# 自由黎曼氣體

## 模型

${\displaystyle \epsilon _{1}<\epsilon _{2}<\epsilon _{3}...}$，且：
${\displaystyle a_{1},a_{2},a_{3}...}$是与之对应的湮灭算子。则真空态${\displaystyle |\Omega \rangle }$和所有粒子态：
${\displaystyle |k_{1},k_{2},k_{3}...\rangle \equiv (a_{1}^{\dagger })^{k_{1}}(a_{2}^{\dagger })^{k_{2}}...|\Omega \rangle }$${\displaystyle k_{i}\in \mathbb {N} }$

${\displaystyle p_{1}

${\displaystyle |k_{1},k_{2},k_{3}...\rangle \mapsto N\equiv p_{1}^{k_{1}}p_{2}^{k_{2}}p_{3}^{k_{3}}...}$

### 能级和正则配分函数

${\displaystyle \epsilon _{i}=\ln p_{i}}$

${\displaystyle E_{N}=\sum _{i=1}^{\infty }k_{i}\epsilon _{i}=\sum _{i=1}^{\infty }k_{i}\ln p_{i}=\ln N}$

${\displaystyle Z(\beta )=\sum _{N=1}^{\infty }\exp(-\beta E_{N})=\sum _{N=1}^{\infty }{\frac {1}{N^{\beta }}}=\zeta (\beta )}$

${\displaystyle Z(\beta )=\prod _{i=1}^{\infty }{\frac {1}{1-\exp(-\beta \epsilon _{i})}}=\prod _{i=1}^{\infty }{\frac {1}{1-p_{i}^{-\beta }}}}$

## 超素数子

${\displaystyle f_{1},f_{2},f_{3}...}$

${\displaystyle |k_{1},k_{2},k_{3}...l_{1},l_{2},l_{3}...\rangle \equiv (a_{1}^{\dagger })^{k_{1}}(a_{2}^{\dagger })^{k_{2}}...(f_{1}^{\dagger })^{l_{1}}(f_{2}^{\dagger })^{l_{2}}...|\Omega \rangle }$${\displaystyle k_{i}\in \mathbb {N} }$${\displaystyle l_{i}\in \{0,1\}}$

${\displaystyle N\equiv p_{1}^{k_{1}+l_{1}}p_{2}^{k_{2}+l_{2}}p_{3}^{k_{3}+l_{3}}...}$
${\displaystyle d\equiv p_{1}^{l_{1}}p_{2}^{l_{2}}p_{3}^{l_{3}}...}$

${\displaystyle (-1)^{\hat {F}}|N,d\rangle =\mu (d)|N,d\rangle }$

${\displaystyle \mu (d)=+1}$，若${\displaystyle d}$的素因子数目为偶；
${\displaystyle \mu (d)=-1}$，若${\displaystyle d}$的素因子数目为奇。

### 威腾指标与素数定理

${\displaystyle \Delta \equiv \mathbf {Tr} (\exp(-\beta {\hat {H}})(-1)^{\hat {F}})}$

${\displaystyle \Delta =\mathbf {Tr} (\exp(-\beta {\hat {H}}_{f})(-1)^{\hat {F}})\mathbf {Tr} (\exp(-\beta {\hat {H}}_{b}))}$
${\displaystyle \mathbf {Tr} (\exp(-\beta {\hat {H}}_{b}))=Z(\beta )=\zeta (\beta )}$
${\displaystyle \mathbf {Tr} (\exp(-\beta {\hat {H}}_{f})(-1)^{\hat {F}})=\sum _{d}\exp(-\beta {\hat {E}}_{d})\mu (d)=\sum _{d}{\frac {\mu (d)}{d^{\beta }}}}$

${\displaystyle \Delta =\sum _{N=1}^{\infty }\sum _{d|N}\exp(-\beta E_{N})\mu (d)}$

${\displaystyle \sum _{d|N}\mu (d)=\delta _{N,1}}$
${\displaystyle \Delta =\exp(-\beta E_{1})=1}$

${\displaystyle \sum _{d}{\frac {\mu (d)}{d^{\beta }}}=\zeta ^{-1}(\beta )}$

## 参考文献

1. André LeClair, Giuseppe Mussardo. Generalized Riemann hypothesis, time series and normal distributions. Journal of Statistical Mechanics: Theory and Experiment. 2019-02-15, 2019 (2): 023203 [2019-08-07]. ISSN 1742-5468. doi:10.1088/1742-5468/aaf717.[永久失效連結]
2. D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathematical Physics, 1990, (127): 239–252
3. ^ Bernard L. Julia, J. M. Luck, P. Moussa, M. Waldschmidt , 编, Statistical theory of numbers, Number Theory and Physics, Springer Proceedings in Physics (Springer-Verlag), 1990, 47: 276–293
4. ^ I. Bakas, M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys, 1991, (32): 1881
5. ^ D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys, 1998, (39): 1919–1927