The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the critical strip off the critical line?

This question came to my mind considering the sequence of trivial zeros in decreasing order for zeta as a $2$-periodic signal whose Fourier transform would be a $1/2$-periodic signal made of Dirac peaks (the former future physicist in me is speaking, sorry), which if we compactify partially the critical strip by identifying the vertical lines of real parts $0$ and $1$ becomes a single Dirac peak which is supported on $1/2$, hence the real part of the non trivial zeros under RH. So my idea is that adding a Landau-Siegel zero would create a non periodic signal made by the decreasing sequence of real zeros, whose Fourier transform would not be periodic either, suggesting the existence of a complex non trivial zero in the critical strip off the critical line.

So would the existence of a Landau-Siegel zero create such a havoc that the analogue of RH for the considered Dirichlet L-function would fail completely?

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    $\begingroup$ No such implication is known. Our current state of knowledge permits that GRH could hold with the exception of a single Siegel zero for a single primitive Dirichlet L-function. $\endgroup$
    – Wojowu
    Sep 19 at 20:14
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    $\begingroup$ As Wojowu notes this is not known. @Wojowu: that said, the definition of Landau-Siegel zeros only makes sense in the context of an infinite collection, so "a single Siegel zero for a single primitive Dirichlet L-function" doesn't make too much sense. $\endgroup$
    – Mark Lewko
    Sep 19 at 20:46
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    $\begingroup$ @MarkLewko That's true, I suppose I meant a single real zero of such an L-function. But even an infinite family like usually considered in this context isn't known to have any such implications as far as I'm aware. $\endgroup$
    – Wojowu
    Sep 19 at 20:58
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    $\begingroup$ Just curious, why did this question got 3 downvotes? $\endgroup$ Sep 20 at 1:59
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    $\begingroup$ A pedantic note: "Single (or only) real zero" here means both the exceptional zero and its reflection under $s\rightarrow 1-s$. $\endgroup$
    – user334725
    Sep 20 at 13:37

A result of Sarnak and Zaharescu, stated in the contrapositive, implies the existence of a complex zero off the critical line for at least one Dirichlet L-function if one has a sufficiently strong Siegel zero: ProjectEuclid link

  • $\begingroup$ I think their phrase of "$L$-functions under consideration" includes the $L$-function of a specific elliptic curve (and its twist by the exceptional character $\chi$). So it is not a Dirichlet $L$-function necessarily, that has the nonreal zero off the critical line. $\endgroup$
    – user334725
    Sep 20 at 13:06
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    $\begingroup$ This is the case for Theorems 2 and 3 of that paper, but Theorem 1 only involves Dirichlet L-functions (though the catch is that the implied constant in "sufficiently strong" becomes ineffective). $\endgroup$
    – Terry Tao
    Sep 20 at 15:26

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