�A�	�����	�����	�a�%�

�Q�q�E�1�a�(�(�c�4�d}|dkDrd}|dkDrd}|dkDrd}|S)z(Precision needed to compute higher zeros�5i���?lh�]�F�@� �k�S�)r�wps  r"�wpzerosr-7s0��	�B��7�{�
���6�z�
���6�z�
��
�Ir$Nc���|��j}d}t|�}||k�r!||k�r|d}|d}	|g}
|	g}d}tdt|��D]�}||}
||}||	zdkDr#�j	||	z�}||z|
z|dzz}n||
zdz}|dkr;�j
j
|�}t|�|kr#�j
|�}n�j
|�}|	|zdkr|dz
}|
j|�|j|�||}||zdkr|dz
}|
j|
�|j|�|
}|}	��|
}|}|dz
}|tkDr�|dkDr�|dz|k(r�d}d}d}tdt|��D]*}||||dz
z
}||kDr|}|}|}�||ks�#||kDs�)|}�,|d|zkDrt�fd�}||dz
}||}�j|||fddd�	�}�j
|�}	||kr4||kr/|	||zdkr$|j||�|j||	�t|�}||kr||kr��||k(rd
}nd}|||fS)z^Separate the zeros contained in the block T, limitloop
    determines how long one must searchrrr�
�c�*���j|d��S)Nr��
derivative)�rs_z��xrs �r"�<lambda>z)separate_zeros_in_block.<locals>.<lambda>ys���c�h�h�q�A�h�6�r$�illinoisF)�solver�verify�verboseT)�inf�count_variationsrr	�sqrtrr
�absr�ITERATION_LIMIT�findrootr)rrrr �	limitloop�fp_tolerance�
loopnumber�
variationsrr�newT�newVr�b2�u�alphar�w�dtMax�dtSec�kMax�k1�dt�frrr�	separateds`                           r"�separate_zeros_in_blockrSBs�������G�G�	��J�!�!�$�J��*�*��Y�1F�
�a�D��
�a�D���s���s���
��q��Q���	�A��1��B��!��A��!��A������1��
���!�G�B�J��q��)���r�T�1�H���b� ��G�G�O�O�A�&���q�6�,�&����A��A��+�+�a�.����s�1�u��a��
��K�K��N��K�K��N��!��A���s�A�v��a��
��K�K��O��K�K��N��A��A�1	�2
�����Q��
��o�%�*�Q�,�:�a�<�IZ�;Z��E��E��D��A�c�!�f�o�
���r�U�1�R��T�7�]����:��D�!�E��E��%�x�R��Y��E�
��Q�u�W�}�6���T�!�V�9���t�W���,�,�q�B�r�7�J�e�UZ�,�[���K�K��N���q�D�q��t�!�A�d�G�)�A�+��H�H�T�!�$��H�H�T�!�$�%�a�(�
�o
�*�*��Y�1F�p�&�&��	��	�
�a���r$c����d}|d}tdt|��D]!}||}	||	zdkr|dz
}||k(r|}
|}|	}|	}�#|
}
||
dz
}|�_t|�j	|�z�}d�j|�z}�jdzg}d}|dd|zkDr&|dz
}|ddzdzd|zzg|z}|dd|zkDr�&|d|z�_�j
�fd�||
fdd�	�}�jd
|�}|ddD]U}||z�_|�j|��j|d��zz
}�jd
�j|��}�W�j|�S)
zPIf we know which zero of this block is mine,
    the function separates the zerorr�rr0c�&���j|�S)N�r
r5s �r"r7z"separate_my_zero.<locals>.<lambda>�s���c�k�k�!�n�r$r8F)r9r;��?Nr2)
rr	�precr-�log�magrA�mpc�zeta�im)rrrrr rYrErrr�k0�leftv�rightvrr�wpz�guard�precs�index�r�z�znews`                     r"�separate_my_zerori�s�����J�	
�1��B�
�1�S��V�_���
�q�T��
�b�5�1�9���N�J��^�+�������
���
�2��B�	
�2�a�4��B��C�H�
�.�����!8�8�
9�C�
�c�g�g�n�%�%�E�
�X�X�a�Z�L�E�
�E�
��(�Q�s�U�
�
��	���q��Q���!�!�E�'�)�*�U�2����(�Q�s�U�
��Q�x�%��C�H����,�r�"�g�z�SX��Y�A�	�g�g�c�!�n�A��a�b�	�$���%�<����3�8�8�A�;����!���!:�:�:��

