Tagged Questions

Questions about the famous conjecture from Riemann saying that the non-trivial zeroes of the Riemann Zeta function all lie on the so-called critical line $\Re(s)=\dfrac{1}{2}$, its various generalizations and the different approaches towards its solution.

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Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?
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Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?

Numerical evidence suggests that the complex zeros of: $$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$ all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...
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Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?

Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of: $$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$ are on the ...
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Are there infinitely many zeros of $\chi(s)+ \dfrac{2^{s}- 2^{2s-1}}{2^{s}-1}$ on the critical line?

Take $\chi(s)= 2^s\,\pi^{s-1}\,\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma(1-s)$, so that $\zeta(s)=\chi(s)\,\zeta(1-s)$. The zeros of $\chi(s)=-1$ and the non-trivial zeros $\rho$ of $\zeta(s)$, seem ...
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Lexicographic distribution of irreducible polynomials

Let $A = {\mathbb F}_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its ...
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Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime. Let'...
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Are all zeros of $\xi(a\,s) \pm \xi\left(a\,(1-s)\right)$ on the critical line for $\forall a \in \mathbb{R}/0$?

This question expands on this one and seems to have a stronger result. Take the Riemann $\xi$-function $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. We ...
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Are the complex zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$ all on the critical line for $a \lt 0, a \ge 1$?

With $s \in \mathbb{C}, a \in \mathbb{R}$, numerical evidence strongly suggests that the complex zeros in the critical strip of: $$\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$$...