a gd,d?d@gd'dAdBgdCdDdEgd,dFdGgd'dHdIgd,dJdKgd,dLdMgd'dNdOgd,dPdQgd'dRdSgd'dTdUgd,dVdWgd'dXdYgd'dZd[gd,d\d]gd'd^d_gd'd`dagd,dbdcgd'dddegd,dfdggd'dhdigd,djdkgd'dldmgd'dndogd,dpdqgd,drdsgd,dtdugd,dvdwgd'dxdygd'dzd{gd,d|d}gd,d~dgd,ddgd,ddgd,ddgd'ddgd,ddgd'ddgd'ddgd'ddgd,ddgd,ddgd'ddgd,ddgd'ddgd'ddgd,ddgd,ddgd,ddgd'ddgd,ddgd'ddgd'ddgd,ddgd,ddgd'ddgd'ddgd,ddgdCddgd,ddgd'ddgd'ddgd'ddgd'ddgd,ddgd,ddgd'ddgd,ddgd'ddgd'ddgd'ddgd'ddgd,ddgd,ddgd,ddgd'ddgd,ddgd'ddgdddgd,ddgd,ddgd,ddgdddgd'ddgdddgd'ddgd'ddgd,ddgd,ddgd'ddgd,ddgd'ddgd'ddgd'ddgd,ddgd,ddgdddgd,ddgd,ddgd,dd gd'd d gd'd d gd,ddgd,ddgdddgd,ddgd'ddgd,ddgd,ddgd'ddgd'dd gd'd!d"gd'd#d$gd,d%d&gd,d'd(gd,d)d*gd,d+d,gd'd-d.gd'd/d0gd,d1d2gd'd3d4gd,d5d6gd'd7d8gd'd9d:gd'd;dgd'd?d@gdܐdAdBgd'dCdDgd'dEdFgd,dGdHgd'dIdJgd'dKdLgd'dMdNgd,dOdPgd,dQdRgd'dSdTgd,dUdVgd,dWdXgd'dYdZgd'd[d\gd,d]d^gd_d`dagd'dbdcgd,dddegdfdgdhgd'didjgd'dkdlgd,dmdngd,dodpgd'dqdrgd'dsdtgd,dudvgd'dwdxgd'dydzgd'd{d|gd,d}d~gd'ddgd'ddgd,ddgd,ddgd'ddgd,ddgd'ddgd'ddgdܐddgd,ddgd'ddgd,ddgdfddgd'ddgd,ddgd'ddgd,ddgd,ddgd'ddgd,ddgd'ddgd'ddgd,ddgd,ddgd'ddgd,ddgd'ddgd,ddgd,ddgd,ddgd'ddgdddgd,ddgd,ddgd,dÐdgd,dŐdgd,dǐdgd'dɐdgd,dːdgd,d͐dgd'dϐdgd,dѐdgd,dӐdgd'dՐdgd,dאdgd,dِdgd'dېdgd,dݐdgd,dߐdgd,ddgd'ddgd'ddgd,ddgd,ddgd,ddgdddgd,ddgd,ddgd,ddgd,ddgd,ddgd'ddgd,ddgd,ddgd,ddgd,ddgd'ddgd'ddgd'ddgd,d d gd,d d gdd dgd'ddgd,ddgd,ddgd,ddgd,ddgd,ddgd'ddgd,ddgd,dd gd'd!d"gd,d#d$gd'd%d&gdCd'd(gd,d)d*gd,d+d,gd'd-d.gd,d/d0gd,d1d2gd,d3d4gdfd5d6gd,d7d8gd'd9d:gd,d;dgd'd?d@gd'dAdBgd'dCdDgd,dEdFgd,dGdHgd,dIdJgd'dKdLgd,dMdNgd'dOdPgd'dQdRgd,dSdTgd'dUdVgd'dWdXgd'dYdZgd'd[d\gd'd]d^gdd_d`gd,dadbgd,dcddgd,dedfgd,dgdhgd'didjgd'dkdlgd,dmdngd,dodpgd'dqdrgd'dsdtgd,dudvgd,dwdxgd,dydzgd,d{d|gd,d}d~gd,ddgd'ddgd,ddgd'ddgd,ddgd,ddgd,ddgd,ddgd'ddgd,ddgd'ddgd'ddgdCddgd,ddgd'ddgd,ddgd'ddgd,ddgd,ddgd,ddgd'ddgd,ddgdfddgd'ddgd'ddgd,ddgd'ddgd'ddgd,ddgd,ddgdddgd'ddgd'ddgd,ddgd,dĐdgddƐdgd'dȐdgd'dʐdgd,d̐dgd,dΐdgd,dАdgd'dҐdgd'dԐdgd,d֐dgd,dؐdgd'dڐdgd,dܐdgd,dސdgdfddgd,ddgd,ddgd'ddgdܐddgd,ddgd'ddgd,ddgd,ddgdܐddgd'ddgd,ddgd,ddgd'ddgdddgd,ddgd,ddgd,ddgdddgd'ddgd,dd gd,d d gd'd d gd'ddgd'ddgd,ddgd,ddgd,ddgd,ddgd,ddgdddgdddgddd gd!d"d#gdd$d%gdd&d'gd(d)d*gdd+d,gd-d.d/gdd0d1gd(d2d3gd-gZdS(6a 