So what happens if there is a non-trivial zero of the Riemann zeta function off the critical line? Has there been any work in the following direction: We know from Landaus theorem that there is a positive proportion of the zeros $\alpha$ on the critical line. Suppose that $\rho$ is a a non-trivial zero of $\zeta$ off the critical line. Then can we use this to cook up an argument to show the existence of another zero off the line? I am being highly optimistic here but continuing in this direction maybe we can show there will be infinite set of zeros off the critical line and going even further we may try to show that the density of this set is greater than $1-\alpha$, yielding the desired contradiction.
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asked Oct 8, 2017 at 18:32
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5$\begingroup$ You probably know this, but only mentioning "positive proportion" is a little outdated: as everyone knows, that 'positive proportion' was proved by N. Levinson to be $\geq\frac13$ and J. B. Conrey in 1989 published [J. reine angew. Math. 399 (1989) page 25] that it's $\geq0.4088>\frac25$. $\endgroup$– Peter HeinigCommented Oct 8, 2017 at 18:51
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5$\begingroup$ @PeterHeinig: Feng (Journal of Number Theory 132 (2012), 511-542) proved that the proportion is $\geq 0.4128$. $\endgroup$– GH from MOCommented Oct 8, 2017 at 19:20
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8$\begingroup$ To be pedantic, the nontrivial zeros of $\zeta$ off the critical line come in sets of four, since if $\rho$ is a zero inside the critical strip then so are $\bar\rho$, $1-\rho$, and $1-\bar\rho$. So presumably the question should mean "another zero besides these four". $\endgroup$– Greg MartinCommented Oct 8, 2017 at 19:22
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2$\begingroup$ And thus it would "suffice" to prove that the proportion of "independent zeros" is at least $ (1-\alpha)/4+\varepsilon $ to conclude. $\endgroup$– Sylvain JULIENCommented Oct 8, 2017 at 19:47
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2$\begingroup$ @GHfromMO Kyle Pratt and Nicolas Robles have increased that proportion to $\geq 0.4149$; see arxiv.org/abs/1706.04593 $\endgroup$– Pace NielsenCommented Oct 9, 2017 at 0:55
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2 Answers
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1. First, let us get history right: Hardy (1914) proved there are infinitely many zeros on the critical line, Hardy-Littlewood (1921) proved there are $\gg T$ zeros on the critical line up to height $T$, and Selberg (1942) proved there are $\gg T\log T$ zeros on the critical line up to height $T$ (i.e. positive proportion).
2. I am quite certain that it is not known that if there exists a nontrivial zero off the critical line, then there are infinitely many such zeros.
answered Oct 8, 2017 at 18:55
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4$\begingroup$ To state something obvious : maybe not infinitely many, but at least two (or four) due to the functional equation of Zeta. $\endgroup$ Commented Oct 8, 2017 at 19:22
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Perhaps this is a duplicate of this question If the Riemann Hypothesis fails, must it fail infinitely often?.
The discussion continues here The Hardy Z-function and failure of the Riemann hypothesis, where a hypothesis is proposed, which implies If the Riemann Hypothesis fails, must it fail infinitely often?.
To this hypothesis there are two approximations based on: 1. Zeta function universality 2. GUE hypothesis.
Concerning 1. I can say that even, $\zeta (s)$ for $Re(s)=\frac{1}{2}$ is dense in $\mathbb{C}$, is unknown.
2. it seems to me it will be easier. It is enough to make an inversion for the expression $N(t_{n})=n$, where $N(t_{n})$ the number of non-trivial zeros of the zeta function, which is well-known function and $t_{n}$ the nth non-trivial zero, and define the property of the recurrence.
