[hx(SrSSKJrJr SrSrGS#SjrSrSrS r S r S r S r S r \GS$Sj5rSrSrSr\S5r\S5r/SS/PSPSS/PSPSS/PSPSS/PSPSS/PSPS S!/PSPS"S#/PSPS$S%/PSPS&S'/PSPS(S)/PSPS*S+/PSPS,S-/PSPS.S//PSPS0S1/PS2PS3S4/PSPS5S6/PSPS7S8/PSPS9S:/PSPS;S/PSPS?S@/PSPSASB/PSPSCSD/PSPSESF/PSPSGSH/PSPSISJ/PSPSKSL/PSPSMSN/PSPSOSP/PSPSQSR/PSPSSST/PSPSUSV/PSPSWSX/PSPSYSZ/PSPS[S\/PSPS]S^/PSPS_S`/PSPSaSb/PSPScSd/PSPSeSf/PSPSgSh/PSPSiSj/PSPSkSl/PSPSmSn/PSPSoSp/PSPSqSr/PSPSsSt/PSPSuSv/PSPSwSx/PSPSySz/PSPS{S|/PSPS}S~/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PS2PSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSS/PSPSGS/PGSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS /PSPGS GS /PSPGS GS /PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGS GS!/PSPGS"GS#/PSPGS$GS%/PSPGS&GS'/PSPGS(GS)/PSPGS*GS+/PSPGS,GS-/PSPGS.GS//PSPGS0GS1/PSPGS2GS3/PSPGS4GS5/PSPGS6GS7/PSPGS8GS9/PSPGS:GS;/PSPGSGS?/PSPGS@GSA/PSPGSBGSC/PSPGSDGSE/PSPGSFGSG/PSPGSHGSI/PSPGSJGSK/PSPGSLGSM/PGSNPGSOGSP/PSPGSQGSR/PSPGSSGST/PGSUPGSVGSW/PSPGSXGSY/PSPGSZGS[/PSPGS\GS]/PSPGS^GS_/PSPGS`GSa/PSPGSbGSc/PSPGSdGSe/PSPGSfGSg/PSPGShGSi/PSPGSjGSk/PSPGSlGSm/PSPGSnGSo/PSPGSpGSq/PSPGSrGSs/PSPGStGSu/PSPGSvGSw/PSPGSxGSy/PSPGSzGS{/PSPGS|GS}/PSPGS~GS/PSPGSGS/PSPGSGS/PSPGSGS/PGSUPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PGSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS /PSPGS GS /PSPGS GS /PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PS2PGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGS GS!/PSPGS"GS#/PGSUPGS$GS%/PSPGS&GS'/PSPGS(GS)/PSPGS*GS+/PSPGS,GS-/PSPGS.GS//PSPGS0GS1/PSPGS2GS3/PSPGS4GS5/PSPGS6GS7/PSPGS8GS9/PSPGS:GS;/PSPGSGS?/PSPGS@GSA/PSPGSBGSC/PSPGSDGSE/PSPGSFGSG/PSPGSHGSI/PSPGSJGSK/PSPGSLGSM/PSPGSNGSO/PSPGSPGSQ/PSPGSRGSS/PSPGSTGSU/PSPGSVGSW/PSPGSXGSY/PSPGSZGS[/PSPGS\GS]/PSPGS^GS_/PSPGS`GSa/PSPGSbGSc/PSPGSdGSe/PSPGSfGSg/PSPGShGSi/PSPGSjGSk/PSPGSlGSm/PSPGSnGSo/PSPGSpGSq/PSPGSrGSs/PSPGStGSu/PSPGSvGSw/PSPGSxGSy/PSPGSzGS{/PSPGS|GS}/PSPGS~GS/PSPGSGS/PSPGSGS/PSPGSGS/PS2PGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PGSUPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PGSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PGSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PGSUPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PSPGSGS/PGSPGS GS /PGSPGS GS /PGS PGSGS/PGSPGSGS/PGS PGSGS/PGSPGSGS/PGSPGSGS/PGSPGSGS/PGSPGSGS/PGS PGSGS /PGSPGS!GS"/PGSPrg(%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_wrappedcl[[[5S-5Hn[SU-Sn[SU-SnX1S- ::dM+US- U::dM6URU5nURU5nURR U5nURR U5nX- S- n XC- n [SU-S-n XU/XV/Xx/4s $ US- n[ X5upnU /nU /nUS:a?US-n[ X5upnURSU 5 URSU 5 US:aM?X- S- n US- n[ UU5upnURU 5 URU 5 US:a>US- n[ UU5upnURU 5 URU 5 US:aM>XU/X4$)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 R/var/www/auris/envauris/lib/python3.13/site-packages/mpmath/functions/zetazeros.pyfind_rosser_block_zeror#s 3)*A- . QqS !! $ QqS !! $ 1W1Q3!8q!Bq!B$B$BSUN ! (1Q/G"qEB7RG< </ !A s &EA A A a% Q"3*A 1  1 a% SUN !A sA &EAHHQKHHQK a% Q"3*A    a% qE1 ((c:SnUS:aSnUS:aSnUS:aSnU$)z(Precision needed to compute higher zeros5i?lh]F@ kS)rwps r"wpzerosr-7s0 B7{ 6z 6z  Ir$Nc^Uc