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I want to know what the sign of the real part and imaginary part of $\zeta(1/2+x+iy)$ and $\zeta(1/2-x+iy)$ are ,are they the same? for example in this case they are the same

zeta(0.25+I 10)=0.74513-0.0507106 I

zeta(0.75+I 10)=0.66-0.0517 I

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    $\begingroup$ If you like my answer, please accept it officially (so that it turns green). Thanks in advance! $\endgroup$
    – GH from MO
    Commented Aug 6, 2023 at 19:51

1 Answer 1

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Let $\rho=1/2+iy$ be the first zero of $\zeta(s)$. Then $\zeta'(\rho)$ has nonzero real and imaginary parts, and $$\zeta(\rho+h)\sim\zeta'(\rho)h\qquad\text{as}\qquad h\to 0.$$ That is, if $x>0$ is sufficiently small, then both the real parts and the imaginary parts of $\zeta(1/2\pm x+iy)$ have opposite signs.

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  • $\begingroup$ it seems to me that the real and imaginary parts of $\zeta(1/2-x+iy)$ and $\zeta(1/2+x+iy)$ have the same sign. $\endgroup$
    – Kevin67
    Commented May 5, 2023 at 13:46
  • $\begingroup$ a counterexample will suffice. $\endgroup$
    – Kevin67
    Commented May 5, 2023 at 13:54
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    $\begingroup$ Well, I proved that sometimes both the real parts and the imaginary parts of $\zeta(1/2\pm x+iy)$ have opposite signs. For a concrete counterexample, consider $\zeta(0.1+14.1 i)\approx -0.358787 - 0.10908 i$ and $\zeta(0.9+14.1 i)\approx 0.271012 + 0.0147528 i$. $\endgroup$
    – GH from MO
    Commented May 5, 2023 at 14:00
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    $\begingroup$ thanks for your answer $\endgroup$
    – Kevin67
    Commented May 5, 2023 at 15:53

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