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The link http://mathworld.wolfram.com/Xi-Function.html claims "$\xi(1/2+it)$ is purely real".

Looking at the definition of Landau $\xi$ function, this doesn't seem trivial. I have been trying to find a proof and haven't found one yet.

Can someone point me to a resource with such proof?

Thanks

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    See whether this helps you: https://terrytao.wordpress.com/tag/riemann-zeta-function/ – 2017-01-07
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    It looks like many of the citations in that part of the page point to "Riemann's Zeta Function" by Edwards. I'd try looking in that if you can borrow it from a library (I wasn't able to find it online). – 2017-01-07
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    @Rohan, Thanks. I looked through the link you suggested but failed to locate relevant theorem. If you were thinking of any particular theorem, would you mind pointing it out. – 2017-01-07

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$\xi$ satisfies the following equations: $$ \xi(s)=\xi(1-s), $$ (the functional equation), which corresponds to $ \xi(1/2+it) = \xi(1/2-it) $, and $$ \overline{\xi(\bar{s})} = \xi(s), $$ which follows from the definitions of the functions involved in the definition of $\xi$ (and is inevitable since $\xi$ is real on the real axis, by the Schwarz reflection principle). For real $t$, this corresponds to $ \overline{\xi(1/2-it)} = \xi(1/2+it). $

Hence $$ \overline{\xi(1/2+it)} = \overline{\xi(1/2-it)} = \xi(1/2+it), $$ which is true if and only if $\xi(1/2+it)$ is real.