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Zeta(s) at a complex s through the vouched Hurwitz kernel - #559
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…s its rounding error, repeats in double-double and declines (the raw Euler-Maclaurin sum was 1.6e-9 off at Im s = 1e6); Zeta(s) at |Im s| >= 1e5 stays unevaluated; add mpmath table. Fix cortex-js#552
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Zeta(s)at a complexstakes the Hurwitz kernel ata = 1, so its digits are vouched for or the call stays unevaluated. Fixes #552library/arithmetic.ts,Zetahandler: a finite complex literalsunderN()(or a float operand) callshurwitzZetaComplexWithError(s, 1), the kernelHurwitzZeta(s, 1)already uses. A declined kernel (undefinedor a NaN value) leaves the call as it is. A realskeepszeta/bigZeta.zetaComplexandriemannZetaComplexare unchanged: the polylog Crandall expansion reads them.Accuracy, against mpmath 1.3.0 (decimal operands), 64 points,
Re sfrom -5.5 to 3 (including 0, 1 and 0.1),|Im s|from 1 to 10⁶:Im s= 10⁶); 7.3e-11 at 10⁵; 5.9e-13 at 400 and 1000; 26 points above 1e-13The 16 points with
|Im s| >= 10⁵are declined, asHurwitzZeta(s, 1)already declines them: those calls, which used to print a value with 9 to 13 correct digits, are now unevaluated. A call takes about 2 ms atIm s= 400.Tests:
test/compute-engine/zeta-complex-vouched.test.tshas the 48 answered rows at precisions 15, 21 and 30, each also checked to equalHurwitzZeta(s, 1)exactly, the 16 declined rows, exact operands symbolic underevaluate(), a float operand underevaluate(), and the pole and real arguments unchanged.Left alone:
Zetais a real-only lowering in compiled code (JS_REAL_ONLY_LOWERINGS), so a complex operand is NaN there, by that design. The Hurwitz kernel's own double-double pass is what vouches here; I did not widen its bound.Merge:
git merge-treeof this branch with #535 and #536 conflicts only inCHANGELOG.md(each adds the## [Unreleased]heading);arithmetic.tsmerges cleanly.