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Hi

is there any lower bound for $\Re\zeta(1+it)$.

I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.

If it is true, is there any reference to prove it. thanks

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There is no lower bound for Re$(\zeta(1+it))$. This can be found in Lamzouri's paper http://www.math.uiuc.edu/~lamzouri/distribzeta.pdf (see e.g. page 3, formula (5)). (Choose $\tau$ large enough, $\theta_1=3\pi/4$ and $\theta_2=5\pi/4$ for and substract). The results of Lamzouri however also implies that on average the argument of $\zeta(1+it)$ is small and that Re$(\zeta(1+it))$ is positive more often than it is negative.

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No, this is not true; see Table 5 in http://arxiv.org/abs/1001.2962 and conclusions. In particular the real part is negative for $t=682112.9$ ; and this the smallest value given there (and it was found via testing at steps of size $.1$ so perhaps no much smaller ones were missed).

You might also be interested in this question's answers for related information for the critical line $1/2 + it$

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