"`x$dZddlmZmZdZdZ d#dZdZdZd Z d Z d Z d Z d Z ed$dZdZdZdZedZedZ gddgdddgdddgdddgdddgdd d!gdd"d#gdd$d%gdd&d'gdd(d)gdd*d+gdd,d-gdd.d/gdd0d1gd2d3d4gdd5d6gdd7d8gdd9d:gdd;dgdd?d@gddAdBgddCdDgddEdFgddGdHgddIdJgddKdLgddMdNgddOdPgddQdRgddSdTgddUdVgddWdXgddYdZgdd[d\gdd]d^gdd_d`gddadbgddcddgddedfgddgdhgddidjgddkdlgddmdngddodpgddqdrgddsdtgddudvgddwdxgddydzgdd{d|gdd}d~gdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgd2ddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdˑddgdddgdddgdddgdԑddgdddgdˑddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdԑddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgddd gdd d gdd d gdddgdddgdddgdddgdddgdddgdddgdddgdddgdd d!gdd"d#gdd$d%gdd&d'gdd(d)gdd*d+gdd,d-gdd.d/gdˑd0d1gdd2d3gdd4d5gdd6d7gdd8d9gdd:d;gdd<d=gdd>d?gdd@dAgddBdCgddDdEgddFdGgddHdIgddJdKgddLdMgdNdOdPgddQdRgddSdTgdUdVdWgddXdYgddZd[gdd\d]gdd^d_gdd`dagddbdcgddddegddfdggddhdigddjdkgddldmgddndogddpdqgddrdsgddtdugddvdwgddxdygddzd{gdd|d}gdˑd~dgdddgdddgdddgdUddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdԑddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgddĐdgddƐdgddȐdgddʐdgdd̐dgddΐdgddАdgddҐdgddԐdgdd֐dgddؐdgddڐdgdԑdܐdgddސdgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgddd gdd d gdd d gdddgdddgdddgdddgd2ddgdddgdddgdddgdddgdd d!gdd"d#gdUd$d%gdd&d'gdd(d)gdd*d+gdd,d-gdd.d/gdd0d1gdd2d3gdd4d5gdd6d7gdd8d9gdd:d;gdd<d=gdd>d?gdd@dAgddBdCgddDdEgddFdGgddHdIgddJdKgddLdMgdԑdNdOgddPdQgddRdSgddTdUgddVdWgddXdYgddZd[gdd\d]gdd^d_gdd`dagddbdcgddddegddfdggddhdigddjdkgddldmgddndogddpdqgddrdsgddtdugddvdwgddxdygddzd{gdd|d}gdd~dgdddgdddgdddgd2ddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdUddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgddÐdgddŐdgddǐdgddɐdgddːdgdd͐dgdUdϐdgddѐdgddӐdgddՐdgdˑdאdgddِdgddېdgddݐdgddߐdgdˑddgdddgdddgdddgdddgdԑddgdddgdddgdddgdԑddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdddgdd d gdd d gd ddgdddgd ddgdddgdddgdddgdddgd dd gdd!d"gdZy(%a The function zetazero(n) computes the n-th nontrivial zero of zeta(s). The general strategy is to locate a block of Gram intervals B where we know exactly the number of zeros contained and which of those zeros is that which we search. If n <= 400 000 000 we know exactly the Rosser exceptions, contained in a list in this file. Hence for n<=400 000 000 we simply look at these list of exceptions. If our zero is implicated in one of these exceptions we have our block B. In other case we simply locate the good Rosser block containing our zero. For n > 400 000 000 we apply the method of Turing, as complemented by Lehman, Brent and Trudgian to find a suitable B. )defun defun_wrappedcttttdzD]}td|zd}td|zd}||dz ks*|dz |ks3|j|}|j|}|jj |}|jj |}||z dz } ||z } td|zdz} | ||g||g||gfcS|dz }t ||\} } }| g}| g}|dkr?|dz}t ||\} } }|jd| |jd| |dkr?||z dz } |dz }t ||\} } }|j| |j| |dkr=|dz }t ||\} } }|j| |j| |dkr=| ||g||fS)z;for n<400 000 000 determines a block were one find our zeror) rangelen_ROSSER_EXCEPTIONS grampoint_fpsiegelzcompute_triple_tvbinsertappend)ctxnkabt0t1v0v1my_zero_numberzero_number_blockpatterntvTVms z)separate_zeros_in_block..yschhqAh6r$illinoisF)solververifyverboseT) infcount_variationsrr sqrtr r absrITERATION_LIMITfindrootr)rrrr limitloop fp_tolerance loopnumber variationsrrnewTnewVrb2ualpharwdtMaxdtSeckMaxk1dtfrrr separateds` r"separate_zeros_in_blockrSBsGG J!!