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Suppose the Riemann zeta function has infinitely many zeros $\rho$ with $\Re(\rho)=\sigma$. Does it follow that for every large $T>0$, there exists some $t$ such that $T<t<3T$, where $t=\Im(\rho)$ ?

Existing literature seems to support this for $\sigma=1/2$.

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    $\begingroup$ I would assume nothing is known. There are no known implications of the form "if there are zeros off the line, then there are many of them". $\endgroup$
    – Wojowu
    Jan 20, 2019 at 15:11
  • $\begingroup$ @Wojowu, it seems so, but doesn't the fact that $t_n \sim \frac{2\pi n}{\log n}$ answer my question in the affirmative ? $\endgroup$
    – neutrino
    Jan 20, 2019 at 15:13
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    $\begingroup$ I assumed with your question you mean a zero $\rho$ with real part $\sigma$ and imaginary part in the interval. This is true for $\sigma=1/2$ but most likely wide open for other $\sigma$. $\endgroup$
    – Wojowu
    Jan 20, 2019 at 15:20
  • $\begingroup$ @Wojowu, indeed, your assumptions are correct. Do you have any idea/reference of the proof for $\sigma=1/2$ ? I guess it wiil be interesting to explore its applicability (or lack thereof) for other $\sigma$. $\endgroup$
    – neutrino
    Jan 20, 2019 at 15:22
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    $\begingroup$ Perhaps the closest thing to this sort of result in the literature is this paper of Sarnak and Zaharescu projecteuclid.org/euclid.dmj/1087575083 , which in the contrapositive asserts that sufficiently poor Landau-Siegel zeroes exist, then there must be complex zeroes of L-functions off of the critical line. $\endgroup$
    – Terry Tao
    Jan 20, 2019 at 18:33

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On the critical line, zeros are analyzed with the Hardy function which is real and has very good approximation formulas (Riemann-Siegel)- tons of references online including a fairly good book by Ivic on this subject (Hardy function), or a more recent one by Iwaniec, so lots of things are doable

Nothing is really known about critical strip zeros off the critical line (they are generally assumed not to exist of course) beyond some classical density theorems which are fairly weak; while there are approximation formulas for the zeta everywhere in the critical strip, they are weaker outside of the critical line as the two sums that are involved are not symmetrical and the approximation is not (essentially - up to function of constant modulus one in fact) real anymore

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To expand on Conrad's reference to 'some classical density theorems', these are generally called Zero Density Theorems, Titchmarsh's book is a good place to start. See also the MathOverflow question Zeta Function: Zero Density Theorems

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