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_wrappedcCstttdD]}td|d}td|d}||dkr|d|kr||}||}|j|}|j|}||d} ||} td|d} | ||g||g||gfSq|d}t||\} } }| g}| g}|dkr|d8}t||\} } }|d| |d| q||d} |d}t||\} } }|| || |dkr|d7}t||\} } }|| || qR| ||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 j/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/mpmath/functions/zetazeros.pyfind_rosser_block_zeros@            r"cCs,d}|dkrd}|dkrd}|dkr(d}|S)z(Precision needed to compute higher zeros5i?lh]F@ kSr )rwpr r r!wpzeros7sr)Ncs|durj}d}t|}||krv||krv|d}|d} |g} | g} d}tdt|D]} || } || }|| dkr|| }||| |d}n || d}|dkr؈j|}t||kr|}n |}| |dkr|d7}| || ||| }||dkr(|d7}| | | || }|} q\| }| }|d7}|t krl|dkrl|d|krld}d}d}tdt|D]J}||||d}||kr|}|}|}n||kr||kr|}q|d|krlfdd}||d}||}j |||fd d d d }|} ||krl||krl| ||dkrl| ||| || t|}q||krd }nd }|||fS) z^Separate the zeros contained in the block T, limitloop determines how long one must searchNrrr csj|ddS)NrZ derivative)Zrs_zxrr r!yz)separate_zeros_in_block..illinoisF)solververifyverboseT) infcount_variationsrrsqrtr r absrITERATION_LIMITfindrootr )rrrr limitloop fp_toleranceZ loopnumber variationsrrZnewTZnewVrb2ualpharwZdtMaxZdtSecZkMaxZk1dtfrrr separatedr r/r!separate_zeros_in_blockBs|           "    &    rFcsdd}|d}tdt|D]8}||} || dkrN|d7}||krN|} |} | } | }q|| } || d}|_t||}d|}jdg}d}|dd|kr|d7}|dddd|g|}q|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-r/r r!r0r1z"separate_my_zero..r2F)r3r5?Nr,) rrprecr)logmagr;mpczetaZim)rrrrrrJr>rrrk0ZleftvZrightvrrwpzZguardZprecsindexrzZznewr r/r!separate_my_zeros:      rTcCsj|dkr dS||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)rrglgZbrentZtrudgianNr r r!sure_number_blocks r]cCsR||}|j|}|t|||dkr<||}|d|}|||fS)N-)r r r rLr9)rrrrrr r r!r s     r rGc&Cs$t||}d}|d}t||\}}}|g} |g} |dkrh|d7}t||\}}}| || |q2|g} |g} |g} |d|kr`|d7}t||\}}}| || ||dkr|d7}t||\}}}| || |q| |t| d}t||| | t|d\}}}| | || | ||rN|d7}nd}|g} |g} qzd}|d}t||\}}}| d|| d||dkr|d8}t||\}}}| d|| d|q| d||g} |g} |d|kr|d8}t||\}}}| d|| d||dkrf|d8}t||\}}}| d|| d|q(| d|t| d}t||| | t|d\}}}||| } ||| } |r|d7}nd}|g} |g} q| d|}t| }| |d|d}t||\}}}| |}t||\}}}| |}| ||d} | ||d} ||}t||| | t|d\}}}|r||d||g||fS| |}t| }| ||d}t||\}}} | |}!t||\}"}#}$| |"}%| |!|%d} | |!