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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_wrappedc\\\4^,4Fp\^V,,^,p\^V,,^,pW1^, 8:gKHV^, V8:gKXVPV4pVPV4pVPP V4pVPP V4pW, ^, p WC, p \^V,^,,p WV.WV.Wx.3u# V^, p\ W4wrpV .pV .pV^8d>V^,p\ W4wrpVP^V 4VP^V 4KDW, ^, p V^, p\ VV4wrpVPV 4VPV 4V^8d=V^, p\ VV4wrpVPV 4VPV 4KCWV.W3#)z;for n<400 000 000 determines a block were one find our zero) rangelen_ROSSER_EXCEPTIONS grampoint_fpsiegelzcompute_triple_tvbinsertappend)ctxnkabt0t1v0v1my_zero_numberzero_number_blockpatterntvTVms&& [/home/luca/screen-checking/.venv/lib/python3.14/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 SUN !A sA &EAHHQKHHQK a% Q"3*A    qE1 ((c@^5pVR8d^?pVR8d^FpVR8d^SpV#)z(Precision needed to compute higher zerosilh]@ k)rwps& rwpzerosr&7s0 B7{ 6z 6z  Ir!NcaVf SPp^p\V4pWq8EdWd8EdV^,pV^,p V.p V .p ^p\^\V44EF8p W,,p W<,pW,^8d6SP W, 4pW,V ,V^,, pMW,^, pV^ 8d>SP P V4p\V4V8dSP V4pMSP V4pV V,^8d V^, pV PV4V PV4W<,pVV,^8d V^, pV PV 4V PV4T pTp EK; T pT pV^, pV\8EdV^8EdV^,V8XEd^p^p^p\^\V44FDpVV,VV^, ,, pVV8d TpTpTpK0VV8gK9VV8gKBTpKF V^V,8dV3RlpVV^, ,pVV,pSPVVV3RRRR7pSP V4p VV8d@VV8d9WV,,^8d%VPVV4VPVV 4\V4pEKWq8XdRpMRpW#V3#)zZSeparate the zeros contained in the block T, limitloop determines how long one must searchc*<SPV^R7#) derivative)rs_zxrs&r)separate_zeros_in_block..yschhqAh6r!illinoisF)solververifyverboseT) infcount_variationsrrsqrtr r absr ITERATION_LIMITfindrootr )rrrr limitloop fp_tolerance loopnumber variationsrrnewTnewVrb2ualpharwdtMaxdtSeckMaxk1dtfrrr separatedsf&&&&& rseparate_zeros_in_blockrLBsGG J!!$J  *1F aD aDss qQABAA GBJq)T1Hb GGOOA&q6,& AA++a.s1ua KKN KKNAsAva KKO KKNAA1!2  Q o %*Q,:a.s ckk!nr!r1F)r2r4?:r)NNr*) rrprecr&logmagr:mpczetaim)rrrrrrRr>rrrk0leftvrightvrrwpzguardprecsindexrzznewsf&&&&& rseparate_my_zerorbsJ 1B 1SV_ T 519 NJ+  2B a4BCH .!88 9C cggn% %E XXaZLE E (QsU   qQ!!E')*U2Qx%CH ,rgzSX YA ggc!nAb %<388A;!!::: ''#cffTl #  66!9r!c@VR8d^#VPV^d, 4pVPPV4pRV^,,RV,,pRV^,,RV,,pVP\ WE44p\ V4pV#)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.gHPx?g{Gz?ga+ei?g)\(?i )rr lnceilminint)rrglgbrenttrudgianNs&& rsure_number_blockrms~ 7{ aeA AB RUNDG #EA~tBw&H U$%A AA Hr!cVPV4pVPPV4pVP\ V44VPV4^-, 8dVPV4pVRV,,pW#V3#)-)rr r rTr8)rrrrrs&& rr r se aA A wws1vswwqz"}$ KKN 2' A q5Lr!c d \W4p^pV^, p\W4wrgpV.p V.p V^8d<V^, p\W4wrgpV PV4V PV4KBV.p V.p V.p V^V,8EdV^, p\W4wrgpV PV4V PV4V^8d<V^, p\W4wrgpV PV4V PV4KBV PV4\V 4^, p\ WW\ VR7wpppV P 4V PV4V P 4V PV4V'd V^, pM^pV.p V.p EK#^pV^, p\W4wrgpV P^V4V P^V4V^8d>V^,p\W4wrgpV P^V4V P^V4KDV P^V4V.p V.p V^V,8Ed V^,p\W4wrgpV P^V4V P^V4V^8d>V^,p\W4wrgpV P^V4V P^V4KDV P^V4\V 4^, p\ WW\ VR7wpppVP 4W,p VP 4VV ,p V'd V^, pM^pV.p V.p EKV ^V,,p\V 4pV V^V,, ^, ,p\VV4wpppV PV4p\VV4wpppV PV4pV VV^,p V VV^,p VV, p\ WW\ VR7wpppV'dVV, ^, VV.VV3#W,p\V 4pV VV, ^, ,p\VV4wppp V PV4p!