Is anything known about existence and/or location of such zeroes ?
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1$\begingroup$ Haseo Ki and Steven M. Gonek are writing a paper on pair correlation of zeroes of the real part of $\zeta$. $\endgroup$– Sylvain JULIENCommented May 10, 2017 at 23:37
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$\begingroup$ Very grateful to Sylvain, Stopple, Juan for their authoritative responses. $\endgroup$– TPTWCommented May 11, 2017 at 22:36
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$\begingroup$ May I ask further, the two papers quoted not seeming to go into any detail about the same question for zeta-derivative(1+it), whether similar facts exist for this ? Arises out of question 'where does d/dt(|zeta(1+it)|^2) =0' ? $\endgroup$– TPTWCommented May 11, 2017 at 23:46
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$\begingroup$ @TPTW There are many zeros o f $\zeta'(s)$ with real part $>1$. For each one of these zeros there are two zeros of $\Re \zeta'(1+it)$ and $\Im\zeta'(1+it)$. For the derivative there are parallel real and imaginary lines (to the real axis) separated by $\pi/\log2$. Each one of these parallel lines gives a zero of $\Re \zeta'(1+it)$ or $\Im\zeta'(1+it)$ respectively. The x-ray of $\zeta'(s)$ show this, but I have not published this. $\endgroup$– juanCommented May 12, 2017 at 7:45
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$\begingroup$ @juan V. many thanks juan. I knew about zeroes zeta-dash from Titchmarsh but yr research on the x-ray quite new to me and v.interesting. $\endgroup$– TPTWCommented May 12, 2017 at 21:05
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2 Answers
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Theorem 11.9 in Titchmarsh (with $\sigma_0=1$) tells us the values of $\log(\zeta(1+it))$, $t>1$ are everywhere dense in the plane. Here $\log(\zeta(1+it))$ is defined by continuation along this line from $\sigma>1$. From this one can see the imaginary part of $\log(\zeta(1+it))$ assumes all integer multiples of $\pi$, and so the imaginary part of $\zeta(1+it)$ is zero infinitely often, and similarly for the real part.
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The solutions of $\Re\zeta(1+it)=0$ are scarce. They limit small intervals where $\Re\zeta(1+it)<0$. The probability in the sense of the limit of the quotient of the measure of the set $\{0<t<T: \Re\zeta(1+it)<0\}$ by $T$ is $d(1)=(3.80\pm0.01)\times 10^{-7}$. R. Brent, J. van de Lune and I show it in https://arxiv.org/abs/1112.4910
We compute the first 50 such small intervals the first was situated around $t=682112.9$ and have length $0.0529$ and the last of these 50 is around $t=8299958.2327$ of length 0.0432. These are the first 100 zeros of $\Re\zeta(1+it)=0$.
The solutions of $\Im\zeta(1+it)=0$ are at least $cT$ with $c\ge \pi/\log 2$, because there are real lines parallel to the real axis that cut the line $\sigma=1$ in the x-ray of $\zeta(s)$ at distances $\pi/\log 2$, and each line gives a zero. There are other real lines that cut the line $\sigma=1$, they are increasing in number with $t$.
Apart from this an intuitive idea can be obtained from my paper on x-ray's of $\zeta(s)$ https://arxiv.org/abs/math/0309433
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$\begingroup$ Hence the two first solutions of $\Re\zeta(1+it)=0$ are (with 30 decimal digits) $$t_1=682112.891338239941159556828817,\quad t_2=682112.944250491762439022676048$$ $\endgroup$– juanCommented May 11, 2017 at 19:35