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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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Minimum and Maximum Mass from the Cosmic Horizon

 

A positive cosmological constant sets the future cosmic event-horizon radius of an expanding FLRW universe:

$$l_\Lambda=\sqrt{\frac{3}{\Lambda}}.$$

Associate two lengths with a positive mass $m$:

$$\bar\lambda_C=\frac{\hbar}{mc} \qquad r_g=\frac{2Gm}{c^2}$$

where $\bar\lambda_C$ is the reduced Compton wavelength and $r_g$ is the gravitational radius.

Requiring both lengths not to exceed the cosmic horizon,

$$\bar\lambda_C\le l_\Lambda \qquad r_g\le l_\Lambda,$$

gives

$$\boxed{ m_{\min}= m_s=\frac{\hbar}{cl_\Lambda} =\frac{\hbar}{c}\sqrt{\frac{\Lambda}{3}} }$$

$m_s$ is the minimum mass first pointed out by Wesson, 2003. This was also adopted by Bohmer and Harko, 2005 to derive a minimum density for static bodies  using the cosmological Buchdahl inequality.

and

$$\boxed{ m_{\max}=\frac{c^2l_\Lambda}{2G} =\frac{c^2}{2G}\sqrt{\frac{3}{\Lambda}} \equiv m_{\rm CEH}. }$$

Wesson also conjectured a maximum mass, but was off by a factor of 2, as well as incorrectly assigning it as the mass of the observable universe, rather than the cosmic event horizon. In an FLRW universe, matter and radiation dilute with time, while the apparent and event horizons approach $l_{\Lambda}$. The mass inside the limiting causal region therefore approaches $m_{CEH}$. 

Barrow, 2014 also re-derived the minimum and maximum masses to be $m_{min}=m_s$ and $m_{max}=m_{CEH}$, keep in mind there is a factor-of-two error in their published final result (Eqn. 26) for $m_{max}$.  

Thus

$$\boxed{ \frac{\hbar}{c}\sqrt{\frac{\Lambda}{3}} \le m\le \frac{c^2}{2G}\sqrt{\frac{3}{\Lambda}} }.$$

The upper endpoint is the vacuum mass enclosed by the cosmic horizon. The vacuum mass density is $\rho_\Lambda=\Lambda c^2/8\pi G$, so

$$\frac{4\pi}{3}\rho_\Lambda l_\Lambda^3 = \frac{\Lambda c^2l_\Lambda^3}{6G} = \frac{c^2l_\Lambda}{2G} = m_{\rm CEH}.$$

$m_{CEH}$ is a horizon-scale quasi-local mass.

The two endpoint masses satisfy

$$m_{\min}m_{\rm CEH} = \frac{\hbar c}{2G} = \frac{m_{\rm Pl}^2}{2}, \qquad m_{\rm Pl}=\sqrt{\frac{\hbar c}{G}}.$$

Therefore,

$$\boxed{ \bar\lambda_C(m_{\min}) = l_\Lambda = r_g(m_{\rm CEH}) \qquad 2 \ m_{s}m_{\rm CEH} = m_{\rm Pl}^2 }$$

The same cosmic horizon therefore defines a minimum mass whose reduced Compton wavelength fits inside it and a maximum enclosed mass whose gravitational radius reaches it. 

Of course, particles less than the minimum mass can exist (indeed even massless particles). The maximum is the mass associated with the cosmic horizon (the Misner–Sharp quasi-local mass of the static patch evaluated at its horizon) not the maximum mass of a black hole inside de Sitter space.

Furthermore, the  Brown-York energy associated with the length of the CEH is 

$$E_{BY}=\frac{c^4l_\Lambda}{G} = 2m_{CEH}c^2$$ i.e. the Brown-York mass is twice the CEH mass. 

The Friedmann equation emerges!

The observed vacuum energy scale,
\begin{equation}
M_\Lambda \approx 2.35 \times 10^{-12}\ \text{GeV}
\end{equation} can be written in a seesaw form closely related to the previous relations:
\begin{equation}
M_\Lambda
= \left(\frac{3}{8\pi}\right)^{1/4}
\sqrt{M_p\, m_s}
\end{equation} Equivalently, in terms of the vacuum energy density,
\begin{equation}
\rho_\Lambda = M_\Lambda^4
= \frac{3}{8\pi}\frac{M_p^2}{\ell_\Lambda^2}
= 3\, M_{Pl}^2\, L^{-2}
\end{equation} where $M_{Pl} = M_p/\sqrt{8\pi}$ is the reduced Planck mass and $L = \ell_\Lambda$. This expression is exactly consistent with the Friedmann equation
\begin{equation}
H_\Lambda^2 = \frac{\rho_\Lambda}{3 M_{Pl}^2}
\end{equation} Writing the Friedmann equation as a seesaw,
\begin{equation}
M_\Lambda \sim \sqrt{M_p H_\Lambda}
\end{equation} has inspired a lot of different ideas, e.g.

  • running vacuum models, e.g. Espana-Bonet et al, 2004,  Testing the running of the cosmological constant with Type Ia Supernovae at high z
  • holographic dark energy, Hsu, 2004,  A speculative relation between the cosmological constant and the Planck mass
  • string theory, Berglund et al, 2023, String Theory Bounds on the Cosmological Constant, the Higgs Mass, and the Quark and Lepton Masses 

In this sense, spacetime itself has a clock, and its tick rate is set by the Hubble parameter of de Sitter space.  

 So can we get a Minimum length? 

$$ \frac{\hbar}{c\,l_\Lambda} \le m \le \frac{c^2 l_\Lambda}{2G}$$ That mass $m$ requires $$ \frac{\hbar}{c\,l_\Lambda} \le \frac{c^2 l_\Lambda}{2G}$$ which means $$ l_\Lambda^2 \ge 2\,l_{\mathrm{Pl}}^2$$ so $$l_\Lambda \ge \sqrt{2}\,l_{\mathrm{Pl}} \ge \frac{l_\Lambda}{\sqrt{2}}$$ We might call this a "minimum length", where the reduced Compton wavelength equals the Schwarzschild radius. 

Well, not so fast, because the minimum position uncertainty $\Delta x_{UV}$ of a relativistic probe having to contend with both quantum mechanics and gravity is actually $\Delta x_{UV}=2L_p$.  $\Delta x \gtrsim \frac{\hbar}{\Delta p} + \frac{G\,\Delta p}{c^3}$ so $\Delta x_{UV} = 2\ell_{\mathrm{Pl}}$. This also is a "minimum length". Actually Mead who in 1964 first made this argument, multiplied the uncertainties, but I think it is more physical to add them.  

All in one horizon!

$\boxed{ m_s m_{\mathrm{CEH}}=\frac{m_{\mathrm{Pl}}^2}{2}, \qquad \frac{m_{\mathrm{CEH}}}{m_s}=\frac{S_\Lambda}{2\pi k_B}, \qquad m_{\mathrm{CEH}}c^2=T_\Lambda S_\Lambda }$

The same horizon ties together UV localisation, IR gravitation, horizon entropy, and vacuum thermodynamics!


 

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