\W4wp"p#p$V PV"4p%V V!V%^,p V V!V%^,p VV, ^, VV.W3#)zTo use for n>400 000 000r;r<) rmr r rrLr9popextendr r^)&rrr<sbnumber_goodblocksm2rrrTfVf goodpointsrrznABrKr_ristrvrbrartsvsbsas1qtqvqbqaqttvtbtats&&&& rsearch_supergood_blockrsL 3 "B 1B )GA! B B a% a"3+A !  ! J A A ad " a$S-a    !e !GB&s/EA HHQK HHQK" VAX "3AO) + 1i  !   !   "  !  C C 1B )GA!IIaNIIaN a% a"3+A !A !Aa A A ad " a$S-a 1  1 !e !GB&s/EA HHQqM HHQqM!B VAX "3AOZf g 1i  T  rT   "  !  C C1R4A ZB2ad719A#C+JBB "B#C+JBB ((2,C 2c!e A 2c!e A 1Bsq_S_`Aq)!AqeAa  A ZB2b57A#C+JBB "B#C+JBB "B 2bd A 2bd A aCE1Q% r!c^pV^,p\^\V44F#pW,pW$,^8d V^, pTpK% V#)rr)rcountvoldrvnews& rr6r62sJ E Q4D 1c!f t 9q= AIE  Lr!cRpV^,pV^,p\W4wrxp ^p ^p \V^,V^,4Fp \W 4wrp\V4pV V8dW*,V 8:d V ^, p KW;V pVPV4VP ^V4\ V4pVRV,,pV^8d VR,pT p YTrpK VRRpV#)(z%sz)(Nrp)r rrr r r6)rblockrrrrrrrb0rrXrrrb1lgTLrs&&&& rpattern_constructr<sG aA aA!#)HB" A B 1Q3qs^%c-b V3wQTRZ FA G   2 #TE\* 6nG bbcrlG Nr!c L\V4pV^8d!VPV)4P4#V^8Xd \R4hVPp\ W4wrVWPnVR8d\ W4wrxrM\WV4wrxrV^,V^,, p \W WVPVR7wrp V'd \WW4p \WE4p\WWW4pVPRV4pW@nV'dV5pV'dVWX 3#V# Y@ni;i)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. zn must be nonzerorrrQ)rgzetazero conjugate ValueErrorrRcomp_fp_tolerancer rrLr5rmaxrbrU)rrinforound wpinitialr[r<rrrrrrKrrRrrs&&&& rrrTs#x AA1u||QB))++Av,--I-c5 y= #C + (N1a$CL 9 (N1!!HU1X-1#!ggL:i '!6G9" S2Ca M GGCN 2 %w// s B)DD#cVR8d^VPV^ 4,pM^pVPpV;PV, un\VPV4VP, 4pW0nV# Y0ni;i) l a$)rSrRrg siegelthetapi)rrr%rRhs&& r gram_indexrsl6z swwq"~   88D B "366) * Is AA==Bc^pV^,pV^,pV^,p^pWq8d4W8,p WY,^8d V^, pT pV^, pW(,pK9VPV4p W,^8d V^, pV#rrO) rrrrrrtoldtnewrrrs &&&& rcount_tors~ E Q4D Q4D Q4D A (t 9q= QJE Qt AAvz   Lr!c|\WPV4,4pVR8dRpW#3#VR8:dRpW#3#^dpW#3#)gMb@?g?i/hYr#)r&rS)rrr[r<s&& rrrsV !GGAJ, C8|   f     r!c VR8d^#\W4p\VPV44pVPp\ W4wrVWPnVP V4pVR8Xd V^8d^#VR8Xd V^8d^#V^,R8d\ W^,4pM\W^,V4pV^,wrW, ^8Xd<V^,^,p W{,^8dW@nV^,#W@nV^,#VwrrW, p\VVWVPVR7wrp\WW4pW@nVV ,^,#)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,@rrrrp) rrgfloorrRrr r rrLr5r)rrr.rrr[r<rRblockn1n2rrrrrrrKrs&& rnzerosrs6J 3A CIIaLAI)#1CH AABw1q5 bQUsY'qS1'qS,? AYFB uz 1IaL 37 HQ3J HQ3J!'N!-c.?8;9EGOA) AH R46Mr!cVPV4^, VPV4VP, , #)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. 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