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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_wrappedctttdzD]}td|zd}td|zd}||dz kr|dz |kr||}||}|j|}|j|}||z dz } ||z } td|zdz} | ||g||g||gfcS|dz }t ||\} } }| g}| g}|dkrK|dz}t ||\} } }|d| |d| |dkK||z dz } |dz }t ||\} } }|| || |dkrI|dz }t ||\} } }|| || |dkI| ||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_blockpatterntvTVms V/var/www/html/ai-engine/env/lib/python3.11/site-packages/mpmath/functions/zetazeros.pyfind_rosser_block_zeror#s) 3)**A- . . = = QqS !! $ QqS !! $ 1WW1Q3!88q!!Bq!!B$$B$$BqSUN !! (1Q/G"QqEBr7RG< < < < !A sA & &EAa A A a%% Q"3**!A 1  1 a%% qSUN !A sA & &EAaHHQKKKHHQKKK a%% Q"3**!A    a%% QqE1a ((c:d}|dkrd}|dkrd}|dkrd}|S)z(Precision needed to compute higher zeros5i?lh]F@ kS)rwps r"wpzerosr-7s6 B7{{ 6zz 6zz  Ir$Nc|j}d}t|}||kr||kr|d}|d} |g} | g} d}tdt|D]%} || } || }|| zdkr'|| z }||z| z|dzz }n|| zdz }|dkrCj|}t||kr|}n|}| |zdkr|dz }| || ||| }||zdkr|dz }| | | || }|} '| }| }|dz }|tkr|dkr|dz|krd}d}d}tdt|D]1}||||dz z }||kr|}|}|}#||kr||kr|}2|d|zkrfd}||dz }||} |||fdd d }|} ||krA||kr;| ||zdkr,| ||| || t|}||kr||k||krd }nd }|||fS) z^Separate the zeros contained in the block T, limitloop determines how long one must searchNrrr c2|dS)Nr derivative)rs_zxrs r"z)separate_zeros_in_block..yschhqAh66r$illinoisF)solververifyverboseT) infcount_variationsrr sqrtr r absrITERATION_LIMITfindrootr)rrrr limitloop fp_tolerance loopnumber variationsrrnewTnewVrb2ualpharwdtMaxdtSeckMaxk1dtfrrr separateds` r"separate_zeros_in_blockrSBs<G J!!$$J * * *Y1F1F aD aDss qQ  A1B!A!A1 !GBJq)rT1Hb  GOOA&&q66,&& AA++a..s1uua KKNNN KKNNN!AsAvva KKOOO KKNNNAAA  Q o % %*Q,,:a.sckk!nnr$r8F)r9r;?Nr2) rr precr-logmagrAmpczetaim)rrrrr rYrErrrk0leftvrightvrrwpzguardprecsindexrzznews` r"separate_my_zerorisJ 1B 1SVV__ qT b5199 NJ^++  2B 2a4BCH .!8!88 9 9C cggn%% %E XaZLE E (QsU    qQ!!E')*U2 (QsU  Qx%CH ,,,,r"gzSX YYA ggc!nnAabb $$%<388A;;!!:!::: ''#cffTll # # 66!99r$c|dkrdS||dz }|j|}d|dzzd|zz}d|dzzd|zz}|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)rrglgbrenttrudgianNs r"sure_number_blockrus 7{{q aeA AB RUNDG #EA~tBw&H U8$$%%A AA Hr$c||}|j|}|t |||dz kr||}|d|zz}|||fS)N-)r r r r[r?)rrrrrs r"rrsv aA A wws1vvswwqzz"}$$ KKNN 2' A Qq5Lr$rUcL t||}d}|dz }t||\}}}|g} |g} |dkrI|dz }t||\}}}| || ||dkI|g} |g} |g} |d|zkrC|dz }t||\}}}| || ||dkrI|dz }t||\}}}| || ||dkI| |t| dz }t ||| | t |\}}}| | || | ||r|dz }nd}|g} |g} |d|zkCd}|dz }t||\}}}| d|| d||dkrK|dz}t||\}}}| d|| d||dkK| d||g} |g} |d|zkr(|dz}t||\}}}| d|| d||dkrK|dz}t||\}}}| d|| d||dkK| d|t| dz }t ||| | t |\}}}||| z} ||| z} |r|dz }nd}|g} |g} |d|zk(| d|z}t| }| |d|zz dz }t||\}}}| |}t||\}}}| |}| ||dz} | ||dz} ||z }t ||| | t |\}}}|r||z dz ||g||fS| |}t| }| ||z dz }t||\}}} | |}!t||\}"}#}$| |"}%| |!