Fix zeta sign flipping in some input regions (#420) (#422)

Author: quytelda

src/gamma.jl

@@ -439,7 +439,7 @@ function _zeta(s::ComplexOrReal{Float64})
             lg = loggamma(1 - s)
             rehalf = real(s)*0.5
             return zeta(1 - s) * exp(lg + absim*halfπ + s*log2π) * inv2π * Complex(
-                sinpi(rehalf), copysign(cospi(rehalf), imag(s))
+                sinpi(rehalf), flipsign(cospi(rehalf), imag(s))
             )
         else
             return zeta(1 - s) * gamma(1 - s) * sinpi(s*0.5) * twoπ^s * invπ

test/gamma.jl

@@ -262,6 +262,12 @@ end
     @test 1e-10 > relerr(zeta(.4 + 4053.0im), -0.1248993234383550+0.9195498409364987im)
     @test 1e-13 > relerr(zeta(.4 + 12.01im), 1.0233184799021265846512208845-0.8008078492939259287905322251im)
     @test zeta(.4 + 12.01im) == conj(zeta(.4 - 12.01im))
+
+    # issue #420
+    @test zeta(-2+13im) ≅ conj(zeta(-2-13im)) ≅ -0.30019019877262619754737023564024299182018857012958761814433485-5.5583626885487917197617298283836431070419020764882132809770386im
+    @test zeta(-6+13im) ≅ conj(zeta(-6-13im)) ≅ 133.4764526350263089084083707864441932569167866714712267139316498-54.15465727586582149098585229287107039070546786014930791081909684im
+    @test 1e-12 > relerr(zeta(-2+13im, 3), 2.3621038290867825837364823054-3.9497600485207119519185591345im)
+    @test 1e-12 > relerr(zeta(-2-13im, 3), 2.3621038290867825837364823054+3.9497600485207119519185591345im)
 end
 
 @testset "logabsbinomial" begin