$J * *Y1F aD aDss qQ A1B!A!A1 !GBJq)rT1Hb GGOOA&q6,& AA++a.s1ua KKN KKN!AsAva KKO KKNAA1 2  Q o %*Q,:a.sckk!nr$r8F)r9r;?Nr2) rr precr-logmagrAmpczetaim)rrrrr rYrErrrk0leftvrightvrrwpzguardprecsindexrzznews` r"separate_my_zerorisJ 1B 1SV_ qT b519 NJ^+  2B 2a4BCH .!88 9C cggn% %E XXaZLE E (QsU   qQ!!E')*U2 (QsU Qx%CH ,r"gzSX YA ggc!nAab $%<388A;!!::: ''#cffTl # $ 66!9r$c|dkry|j|dz }|jj|}d|dzzd|zz}d|dzzd|zz}|jt ||}t |}|S)aThe number of good Rosser blocks needed to apply Turing method References: R. P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Math. Comp. 33 (1979) 1361--1372 T. Trudgian, Improvements to Turing Method, Math. 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B B a% a"3+!A !  ! a% J A A ad " a$S"-1a    !e !GB&sB/EAa HHQK HHQK !e " VAX "3AqO) + 1i  !   !   "  !  C C1 ad "4 1B b)GAq!IIaNIIaN a% a"3+!A !A !A a% a A A ad " a$S"-1a 1  1 !e !GB&sB/EAa HHQqM HHQqM !e !B VAX "3AqOZf g 1i  rT  rT   "  !  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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"rrTsLx AA1u||QB))++Av,--I-c15\ y= #C + (NE1a$CL 9 (NE1a!!HU1X-1#7H!QggL:1i 'E!A6G9c" S.2CAa M GGCN 2 %w// s B'D Dc|dkDrd|j|dz}nd}|j} |xj|z c_t|j||jz }||_|S#||_wxYw)Nl a$r0r/r)rZrYro siegelthetapi)rrr,rYhs r" gram_indexrsp6z swwq"~   88D B "366) * Is >> 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"nzerosrs\J 3A CIIaLAI)#q1CCH AABw1q5 bQUsY'QqS1'QqS,? AYFB "uz 1IaL Q37 CHQ3J CHQ3J!'N5!Q2-c.?A8;9EGOAq) aAACH R46Mr$ch|j|dz |j||jz z S)aw Computes the function `S(t) = \operatorname{arg} \zeta(\frac{1}{2} + it) / \pi`. See Titchmarsh Section 9.3 for details of the definition. **Examples** >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> backlunds(217.3) 0.16302205431184 Generally, the value is a small number. At Gram points it is an integer, frequently equal to 0:: >>> chop(backlunds(grampoint(200))) 0.0 >>> backlunds(extraprec(10)(grampoint)(211)) 1.0 >>> backlunds(extraprec(10)(grampoint)(232)) -1.0 The number of zeros of the Riemann zeta function up to height `t` satisfies `N(t) = \theta(t)/\pi + 1 + S(t)` (see :func:nzeros` and :func:`siegeltheta`):: >>> t = 1234.55 >>> nzeros(t) 842 >>> siegeltheta(t)/pi+1+backlunds(t) 842.0 r)rrr)rrs r" backlundsr!s.H ::a=?3??1-cff4 44r$iiz(00)3iaidiiz3(00)i=i=i oioiKiNi'i'iDiDi5i5i"i"i͜JiМJi+di.diOeiRei6٧i:٧z(00)40i (i(iR4yiU4yiýiýieieiii2i5i^RiaRi(i(iL=iO=iGi"Givi"viiiii Ui Ui_i_ieieihihi$.i'.i;i;iii~)i~)i<i <i@i@iZDi]DipNisNibibi(i(iiivxiyxiiikini7/i:/i(7i(7ioEirEiOeIiReIipipi5i5i iiEŵiHŵi: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= iv iv i i id ig iL iO i; i; i0 i0 iS' iV' i, i, i@ i@ i1T i1T i[ i[ i` i` i`c i`c if if i!y i!y iՊ iՊ i i i i iWC iZC i>h iAh i[b i^b i i z22(00)iǂ iʂ it4d iw4d id id if if z(00)22iy iy i i iך i ך i׬ i׬ is is i i i i iZ i] it it i i i! 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