�'�'�#�c�f�f�T�l�
#��$��6�6�!�9�r$c���|dkry|j|dz
�}|jj|�}d|dzzd|zz}d|dzzd|zz}|jt	||��}t|�}|S)aThe number of good Rosser blocks needed to apply
    Turing method
    References:
    R. P. Brent, On the Zeros of the Riemann Zeta Function
    in the Critical Strip, Math. Comp. 33 (1979) 1361--1372
    T. Trudgian, Improvements to Turing Method, Math. Comp.i��
r�dg�HP�x?g{�G�z�?ga��+ei?g)\��(�?)rr�ln�ceil�min�int)rr�g�lg�brent�trudgian�Ns       r"�sure_number_blockru�s���	�7�{���
�
�a��e��A�	�����A��B��R��U�N�D��G�#�E���A��~�t�B�w�&�H�����U�8�$�%�A��A��A��Hr$c���|j|�}|jj|�}|jt	|��|j|�dz
kr|j|�}|d|zz}|||fS)N�-���)rrr
r[r?)rrrrrs     r"rr�sg���
�
�a��A��������A�
�w�w�s�1�v��s�w�w�q�z�"�}�$��K�K��N��	�2��'�	�A��Q�q�5�Lr$rUc�~�t||�}d}|dz
}t||�\}}}|g}	|g}
|dkr=|dz
}t||�\}}}|	j|�|
j|�|dkr�=|g}|g}|g}
|d|zk�r
|dz
}t||�\}}}|j|�|
j|�|dkr=|dz
}t||�\}}}|j|�|
j|�|dkr�=|j|�t|�dz
}t	||||
t
|��\}}}|	j
�|	j|�|
j
�|
j|�|r|dz
}nd}|g}|g}
|d|zkr��
d}|dz
}t||�\}}}|	jd|�|
jd|�|dkr?|dz}t||�\}}}|	jd|�|
jd|�|dkr�?|jd|�|g}|g}
|d|zkr�|dz}t||�\}}}|jd|�|
jd|�|dkr?|dz}t||�\}}}|jd|�|
jd|�|dkr�?|jd|�t|�dz
}t	||||
t
|��\}}}|j
�||	z}	|j
�||
z}
|r|dz
}nd}|g}|g}
|d|zkr��|d|z}t|�}||d|zz
dz
}t||�\}}}|	j|�}t||�\}}}|	j|�}|	||dz}|
||dz}
||z
}t	||||
t
|��\}}}|r||z
dz
||g||fS||}t|�}|||z
dz
}t||�\}}} |	j|�}!t||�\}"}#}$|	j|"�}%|	|!|%dz}|
|!|%dz}
||z
dz
||g||
fS)zTo use for n>400 000 000rrr�rBrC)
rurrr	rSr@�pop�extendrre)&rrrC�sb�number_goodblocks�m2rrr�Tf�Vf�
goodpointsrr �zn�A�BrRrfrq�s�tr�vr�br�ar�ts�vs�bs�as1�q�tq�vq�bq�aq�tt�vt�bt�ats&                                      r"�search_supergood_blockr��s���	�3��	"�B���	
�1��B� ��b�)�G�A�q�!�
��B�
��B�
�a�%�
�a���"�3��+���!�A�
�	�	�!��
�	�	�!��	�a�%�
��J�	
��A�	
��A�
�a��d�
"�
�a���$�S�"�-���1�a�	�����	������!�e��!�G�B�&�s�B�/�E�A�a��
�H�H�Q�K�
�H�H�Q�K�	�!�e�
	���"��
��V�A�X��"�3��A�q�O�)�+�	��1�i�	����
�	�	�!��
����
�	�	�!�����"�� !��
�C��
�C��1�a��d�
"�4��	
�1��B� ��b�)�G�A�q�!��I�I�a��N��I�I�a��N�
�a�%�
�a���"�3��+���!�A�
�	�	�!�A��
�	�	�!�A��	�a�%�
���a���	
��A�	
��A�
�a��d�
"�
�a���$�S�"�-���1�a�	����1�
�	����1�
��!�e��!�G�B�&�s�B�/�E�A�a��
�H�H�Q�q�M�
�H�H�Q�q�M�	�!�e�
	���!�B��
��V�A�X��"�3��A�q�O�Zf�g�	��1�i�	����
�r�T��	����
�r�T�����"�� !��
�C��
�C��/�a��d�
"�0	�1�R�4��A�	�Z��B��2�a��d�7�1�9��A�#�C��+�J�B��B�	���"��B�#�C��+�J�B��B�
�(�(�2�,�C�
�2�c�!�e��A�
�2�c�!�e��A�	
�1��B��s�B�q��_�S_�`��A�q�)���!��A��q��e�A�a� � ��2��A�	�Z��B��2�b�5��7��A�#�C��+�J�B��B�	���"��B�#�C��+�J�B��B�	���"��B�
�2�b��d��A�
�2�b��d��A�
�a�C��E�1�Q�%��!��r$c�p�d}|d}tdt|��D]}||}||zdkr|dz
}|}�|S�Nrr)rr	)r �count�voldr�vnews     r"r=r=2sS��
�E��Q�4�D�
�1�c�!�f�
�����t����9�q�=��A�I�E���	�
�Lr$c��d}|d}|d}t||�\}}}	d}
d}t|dz|dz�D]�}t||�\}
}}t|�}|
|kr||
|
kr|
dz
}
|
|kr	||
|
kr�|||
}|j|�|j	d|�t|�}|d|zz}|dkDr|dz}|
}|
||}	}}��|dd}|S)N�(rrz%sz)(rx)rrr	rrr=)r�blockrr rrrrr�b0rr_rrr�b1�lgT�Lr�s                   r"�pattern_constructr�<s	���G�
�a��A�
�a��A�!�#�q�)�H�B�r�"�	�A�	
�B�
�1�Q�3�q��s�^�
��%�c�1�-���2�b�
��V���3�w�Q�q�T�R�Z�
��F�A��3�w�Q�q�T�R�Z�
�b��G��	�����	����2�� ��#���T�E�\�*��
��6���n�G�
���b��b�2��
��c�r�l�G��Nr$c�.�t|�}|dkr |j|�j�S|dk(rtd��|j}	t||�\}}||_|dkrt
||�\}}}	}
nt|||�\}}}	}
|d|dz
}t|||	|
|j|��\}	}
}|rt|||	|
�}
t||�}t||||	|
|�}|jd|�}||_|r|�}|r|||
fS|S#||_wxYw)a�
    Computes the `n`-th nontrivial zero of `\zeta(s)` on the critical line,
    i.e. returns an approximation of the `n`-th largest complex number
    `s = \frac{1}{2} + ti` for which `\zeta(s) = 0`. Equivalently, the
    imaginary part `t` is a zero of the Z-function (:func:`~mpmath.siegelz`).

