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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_wrappedcCstttdD]T}td|d}td|d}||dkr\|d|kr\||}||}|j|}|j|}||d} ||} td|d} | ||g||g||gfSq|d}t||\} } }| g}| g}|dkr|d8}t||\} } }|d| |d| |dkss||d} |d}t||\} } }|| || |dkr|d7}t||\} } }|| || |dks| ||g||fS)z;for n<400 000 000 determines a block were one find our zeror) rangelen_ROSSER_EXCEPTIONS grampoint_fpsiegelzcompute_triple_tvbinsertappend)ctxnkabt0t1v0v1my_zero_numberzero_number_blockpatterntvTVmr n/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/mpmath/functions/zetazeros.pyfind_rosser_block_zerosF            r"cCs,d}|dkrd}|dkrd}|dkrd}|S)z(Precision needed to compute higher zeros5i?lh]F@ kSr )rwpr r r!wpzeros7sr)Ncs|durj}d}t|}||kr>||kr>|d}|d} |g} | g} d}tdt|D]s} || } || }|| dkrP|| }||| |d}n|| d}|dkrlj|}t||krk|}n|}| |dkr{|d7}| || ||| }||dkr|d7}| | | || }|} q.| }| }|d7}|t kr1|dkr1|d|kr1d}d}d}tdt|D]!}||||d}||kr|}|}|}q||kr||kr|}q|d|kr1fdd}||d}||}j |||fd d d d }|} ||kr1||kr1| ||dkr1| ||| || t|}||kr>||ks||krFd }nd }|||fS) z^Separate the zeros contained in the block T, limitloop determines how long one must searchNrrr csj|ddS)Nr derivative)rs_zxrr r!ysz)separate_zeros_in_block..illinoisF)solververifyverboseT) infcount_variationsrrsqrtr r absrITERATION_LIMITfindrootr )rrrr limitloop fp_tolerance loopnumber variationsrrnewTnewVrb2ualpharwdtMaxdtSeckMaxk1dtfrrr separatedr r1r!separate_zeros_in_blockBs            "   &   8 rNcspd}|d}tdt|D]}||} || dkr'|d7}||kr'|} |} | } | }q || } || d}|_t||}d|}jdg}d}|dd|krr|d7}|dddd|g|}|dd|ksW|d|_jfdd|| fdd d }d |}|dd D]}||_||j|dd }d |}q |S)zPIf we know which zero of this block is mine, the function separates the zerorrrr+cs |S)Nr r/r1r r!r2s z"separate_my_zero..r3F)r4r6?Nr,) rrprecr)logmagr<mpczetaim)rrrrrrRr@rrrk0leftvrightvrrwpzguardprecsindexrzznewr r1r!separate_my_zeros<      rbcCsj|dkrdS||d}|j|}d|dd|}d|dd|}|t||}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 rdgHPx?g{Gz?ga+ei?g)\(?)r r lnceilminint)rrglgbrenttrudgianNr r r!sure_number_blocks rmcCsR||}|j|}|t|||dkr||}|d|}|||fS)N-)r r r rTr:)rrrrrr r r!r s     r rOc&CsHt||}d}|d}t||\}}}|g} |g} |dkr7|d7}t||\}}}| || ||dks|g} |g} |g} |d|kr|d7}t||\}}}| || ||dkrz|d7}t||\}}}| || ||dks`| |t| d}t||| | t|d\}}}| | || | ||r|d7}nd}|g} |g} |d|ksFd}|d}t||\}}}| d|| d||dkr|d8}t||\}}}| d|| d||dks| d||g} |g} |d|kr|d8}t||\}}}| d|| d||dkr@|d8}t||\}}}| d|| d||dks#| d|t| d}t||| | t|d\}}}||| } ||| } |rq|d7}nd}|g} |g} |d|ks| d|}t| }| |d|d}t||\}}}| |}t||\}}}| |}| ||d} | ||d} ||}t||| | t|d\}}}|r||d||g||fS| |}t| }| ||d}t||\}}} | |}!t||\}"}#}$| |"}%| |!|%d} | |!|%d} ||d||g| | fS)zTo use for n>400 000 000rrrr=r>) rmr rrrNr;popextendr r^)&rrr>sbnumber_goodblocksm2rrrTfVf goodpointsrrznABrMr_ristrvrbrartsvsbsas1qtqvqbqaqttvtbtatr r r!search_supergood_blocks                                 rcCsDd}|d}tdt|D]}||}||dkr|d7}|}q |SNrr)rr)rcountvoldrvnewr r r!r82s r8cCsd}|d}|d}t||\}}} d} d} t|d|dD]S} t|| \} }}t|}| |krE|| | krE| d7} | |krE|| | ks7|| | }|||d|t|}|d|}|dkrh|d}| } | ||}}} q|dd}|S)N(rrz%sz)(ro)r rrrr r8)rblockrrrrrrrb0rrXrrrb1lgTLrr r r!pattern_construct<s.     rFTcCst|}|dkr|| S|dkrtd|j}zZt||\}}||_|dkr4t||\}}} } n t|||\}}} } |d|d} t|| | | |j |d\} } } |r]t ||| | } t ||}t ||| | | |}| d|}W||_n||_w|r| }|r|||| fS|S)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 nonzerorrprQ)rgzetazero conjugate ValueErrorrRcomp_fp_tolerancer"rrNr7rmaxrbrU)rrinforound wpinitialr[r>rrrrrrMrrRrrr r r!rTs:<      rcCs\|dkr d||d}nd}|j}z|j|7_t|||j}W||_|S||_w)Nl a$r+r*r)rSrRrg siegelthetapi)rrr(rRhr r r! gram_indexsrc Csd}|d}|d}|d}d}||kr0||} || dkr"|d7}| }|d7}||}||ks||} | |dkr?|d7}|SrrP) rrrrrrtoldtnewrrrr r r!count_tos"   rcCsFt|||}|dkrd}||fS|dkrd}||fSd}||fS)Ni/hYgMb@?r&g?rc)r)rS)rrr[r>r r r!rsrcCs8|dkrdSt||}t||}|j}t||\}}||_||}|dkr.|dkr.dS|dkr8|dkr8dS|ddkrFt||d}nt||d|}|d\} } | | dkrt|dd} || dkrm||_|dS||_|dS|\} } }}| | }t|||||j |d\}}}t ||||}||_|| dS) 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. 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