Question on $\xi(\sigma + it)$

Question

I am aware that $\xi(\sigma + it)$ is real for $\sigma = 1/2$. Are there other known values of $\sigma$ for which $\xi(\sigma + it)$ is also real?

Thanks

why has someone recommended that we close this. Could you please explain. Isn't it a fair question? Thanks — sku, Sep 26 '17 at 16:22
Any non-constant analytic function has infinitely many curves where it is real... See any complex analysis course — reuns, Sep 26 '17 at 17:24
Is there another value of $\sigma$ in the range $0 < \sigma < 1$ where the function in question is real? Thanks — sku, Sep 26 '17 at 17:40

gammatester · Accepted Answer · 2017-09-26T20:45:10.660

1

$\xi(\sigma+it) $ is real for $\sigma \in \mathbb{R}, t=0$.

There are other real values: For the X-rays (lines of pure real or pure imaginary values) of $\xi(s)$ I found https://www.numbertheory.org/pdfs/xrays.pdf. In the picture below you see the imaginary part of $\xi(x + 10i)$ for $x=-10\dots 10.$

edited Sep 26 '17 at 20:45

answered Sep 26 '17 at 16:25

gammatester

18,827

I didn't see the $t = 0$ restriction. Anything for non-zero $t$? Thanks – sku Sep 26 '17 at 16:38
See the updated answer. – gammatester Sep 26 '17 at 20:47

Question on $\xi(\sigma + it)$

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