|%dz} | |!|%dz} ||z dz ||g| | fS)zTo use for n>400 000 000rrrrBrC) rurrr rSr@popextendrre)&rrrCsbnumber_goodblocksm2rrrTfVf goodpointsrr znABrRrfrqstrvrbrartsvsbsas1qtqvqbqaqttvtbtats& r"search_supergood_blockrs 3 " "B 1B b))GAq! B B a%% a"3++!A !  ! a%% J A A ad " " a$S"--1a    !ee !GB&sB//EAa HHQKKK HHQKKK !ee " VVAX "3AqO) + + + 1i  !   !  "  "   !  C C1 ad " "4 1B b))GAq!IIaNNNIIaNNN a%% a"3++!A !A !A a%% a A A ad " " a$S"--1a 1  1 !ee !GB&sB//EAa HHQqMMM HHQqMMM !ee !B VVAX "3AqOZf g g g 1i  rT  rT  "  "   !  C C/ ad " "0 1R4A ZB2ad719A#C++JBB "B#C++JBB ((2,,C 2c!e8 A 2c!e8 A 1BsBq_S_```Aq)!!AqeAa  2A ZB2b57A#C++JBB "B#C++JBB "B 2bd7 A 2bd7 A aCE1Q%! r$cd}|d}tdt|D]}||}||zdkr|dz }|}|SNrr)rr )r countvoldrvnews r"r=r=2sZ E Q4D 1c!ff  t 9q== AIE Lr$cd}|d}|d}t||\}}} d} d} t|dz|dzD]} t|| \} }}t|}| |kr#|| | kr| dz } | |kr || | k|| | }|||d|t |}|d|zz}|dkr|dz}| } | ||} }}|dd}|S)N(rrz%sz)(rx)rrr rrr=)rblockrr rrrrrb0rr_rrrb1lgTLrs r"pattern_constructr<s,G aA aA!#q))HBr" A B 1Q3qs^^  %c1--2b VV3wwQqTRZZ FA3wwQqTRZZ bdG   2 ##TE\* 66nG bb2crclG Nr$FTct|}|dkr(|| S|dkrtd|j} t ||\}}||_|dkrt ||\}}} } nt|||\}}} } |d|dz } t|| | | |j |\} } } |rt||| | } t||}t||| | | |}| d|}||_n #||_wxYw|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 nonzerorrzrX)rozetazero conjugate ValueErrorrYcomp_fp_tolerancer#rrSr<rmaxrir\)rrinforound wpinitialrbrCrrrr rrRrrYrrs r"rrTsx AA1uu||QB))+++Avv,---I-c155\ y== #C + + (NE1aa$CL 9 9 (NE1a!!HU1X-1#7H!QgL:::1i  7'E!A66G9c"" S.2CAa M M GGCNN9  2 %w//s C D,, D5c|dkrd||dz}nd}|j} |xj|z c_t|||jz }||_n #||_wxYw|S)Nl a$r0r/r)rZrYro siegelthetapi)rrr,rYhs r" gram_indexrs6zz swwq"~~   8D B ""36) * *4 Is :A-- A6cd}|d}|d}|d}d}||kr+||} || zdkr|dz }| }|dz }||}||k+||} | |zdkr|dz }|SrrW) rrrr rrtoldtnewrrrs r"count_tors E Q4D Q4D Q4D A ((t 9q== QJE Qt (( AAvzz   Lr$c|t|||z}|dkrd}n |dkrd}nd}||fS)Ni/hYgMb@?r)g?rk)r-rZ)rrrbrCs r"rrsM !CGGAJJ,  C8|| f   r$c|dkrdSt||}t||}|j}t ||\}}||_||}|dkr|dkrdS|dkr|dkrdS|dzdkrt ||dz}nt||dz|}|d\} } | | z dkr/|dd} || zdkr ||_|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%fD,@rrxrrrr0rz) rrofloorrYrr r#rrSr<r)rrr6rrrbrCrRblockn1n2rrrrr rrRrs r"nzerosrsJ q3A CIIaLLAI)#q11CCH AABww1q55q bQUUqsY'QqS11'QqS,?? AYFB "uzz 1IaL Q377 CHQ3J CHQ3J!'N5!Q2-c.?A8;9EGGGOAq) aAACH R46Mr$cn||dz |||jz z 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. 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