TRnSn[U5nXq:Ga.Xd:Ga(USnUSn U/n U /n Sn[S[U55Hn X,n X<nX-S:a!TR X- 5nX-U -US-- nOX-S- nUS:a<TR R U5n[U5U:aTR U5nOTR U5nU U-S:aUS- nU RU5 U RU5 X<nUU-S:aUS- nU RU 5 U RU5 U nUn M U nU nUS- nU[:aUS:aUS-U:XaSnSnSn[S[U55H1nUUUUS- - nUU:aUnUnUnMUU:dM'UU:dM/UnM3 USU-:atU4SjnUUS- nUUnTRUUU4SSSS 9nTR U5n UU:a5UU:a/XU-S:a$URUU5 URUU 5 [U5nXq:aXd:aGM(Xq:XaS nOSnX#U4$) zZSeparate the zeros contained in the block T, limitloop determines how long one must searchrrr c$>TRUSS9$)Nr derivative)rs_zxrs r")separate_zeros_in_block..yschhqAh6r$illinoisF)solververifyverboseT) infcount_variationsrr sqrtr r absrITERATION_LIMITfindrootr)rrrr limitloop fp_tolerance loopnumber variationsrrnewTnewVrb2ualpharwdtMaxdtSeckMaxk1dtfrrr separateds` r"separate_zeros_in_blockrTBsGG J!!$J  *1F aD aDss qQABAA GBJq)T1Hb GGOOA&q6,& AA++a.s1ua KKN KKNAsAva KKO KKNAA1!2  Q o %*Q,:aTRU5$)Nr r5s r"r7"separate_my_zero..s ckk!nr$r9F)r:r<?Nr2) rr precr-logmagrBmpczetaim)rrrrr r[rFrrrk0leftvrightvrrwpzguardprecsindexrzznews` r"separate_my_zerorksJ 1B 1SV_ T 519 NJ+  2B a4BCH .!88 9C cggn% %E XXaZLE E (QsU   qQ!!E')*U2 (QsU Qx%CH ,rgzSX YA ggc!nAab %<388A;!!::: ''#cffTl #  66!9r$cUS:agURUS- 5nURRU5nSUS--SU--nSUS--SU--nUR[ XE55n[ U5nU$)zThe 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_blockrws~ 7{ aeA AB RUNDG #EA~tBw&H U$%A AA Hr$cURU5nURRU5nUR[ U55URU5S- :aURU5nUSU--nX#U4$)N-)r r r r]r@)rrrrrs r"rrse aA A wws1vswwqz"}$ KKN 2' A q5Lr$rVc x[X5nSnUS- n[X5upgnU/n U/n US:a=US- n[X5upgnU RU5 U RU5 US:aM=U/n U/n U/n USU-:GaUS- n[X5upgnU RU5 U RU5 US:a=US- n[X5upgnU RU5 U RU5 US:aM=U RU5 [U 5S- n[ XX[ US9unnnU R 5 U RU5 U R 5 U RU5 U(aUS- nOSnU/n U/n USU-:aGMSnUS- n[X5upgnU RSU5 U RSU5 US:a?US-n[X5upgnU RSU5 U RSU5 US:aM?U RSU5 U/n U/n USU-:aUS-n[X5upgnU RSU5 U RSU5 US:a?US-n[X5upgnU RSU5 U RSU5 US:aM?U RSU5 [U 5S- n[ XX[ US9unnnUR 5 X-n UR 5 UU -n U(aUS- nOSnU/n U/n USU-:aMU SU-n[U 5nU USU-- S- n[UU5unnnU RU5n[UU5unnnU RU5nU UUS-n U UUS-n UU- n[ XX[ US9unnnU(aUU- S- UU/UU4$Xn[U 5nU UU- S- n[UU5unnn U RU5n![X5un"n#n$U RU"5n%U U!U%S-n U U!U%S-n UU- S- UU/X4$)zTo use for n>400 000 000rrrrCrD) rwrrr rTrApopextendrrg)&rrrDsbnumber_goodblocksm2rrrTfVf goodpointsrr znABrSrhrsstrvrbrartsvsbsas1qtqvqbqaqttvtbtats& r"search_supergood_blockrs 3 "B 1B )GA! B B a% a"3+A !  ! a% J A A ad " a$S-a    !e !GB&s/EA HHQK HHQK !e " VAX "3AO) + 1i  !   !   "  !  C C1 ad "4 1B )GA!IIaNIIaN a% a"3+A !A !A a% a A A ad " a$S-a 1  1 !e !GB&s/EA HHQqM HHQqM !e !B VAX "3AOZf g 1i  T  rT   "  !  C C/ ad "0 1R4A ZB2ad719A#C+JBB "B#C+JBB ((2,C 2c!e A 2c!e A 1Bsq_S_`Aq)!AqeAa  A ZB2b57A#C+JBB "B#C+JBB "B 2bd A 2bd A aCE1Q% r$crSnUSn[S[U55HnXnX$-S:aUS- nUnM U$Nrr)rr )r countvoldrvnews r"r>r>2sJ E Q4D 1c!f t 9q= AIE  Lr$cSnUSnUSn[X5upxn Sn Sn [US-US-5Hn [X 5upn[U5nU U:aX*U ::aU S- n U U:a X*U ::aMX;U nURU5 UR SU5 [ U5nUSU--nUS:aUS-nU n XUpnM USSnU$)N(rrz%sz)(rz)rrr rrr>)rblockrr rrrrrb0rrarrrb1lgTLrs r"pattern_constructr<sG aA aA!