    **Examples**

    The first few zeros::

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> zetazero(1)
        (0.5 + 14.13472514173469379045725j)
        >>> zetazero(2)
        (0.5 + 21.02203963877155499262848j)
        >>> zetazero(20)
        (0.5 + 77.14484006887480537268266j)

    Verifying that the values are zeros::

        >>> for n in range(1,5):
        ...     s = zetazero(n)
        ...     chop(zeta(s)), chop(siegelz(s.imag))
        ...
        (0.0, 0.0)
        (0.0, 0.0)
        (0.0, 0.0)
        (0.0, 0.0)

    Negative indices give the conjugate zeros (`n = 0` is undefined)::

        >>> zetazero(-1)
        (0.5 - 14.13472514173469379045725j)

    :func:`~mpmath.zetazero` supports arbitrarily large `n` and arbitrary precision::

        >>> mp.dps = 15
        >>> zetazero(1234567)
        (0.5 + 727690.906948208j)
        >>> mp.dps = 50
        >>> zetazero(1234567)
        (0.5 + 727690.9069482075392389420041147142092708393819935j)
        >>> chop(zeta(_)/_)
        0.0

    with *info=True*, :func:`~mpmath.zetazero` gives additional information::

        >>> mp.dps = 15
        >>> zetazero(542964976,info=True)
        ((0.5 + 209039046.578535j), [542964969, 542964978], 6, '(013111110)')

