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The Riemann Hypothesis as a Stability Principle

  The Riemann Hypothesis can be reformulated as an infinite sequence of explicit positivity tests. This is not a proof. It turns a statement about complex zeros into inequalities that can be calculated, falsified at finite order, and compared with operator or physical models.   The completed zeta function is $$\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s),$$ with $$\Xi(t)=\xi\left(\frac12+it\right).$$ The function $\Xi$ is even and real on the real axis. A nontrivial zero $\rho=\beta+i\gamma$ corresponds to $$t_\rho=\gamma-i\left(\beta-\frac12\right).$$ Therefore $$t_\rho\in\mathbb R \quad\Longleftrightarrow\quad \beta=\frac12,$$ and hence $$\boxed{\mathrm{RH}\quad\Longleftrightarrow\quad\text{every zero of }\Xi\text{ is real}.}$$ Folding the Zeros Because $\Xi$ is even, write $$\Xi(t)=\sum_{n\ge0}c_nt^{2n}.$$ Now define $$G(z)=\frac{\Xi(i\sqrt z)}{\Xi(0)}.$$ Although this contains $\sqrt z$, the even expansion makes $G$ entire: $$G(z)=...
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Remarks on the cosmological constant and minimal acceleration

As Lineweaver explained, because our Universe is expanding at an accelerating rate, our Universe has a cosmic event horizon (CEH).  

Its an event horizon Jim, but not as we know it...Image credit: DALL.E2 by SR Anderson

Events beyond the CEH will never be observed. The CEH is also the source of de Sitter radiation, which has a specific temperature $T_{dS}$. It is the minimum possible temperature of the Universe, and, it is not absolute zero (zero Kelvins). Numerically $T_{dS} \approx 2.4 \times 10^{-30}K$, the universal minimum (black body) `absolute cold' local temperature of the future dS state. As a comparison, in 2020 the NASA Cold Atom lab was able to cool an atom to a record low $\sim 2 \times 10^{-7}K$. 

Now, in any theory one may think of temperature as an energy, and from the semi-classical Unruh relationship, temperature $\sim$ acceleration. Therefore:  

\begin{equation}
\notag
E_{dS} = T_{ds} \ k_B= \frac {\hslash a}{2\pi c}
\end{equation}

What we get from Unruh is $a=cH$, the universal background (local) minimum acceleration. $H$ is the future Hubble constant (as you probably know, this constant is not actually...constant). You could also express this as saying that our present Universe has a de Sitter attractor in our infinite far future. Yes, this acceleration scale is also similar to the MOND acceleration, but no-one is quite sure what this coincidence means. 

Gibbons taught us that a theory with a minimum length should have a maximal acceleration and a maximal temperature. Now, we all know that in classical General Relativity (GR) there is no such thing as a minimum length. However, in most approaches to quantum gravity, which includes semi-classical approximations such as Hawking's black hole temperature and the Unruh relationship, there is such a beast. 

 

(Lambda, the symbol used for the cosmological constant) 

Here of course, we are talking about the local minimum universal acceleration, which implies the existence of a maximum length scale, being the de Sitter characteristic length $l_{\Lambda}$ the future cosmic event horizon radius. Numerically $l_{\Lambda} \sim 16 \ Gly$. You can also write $l_{\Lambda} =\sqrt{ {3}/{\Lambda}}$, showing that the cosmological constant is the only parameter in dS space.  

In 2001, Jacob Bekenstein and Avraham Mayo showed that, as far as entropy and information flow are concerned, black hole radiation behaves like a one dimensional channel. For ideal unidirectional bosonic thermal channels $$ P_{\mathrm{ch}} = \frac{\pi k_{B}^{2} T^{2}}{12\hbar} \qquad \dot{S}_{\mathrm{ch}} = \frac{2P_{\mathrm{ch}}}{T}$$ This is radiation entropy throughput, maximal at the specified channel power. If enough independent channels can carry radiation entropy equal to the horizon entropy in one characteristic crossing time, our calculation gives $$P_{\rm transport}=N_{\rm required}P_{\rm ch}=\frac{T_{\rm dS}S_{\rm dS}}{2\tau}=\frac{E_{\rm MS}}{2\tau}=\boxed{\frac{c^5}{4G}}$$ $$
s\equiv\frac{S_{\rm dS}}{k_B}=\frac{\pi R^2}{\ell_P^2},
\qquad
\tau\equiv\frac{R}{c},
\qquad
T_{\rm dS}=\frac{\hbar}{2\pi k_B\tau}
$$ Equal entropy does not establish complete information transfer: the radiation must preserve every initial quantum state, while everything inaccessible loses all dependence on that state. 

This does not independently prove the maximum force conjecture, but it does show a connection between holographic entropy saturation and the associated power scale $c^5/(4G)$.  

Of course, strong gravitational backreaction may be exactly what prevents the nominal capacity of the bosonic thermal channels from being used simultaneously. This certainly happens if the transported energy accumulates as a central spherical mass at fixed $\Lambda$ (it exceeds the static de Sitter black-hole mass limit). 

  


 

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