#)HB" A B 1Q3qs^%c-b V3wQTRZ FA3wQTRZ G   2 #TE\* 6nG bbcrlG Nr$c "[U5nUS:a URU*5R5$US:Xa [S5eURn[ X5upVXPlUS:a[ X5upxpO[XU5upxpUSUS- n [X XURUS9upn U(a [XX5n [XE5n[XXX5nURSU5nX@lU(aU7nU(aUXW 4$U$!X@lf=f)a Computes the `n`-th nontrivial zero of `\zeta(s)` on the critical line, i.e. returns an approximation of the `n`-th largest complex number `s = \frac{1}{2} + ti` for which `\zeta(s) = 0`. Equivalently, the imaginary part `t` is a zero of the Z-function (:func:`~mpmath.siegelz`). **Examples** The first few zeros:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> zetazero(1) (0.5 + 14.13472514173469379045725j) >>> zetazero(2) (0.5 + 21.02203963877155499262848j) >>> zetazero(20) (0.5 + 77.14484006887480537268266j) Verifying that the values are zeros:: >>> for n in range(1,5): ... s = zetazero(n) ... chop(zeta(s)), chop(siegelz(s.imag)) ... (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) Negative indices give the conjugate zeros (`n = 0` is undefined):: >>> zetazero(-1) (0.5 - 14.13472514173469379045725j) :func:`~mpmath.zetazero` supports arbitrarily large `n` and arbitrary precision:: >>> mp.dps = 15 >>> zetazero(1234567) (0.5 + 727690.906948208j) >>> mp.dps = 50 >>> zetazero(1234567) (0.5 + 727690.9069482075392389420041147142092708393819935j) >>> chop(zeta(_)/_) 0.0 with *info=True*, :func:`~mpmath.zetazero` gives additional information:: >>> mp.dps = 15 >>> zetazero(542964976,info=True) ((0.5 + 209039046.578535j), [542964969, 542964978], 6, '(013111110)') This means that the zero is between Gram points 542964969 and 542964978; it is the 6-th zero between them. Finally (01311110) is the pattern of zeros in this interval. The numbers indicate the number of zeros in each Gram interval (Rosser blocks between parenthesis). In this case there is only one Rosser block of length nine. rzn must be nonzerorr|rZ)rqzetazero conjugate ValueErrorr[comp_fp_tolerancer#rrTr=rmaxrkr^)rrinforound wpinitialrdrDrrrr rrSrr[rrs r"rrTs#x AA1u||QB))++Av,--I-c5 y= #C + (N1a$CL 9 (N1!!HU1X-1#!ggL:i '!6G9" S2Ca M GGCN 2 %w// s BDDcUS:aSURUS5-nOSnURnU=RU- sl[URU5UR- 5nX0lU$!X0lf=f)Nl a$r0r/r)r\r[rq siegelthetapi)rrr,r[hs r" gram_indexrsl6z 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,@rrzrrrr0r|) rrqfloorr[rr r#rrTr=r)rrr6rrrdrDrRblockn1n2rrrrr rrSrs r"nzerosrs6J 3A CIIaLAI)#1CH AABw1q5 bQUsY'qS1'qS,? AYFB uz 1IaL 37 HQ3J HQ3J!'N!-c.?8;9EGOA) AH R46Mr$chURU5S- URU5UR- - $)a Computes the function `S(t) = \operatorname{arg} \zeta(\frac{1}{2} + it) / \pi`. See Titchmarsh Section 9.3 for details of the definition. **Examples** >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> backlunds(217.3) 0.16302205431184 Generally, the value is a small number. At Gram points it is an integer, frequently equal to 0:: >>> chop(backlunds(grampoint(200))) 0.0 >>> backlunds(extraprec(10)(grampoint)(211)) 1.0 >>> backlunds(extraprec(10)(grampoint)(232)) -1.0 The number of zeros of the Riemann zeta function up to height `t` satisfies `N(t) = \theta(t)/\pi + 1 + S(t)` (see :func:nzeros` and :func:`siegeltheta`):: >>> t = 1234.55 >>> nzeros(t) 842 >>> siegeltheta(t)/pi+1+backlunds(t) 842.0 r)rrr)rrs r" backlundsr!s.H ::a=?3??1-cff4 44r$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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