|%d} ||d||g| | fS)zTo use for n>400 000 000rrrr<r=) r]r rrrFr:popextendr rQ)&rrr=ZsbZnumber_goodblocksm2rrrZTfZVfZ goodpointsrrZznABrErRr[strZvrbrartsvsbsZas1qZtqZvqbqZaqttZvtZbtatr r r!search_supergood_blocks                               rqcCsDd}|d}tdt|D]$}||}||dkr:|d7}|}q|SNrr)rr)rcountvoldrvnewr r r!r72s r7cCsd}|d}|d}t||\}}} d} d} t|d|dD]} t|| \} }}t|}| |krx|| | krx| d7} qZ|| | }|||d|t|}|d|}|dkr|d}| } | ||}}} q>|dd}|S)N(rrz%sz)(r_)r rrrr r7)rblockrrrrrrrZb0rrOrrrb1ZlgTLrsr r r!pattern_construct<s,      rzFTcCst|}|dkr || S|dkr0td|j}zt||\}}||_|dkrht||\}}} } nt|||\}}} } |d|d} t|| | | |j |d\} } } |rt ||| | } t ||}t ||| | | |}| d|}W||_n||_0|r| }|r|||| fS|SdS)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`rIN)rYzetazero conjugate ValueErrorrJcomp_fp_tolerancer"rqrFr6rzmaxrTrM)rrinforound wpinitialrPr=rrwrrrrErrJrrr r r!r|Ts:<      r|cCs^|dkrd||d}nd}|j}z,|j|7_t|||j}W||_n||_0|S)Nl a$r+r*r)rKrJrY siegelthetapi)rrr(rJhr r r! gram_indexsrc Cs|d}|d}|d}|d}d}||krZ||} || dkrD|d7}| }|d7}||}q ||} | |dkrx|d7}|SrrrH) rrrrrsrtZtoldZtnewrrurr r r!count_tos     rcCs:t|||}|dkr d}n|dkr.d}nd}||fS)Ni/hYgMb@?r&g?rU)r)rK)rrrPr=r r r!rsrcCs8|dkr dSt||}t||}|j}t||\}}||_||}|dkr\|dkr\dS|dkrp|dkrpdS|ddkrt||d}nt||d|}|d\} } | | dkr|dd} || dkr||_|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. J. J. te Riele and D. T. Winter in 1986. g%fD,@rr_rrr{r+r`) rrYfloorrJrr r"rqrFr6r)rrr.rrrPr=rZRblockZn1Zn2rrrwrrrrErr r r!nzerossB%         rcCs||d|||jS)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)rrr r r! backlunds!s$riiz(00)3iaidiiz3(00)i=i=i oioiKiNi'i'iDiDi5i5i"i"i͜JiМJi+di.diOeiRei6٧i:٧z(00)40i (i(iR4yiU4yiýiýieieiii2i5i^RiaRi(i(iL=iO=iGi"Givi"viiiii Ui Ui_i_ieieihihi$.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! i! i{ i{ i" i% iy iy i{ i~ ii ii i=& i=& i|E i|E iwL izL iY iY ioY irY i] i] z3(010)i$` i'` if if iˁg i΁g ib| ie| i` i` i3 i6 i i i߀ i i(> i+> iO iR i i i iÀ i i i i ie ih ij im i.W i1W i0` i3` i)T/ i,T/ i S i S iX iX i]Y i`Y iKiAKi{~i~~iV iY iHiHiii*i*iiiii3i6iiiiiY#iY#i#i#i$i$iNgiQgidligliůiůipipi\i\i׊iڊiNiQiiiiiAiAi .i .i2i2ipipiiiiiii'*i**iZ35i^35i<7i<7i'=i*=iAiAiEiEiEiEiPUiPUi{Xi{Xififiiiniri1i1i@i@iii?i?iiiii*i-i$i$z04(010)i9i<iiiImiLmisisiІ4iԆ4iZ6i]6i;i;iBiBicici vi vidzidzi}i}i8i8i5i5ixi{iʯiʯiGSiJSiiiQNiTNipipiiizi~iii6i6i}i}iViVi/i/iMo%iPo%i*i*i-i-iA@iA@iNAiRAiGCiJCizXizXi'\i*\i\i\i'li'li1li4lifrifriTKxiWKxi|i|i꣗iiљiљiiiiiii2ji5jiۣii ]i]z032(00)i΄c,iӄc,z(010)40iӹ,iӹ,i9i9i%:i%:z(00)410iw;iw;it?it?z(00)230iqHivHi }Ji}JiQiQ)NN)FT)__doc__Z functionsrrr"r)rFrTr]r r:rqr7rzr|rrrrrrr r r r!s:#  F% h [   G U