    This means that the zero is between Gram points 542964969 and 542964978;
    it is the 6-th zero between them. Finally (01311110) is the pattern
    of zeros in this interval. The numbers indicate the number of zeros
    in each Gram interval (Rosser blocks between parenthesis). In this case
    there is only one Rosser block of length nine.
    rzn must be nonzero���rrzrX)ro�zetazero�	conjugate�
ValueErrorrY�comp_fp_tolerancer#r�rSr<r��maxrir\)rr�info�round�	wpinitialrbrCrr�rr rrRrrYrrs                 r"r�r�TsL��x	�A��A��1�u��|�|�Q�B��)�)�+�+��A�v��,�-�-����I��-�c�1�5���\�����y�=�
#�C��
+�
(�N�E�1�a�$�C��L�
9�
(�N�E�1�a�!�!�H�U�1�X�-��1�#�7H�!�Q��g�g�L�:���1�i��'��E�!�A�6�G��9�c�"���S�.�2C�A�a��M���G�G�C��N������
�2����%��w�/�/����
���s
�B'D�	Dc��|dkDrd|j|d�z}nd}|j}	|xj|z
c_t|j|�|jz�}||_|S#||_wxYw)Nl �a$r0r/r)rZrYro�siegeltheta�pi)rrr,rY�hs     r"�
gram_indexr��sp���6�z�
�s�w�w�q�"�~�
��
���8�8�D�����B��������"�3�6�6�)�*������I�����s�<A0�0	A9c��d}|d}|d}|d}d}||kr$||}	||	zdkr|dz
}|	}|dz
}||}||kr�$|j|�}
|
|zdkr|dz
}|Sr�rW)rrrr r�r��told�tnewrr�rs           r"�count_tor��s���
�E��Q�4�D��Q�4�D��Q�4�D�	�A�
��(���t����9�q�=��Q�J�E���	�Q�����t��
��(�	���A��A���v��z�
��
���Lr$c�t�t||j|�z�}|dkrd}||fS|dkrd}||fSd}||fS)Ni/hYg����Mb@?r)g�������?rk)r-rZ)rrrbrCs    r"r�r��s^��
�!�C�G�G�A�J�,�
�C��8�|���
����	
�f�������������r$c�<�|dkryt||�}t|j|��}|j}t	||�\}}||_|j|�}|dk(r|dkry|dk(r|dkDry|dzdkrt
||dz�}nt||dz|�}|d\}	}
|
|	z
dk(r(|dd}||zdkDr||_|dzS||_|dzS|\}}
}}|
|	z
}t|||||j|��\}}}t||||�}||_||	zdzS)	a
    Computes the number of zeros of the Riemann zeta function in
    `(0,1) \times (0,t]`, usually denoted by `N(t)`.

    **Examples**

    The first zero has imaginary part between 14 and 15::

        >>> from mpmath import *
        >>> mp.dps = 15; mp.pretty = True
        >>> nzeros(14)
        0
        >>> nzeros(15)
        1
        >>> zetazero(1)
        (0.5 + 14.1347251417347j)

    Some closely spaced zeros::

        >>> nzeros(10**7)
        21136125
        >>> zetazero(21136125)
        (0.5 + 9999999.32718175j)
        >>> zetazero(21136126)
        (0.5 + 10000000.2400236j)
        >>> nzeros(545439823.215)
        1500000001
        >>> zetazero(1500000001)
        (0.5 + 545439823.201985j)
        >>> zetazero(1500000002)
        (0.5 + 545439823.325697j)

    This confirms the data given by J. van de Lune,
    H. J. J. te Riele and D. T. Winter in 1986.
    g%f���D,@rrxrrr�r0rz)r�ro�floorrYr�r
r#r�rSr<r�)rrr6rr�rbrCr�Rblock�n1�n2rrr�rr rrRrs                   r"�nzerosr��s\��J	�����3���A��C�I�I�a�L��A����I�)�#�q�1��C���C�H����A��A��B�w�1�q�5��	
�b��Q��U����s�Y��'��Q�q�S�1��'��Q�q�S�,�?��
�A�Y�F�B��	�"�u��z��1�I�a�L���Q�3��7� �C�H��Q�3�J� �C�H��Q�3�J�!'��N�5�!�Q��2���-�c�.?��A�8;���9E�G�O�A�q�)�	��a��A��A��C�H��R�4��6�Mr$c�h�|j|�dz
|j|�|jzz
S)aw
    Computes the function
    `S(t) = \operatorname{arg} \zeta(\frac{1}{2} + it) / \pi`.

    See Titchmarsh Section 9.3 for details of the definition.

