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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_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 X/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/mpmath/functions/zetazeros.pyfind_rosser_block_zeror#s 3)*A- . = QqS !! $ QqS !! $ 1W1Q3!8q!Bq!B$B$BqSUN !! 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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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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. 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