    **Examples**

        >>> from mpmath import *
        >>> mp.dps = 15; mp.pretty = True
        >>> backlunds(217.3)
        0.16302205431184

    Generally, the value is a small number. At Gram points it is an integer,
    frequently equal to 0::

        >>> chop(backlunds(grampoint(200)))
        0.0
        >>> backlunds(extraprec(10)(grampoint)(211))
        1.0
        >>> backlunds(extraprec(10)(grampoint)(232))
        -1.0

    The number of zeros of the Riemann zeta function up to height `t`
    satisfies `N(t) = \theta(t)/\pi + 1 + S(t)` (see :func:nzeros` and
    :func:`siegeltheta`)::

        >>> t = 1234.55
        >>> nzeros(t)
        842
        >>> siegeltheta(t)/pi+1+backlunds(t)
        842.0

    r)r�r�r�)rrs  r"�	backlundsr�!s.��H�:�:�a�=��?�3�?�?�1�-�c�f�f�4�4�4r$i���i���z(00)3ia��id��i��i��z3(00)i��=i��=i�oi�oiK��iN��i�'�i�'�iD�iD�i�5�i�5�i�"i�"i͜JiМJi+�di.�diOe�iRe�i6٧i:٧z(00)40i�(i�(iR4yiU4yi�ýi�ýi�e�i�e�i���i���i2�i5�i^RiaRi�(i�(iL�=iO�=i�Gi"�Giv�i"v�i���i���i��i��i	Ui	Ui��_i��_i�ei�ei��hi��hi$.�i'.�i�;�i�;�i��i��i�~)i�~)i�<i
�<i�@i�@iZ�Di]�Dip�Nis�Ni�b�i�b�i(�i(�i���i���ivx�iyx�i���i���ik��in��i7�/i:�/i�(7i�(7io�Eir�EiOeIiReIi��pi��pi5�i5�i��i��iEŵiHŵi:��i=��i#��i&��i�	i�	i.�	i1�	i��	iÁ	i&�&	i)�&	iĺ?	iǺ?	iϴB	iҴB	i��_	i��_	i�_	i�_	i�&g	i�&g	itqo	iwqo	i�	�	i�	�	ic��	if��	i�=�	i�=�	i�v�	i�v�	i���	i���	id��	ig��	iL
�	iO
�	i�;
i�;
i�0
i�0
iS�'
iV�'
i�,
i�,
i�@
i�@
i�1T
i�1T
i�[
i�[
i�`
i�`
i�`c
i�`c
i��f
i��f
i�!y
i�!y
i�Պ
i�Պ
i��
i��
i
��
i��
iWC�
iZC�
i>h�
iAh�
i[b�
i^b�
i���
i���
z22(00)iǂ�
iʂ�
it4diw4di��di��di��fi�fz(00)22i�yi�yi���i���iךiךi�׬i�׬is�is�i���i���i�
�i�
�iZ��i]��i�t�i�t�i��i��i!�i!�i�{�i�{�i"�i%�i�yi�yi{�i~�i�ii�ii=&i=&i�|Ei�|Eiw�Liz�Li�Yi�YioYirYi��]i��]z3(010)i$�`i'�`i��fi��fiˁgi΁gib�|ie�|i�`�i�`�i3��i6��i���i���i߀�i��i(>�i+>�iO�iR�i��i��i���iÀ�i��i��i��

i��

ie�

ih�

ij�
im�
i.W
i1W
i0`
i3`
i)T/
i,T/
i	�S
i�S
i�X
i�X
i]�Y
i`�Y
i<Ng
i@Ng
i��m
i��m
i��
i��
i*�
i-�
iw�
iw�
i_�
ib�
i�٤
i�٤
i��
i��
i=g�
i@g�
i�#�
i�#�
i���
i���
i`~�
ic~�
ib�
ie�
i?�iB�i<�i?�i�7(i�7(z04(00)i//Ni2/Niz|Qi}|Qi�Nci�Ncz(010)3i˾eiξei��hi��hiD�hiG�hihʇikʇi��i��i��i��i��i��ii��il��i�Цi�ЦiS;�iV;�i��i��i%�i(�i���i���i-�i0�i'i*i55i55i�Z7i�Z7i"Mi"Mi��`i��`iS�giW�gi��ti�ti?��iB��i�v�i�v�i���i���i"A�i%A�i�k�i�k�i#��i&��i��i��i��i��i�0�i�0�i[��i^��iX�i[�i�R%i�R%iO'iR'iz�7i}�7iaI?idI?i!�Wi$�WiKYiNYi��hi��hi�}li�}lio�qir�qih�rik�ri
y�iy�i���i���i��i��i(�i(�i$
�i'
�im(�ip(�i[�i^�iѓ�iԓ�iqx�itx�i@Q�iCQ�i�S�i�S�iC�iF�ie,�ih,�iw)�iz)�iߥi�i8�i;�i��i��i��i��i��i��ic�!if�!i��'i��'i��(i��(i�:6i�:6i��Ji��Ji��Oi��Oi]�^i`�^iq�sit�sie�xih�xi&�|i)�|i�"�i�"�i�b�i�b�i��i��i��i���iA��iD��i��i��i�a�i�a�i�i�i��i��i�i�i�1!i�1!iʺ*iκ*iP�2iS�2i�.Ii�.Ii�;Mi�;Mi��Si��Si��di��dij�nim�ni�i�iɭ�i̭�iI�iL�i���i���i!�i!�i�ڜi�ڜi�k�i�k�i�*�i�*�i��i��i��i��ii�il�i'D�i*D�i��i��i~�i��i)��i,��iF�iI�iK\iN\izB-i}B-i*Vi*Vi��mi��mi��xi��xiPyiSyi׸�iڸ�i$Îi'Îiĩ�iǩ�i�ϐi�ϐi���i���i��i��im�ip�i�5�i�5�i�%�i�%�i>K�iAK�i{~�i~~�iV�iY�iH�iH�i���i���i�*�i�*�i�i�i�i�i3�i6�i�i�i�i�i�Y#i�Y#i��#i��#i��$i��$iN�giQ�gid�lig�li�ůi�ůi�p�i�p�i�\�i�\�i׊�iڊ�iN��iQ��i��i��i��i��i�A�i�A�i	.�i.�i�2�i�2�i�p�i�p�i�i�i��i��i�ii'�*i*�*iZ35i^35i�<7i�<7i'�=i*�=i�Ai�Ai��Ei�Ei�Ei�Ei�PUi�PUi�{Xi�{Xi�fi�fi���i���in��ir��i�1�i�1�i@�i@�i���i���i?�i?�i��i��i���i���i*��i-��i$�i$�z04(010)i9�i<�i��i��iImiLmi�si�siІ4iԆ4iZ�6i]�6i�;i�;i�Bi�Bi��ci��ci�
vi�
vi�dzi�dzi��}i��}i�8�i�8�i�5�i�5�ix��i{��i�ʯi�ʯiGS�iJS�i���i���iQN�iTN�i�p�i�p�i���i���iz��i~��i���i���i�6�i�6�i�}i�}i�Vi�Vi�/i�/iMo%iPo%i��*i��*i�-i�-i�A@i�A@iN�AiR�AiG�CiJ�CizXi�zXi'\i*\i�\i�\i�'li�'li1�li4�lifrifriTKxiWKxi��|i��|i꣗i�i�љi�љi���i���i��i��i��i��i2j�i5j�iۣi�i	]i]z032(00)i΄c,iӄc,z(010)40i�ӹ,i�ӹ,i��9i��9i�%�:i�%�:z(00)410i�w�;i�w�;i�t�?i�t�?z(00)230iq�Hiv�Hi}�Ji}�Ji��Qi��Q)NN)FT)�__doc__�	functionsrrr#r-rSrirurr@r�r=r�r�r�r�r�r�r�r
r+r$r"�<module>r�s�.���",�!)�F	�EI��D�L#�J
�"���f�P��0�Y��Y�v��$��E��E�N�#5��#5�L&�^C�(�H��C�w�C�	�8��C��C�	�8��C��C�
�8��C��C�
�8��	C��	C�

�8��C�
�C�
�8��
C��
C�
�8��C��C�
�8��C��C�
�8��C��C�
�8��C��C�
�8��C��C�
�8��C��C�
�8��C��C�
�8��C��C�
�8��C��C� 
�8��!C� �!C�"
�8��#C�"�#C�$
�8��%C�$�%C�&
�8��'C�&�'C�(
�8��)C�(�)C�*
�8��+C�*�+C�,
�8��-C�,�-C�.
�8��/C�.�/C�0
�8��1C�0�1C�2�I��3C�2 �3C�4�I��5C�4 �5C�6�I��7C�6 �7C�8�I��9C�8 �9C�:�I��;C�: �;C�<�I��=C�< �=C�>�I��?C�> �?C�@�I��AC�@ �AC�B�I��CC�B �CC�D�I��EC�D �EC�F�I��GC�F �GC�H�I��IC�H �IC�J�I��KC�J �KC�L�I��MC�L �MC�N�I��OC�N �OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T �UC�V�I��WC�V �WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^ �_C�`�I��aC�` �aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f �gC�h�I��iC�h �iC�j�I��kC�j �kC�l�I��mC�l �mC�n�I��oC�n �oC�p�I��qC�p �qC�r�I��sC�r �sC�t�I��uC�t �uC�v�I��wC�v �wC�x�I��yC�x �yC�z�I��{C�z �{C�|�I��}C�| �}C�~�I��C�~ �C�@�I��AC�@ �AC�B�I��CC�B �CC�D�I��EC�D �EC�F�I��GC�F �GC�H�I��IC�H �IC�J�I��KC�J!�KC�L�I��MC�L �MC�N�I��OC�N �OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T �UC�V�I��WC�V �WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^ �_C�`�I��aC�` �aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f �gC�h�I��iC�h �iC�j�I��kC�j �kC�l�I��mC�l �mC�n�I��oC�n �oC�p�I��qC�p �qC�r�I��sC�r!�sC�t�I��uC�t �uC�v�I��wC�v �wC�x�I��yC�x �yC�z�I��{C�z!�{C�|�I��}C�| �}C�~�I��C�~!�C�@�I��AC�@ �AC�B�I��CC�B �CC�D�I��EC�D �EC�F�I��GC�F �GC�H�I��IC�H �IC�J�I��KC�J �KC�L�I��MC�L �MC�N�I��OC�N �OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T �UC�V�I��WC�V!�WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^ �_C�`�I��aC�` �aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f!�gC�h�I��iC�h �iC�j�I��kC�j �kC�l�I��mC�l �mC�n�I��oC�n �oC�p�I��qC�p �qC�r�I��sC�r �sC�t�I��uC�t �uC�v�I��wC�v �wC�x�I��yC�x �yC�z�I��{C�z �{C�|�I��}C�| �}C�~�I��C�~ �C�@�I��AC�@ �AC�B�I��CC�B �CC�D�I��EC�D �EC�F�I��GC�F �GC�H�I��IC�H �IC�J�I��KC�J �KC�L�I��MC�L �MC�N�I��OC�N �OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T!�UC�V�I��WC�V �WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^ �_C�`�I��aC�` �aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f �gC�h�I��iC�h �iC�j�I��kC�j �kC�l�I��mC�l �mC�n�I��oC�n �oC�p�I��qC�p �qC�r�I��sC�r!�sC�t�I��uC�t �uC�v�I��wC�v �wC�x�I��yC�x!�yC�z�I��{C�z �{C�|�I��}C�| �}C�~�I��C�~ �C�@�I��AC�@ �AC�B�I��CC�B �CC�D�I��EC�D �EC�F�I��GC�F �GC�H�I��IC�H �IC�J�I��KC�J �KC�L�I��MC�L �MC�N�I��OC�N �OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T �UC�V�I��WC�V �WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^ �_C�`�I��aC�`!�aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f �gC�h�I��iC�h!�iC�j�I��kC�j �kC�l�I��mC�l �mC�n�I��oC�n �oC�p�I��qC�p �qC�r�I��sC�r �sC�t�I��uC�t �uC�v�I��wC�v �wC�x�I��yC�x �yC�z�I��{C�z �{C�|�I��}C�| �}C�~�I��C�~ �C�@�I��AC�@ �AC�B�I��CC�B �CC�D�I��EC�D �EC�F�I��GC�F �GC�H�I��IC�H �IC�J�I��KC�J �KC�L�I��MC�L �MC�N�I��OC�N!�OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T �UC�V�I��WC�V �WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^ �_C�`�I��aC�` �aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f �gC�h�I��iC�h �iC�j�I��kC�j �kC�l�I��mC�l �mC�n�I��oC�n �oC�p�I��qC�p �qC�r�I��sC�r �sC�t�I��uC�t �uC�v�I��wC�v �wC�x�I��yC�x �yC�z�I��{C�z �{C�|�I��}C�| �}C�~�I��C�~!�C�@�I��AC�@ �AC�B�I��CC�B �CC�D�I��EC�D �EC�F�I��GC�F �GC�H�I��IC�H �IC�J�I��KC�J �KC�L�I��MC�L �MC�N�I��OC�N �OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T �UC�V�I��WC�V �WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^!�_C�`�I��aC�` �aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f �gC�h�I��iC�h �iC�j�I��kC�j �kC�l�I��mC�l �mC�n�I��oC�n �oC�p�I��qC�p �qC�r�I��sC�r �sC�t�I��uC�t �uC�v�I��wC�v �wC�x�I��yC�x!�yC�z�I��{C�z �{C�|�I��}C�| �}C�~�I��C�~ �C�@�I��AC�@ �AC�B�I��CC�B �CC�D�I��EC�D �EC�F�I��GC�F!�GC�H�I��IC�H �IC�J�I��KC�J �KC�L�I��MC�L �MC�N�I��OC�N �OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T �UC�V�I��WC�V �WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^ �_C�`�I��aC�` �aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f �gC�h�I��iC�h �iC�j�I��kC�j �kC�l�I��mC�l �mC�n�I��oC�n �oC�p�I��qC�p!�qC�r�I��sC�r �sC�t�I��uC�t �uC�v�I��wC�v �wC�x�I��yC�x �yC�z�I��{C�z �{C�|�I��}C�| �}C�~�I��C�~ �C�@	�I��A	C�@	 �A	C�B	�I��C	C�B	 �C	C�D	�I��E	C�D	 �E	C�F	�I��G	C�F	 �G	C�H	�I��I	C�H	 �I	C�J	�I��K	C�J	 �K	C�L	�I��M	C�L	 �M	C�N	�I��O	C�N	 �O	C�P	�I��Q	C�P	 �Q	C�R	�I��S	C�R	 �S	C�T	�I��U	C�T	 �U	C�V	�I��W	C�V	 �W	C�X	�I��Y	C�X	 �Y	C�Z	�I��[	C�Z	 �[	C�\	�I��]	C�\	 �]	C�^	�I��_	C�^	 �_	C�`	�I��a	C�`	 �a	C�b	�I��c	C�b	 �c	C�d	�I��e	C�d	 �e	C�f	�I��g	C�f	 �g	C�h	�I��i	C�h	!�i	C�j	�I��k	C�j	 �k	C�l	�I��m	C�l	 �m	C�n	�I��o	C�n	 �o	C�p	�I��q	C�p	 �q	C�r	�I��s	C�r	 �s	C�t	�I��u	C�t	 �u	C�v	�I��w	C�v	 �w	C�x	�I��y	C�x	 �y	C�z	�I��{	C�z	 �{	C�|	�I��}	C�|	!�}	C�~	�I��	C�~	 �	C�@
�I��A
C�@
 �A
C�B
�I��C
C�B
 �C
C�D
�I��E
C�D
 �E
C�F
�I��G
C�F
 �G
C�H
�I��I
C�H
 �I
C�J
�I��K
C�J
 �K
C�L
�I��M
C�L
"�M
C�N
�I��O
C�N
 �O
C�P
�I��Q
C�P
 �Q
C�R
�I��S
C�R
 �S
C�T
�I��U
C�T
 �U
C�V
�I��W
C�V
!�W
C�X
�I��Y
C�X
 �Y
C�Z
�I��[
C�Z
 �[
C�\
�I��]
C�\
 �]
C�^
�I��_
C�^
 �_
C�`
�I��a
C�`
 �a
C�b
�I��c
C�b
 �c
C�d
�I��e
C�d
 �e
C�f
�I��g
C�f
 �g
C�h
�I��i
C�h
 �i
C�j
�I��k
C�j
 �k
C�l
�I��m
C�l
 �m
C�n
�I��o
C�n
 �o
C�p
�I��q
C�p
!�q
C�r
�I��s
C�r
 �s
C�t
�I��u
C�t
 �u
C�v
�I��w
C�v
 �w
C�x
�I��y
C�x
!�y
C�z
�I��{
C�z
 �{
C�|
�I��}
C�|
 �}
C�~
�I��
C�~
 �
C�@�I��AC�@ �AC�B�I��CC�B!�CC�D�I��EC�D �EC�F�I��GC�F �GC�H�I��IC�H �IC�J�I��KC�J �KC�L�I��MC�L!�MC�N�I��OC�N �OC�P�I��QC�P �QC�R�I��SC�R �SC�T�I��UC�T!�UC�V�I��WC�V �WC�X�I��YC�X �YC�Z�I��[C�Z �[C�\�I��]C�\ �]C�^�I��_C�^ �_C�`�I��aC�` �aC�b�I��cC�b �cC�d�I��eC�d �eC�f�I��gC�f �gC�h�I��iC�h �iC�j�I��kC�j �kC�n�I��oC�n"�oC�p�I��qC�p"�qC�r�I��sC�r"�sC�t�I��uC�t"�uC�v�I��wC�v"�wC�x�I��yC�x"�yC�z�I��{C�z"�{C�|�Z��}C�|$�}C�~�Z��C�~$�C�@�Z��AC�@$�AC�B�Z��CC�B$�CC�D�Z��EC�D$�EC�r$

Youez - 2016 - github.com/yon3zu
LinuXploit