# Arithmetic Gas

Prime numbers can be considered as the building blocks of natural numbers. For example, 12 can be factored in a unique way as $2^{2} \cdot 3$ where 2 and 3 are primes. Thus, prime numbers can be compared to elementary particles, which are the building blocks of matter in physics. The concept of arithmetic gas provides an interesting connection between statistical physics and number theory, for which the Riemann zeta function plays the role of a partition function.

PRIMONS

By the fundamental theorem of arithmetic, every positive integer admits a unique prime decomposition $n=p_{1}^{r_{1}} \ldots p_{k}^{r_{k}}$ where $p_{i}$ denotes the i-th prime number and $r_{i}$ a natural number.

This can be used in the framework of second quantized field theory to define Fock states

$\left|r_{1},\ldots,r_{k}\right\rangle=\left|p_{1}\right\rangle^{r_{1}}\ldots\left|p_{k}\right\rangle^{r_{k}}=\left|n\right\rangle$

which means that $r_{1}$ particles are in state $\left|p_{1}\right\rangle$ , …, $r_{k}$ particles are in state $\left|p_{k}\right\rangle$. Thus, the states can be labeled by positive integers due to the unicity of prime decomposition. The elementary particles associated to the prime numbers $p_{i}$ are called primons.

Alternatively, the creation operators $a_{i}^{\dagger}$ can be associated to the prime numbers $p_{i}$ such that

$\left(a_{1}^{\dagger}\right)^{r_{1}}\ldots\left(a_{k}^{\dagger}\right)^{r_{k}}\left|0\right\rangle =\left|n\right\rangle$

where $\left|0\right\rangle$ denotes the vacuum state.

As such, this model is bosonic because any number of particles can occupy the same state. In order to obtain a fermionic model, it is necessary to consider the Pauli exclusion principle, which corresponds to the constraint $r_{i}\in\left\{ 0,1\right\}$.

RIEMANN GAS

We define the energy of an individual primon as a logarithm

$E_{i}=\log p_{i}$

The energy of a non-interacting multi-particle state $\left|n\right\rangle$ is given by the sum of the energies of the constituents

$\displaystyle E\left(n\right)=\sum_{i=1}^{k}r_{i}E_{i}=\sum_{i=1}^{k}r_{i}\log p_{i}=\sum_{i=1}^{k}\log p_{i}^{r_{i}}=\log p_{1}^{r_{1}}\ldots p_{k}^{r_{k}}=\log n$

Note that the ground state corresponds to $E\left(1\right)=0$.

By definition, the canonical partition function is

$\displaystyle Z\left(s\right)=\sum_{n=1}^{\infty}e^{-sE\left(n\right)}$

where $s$ corresponds to the inverse temperature in physics. A quick computation shows

$\displaystyle Z\left(s\right)=\sum_{n=1}^{\infty}e^{-s\log n}=\sum_{n=1}^{\infty}n^{-s}=\zeta\left(s\right)$

where $\zeta\left(s\right)$ is the famous Riemann zeta function. For this reason, such a system is called a Riemann gas. The zeta function is convergent for $s>1$. It is divergent at $s=1$, which is similar to the Hagedorn temperature. The Euler product formula allows to rewrite the partition function as

$\displaystyle Z\left(s\right)=\prod_{p\textrm{ prime}}\frac{1}{1-p^{-s}}$

DIRICHLET SERIES

In number theory, a function $f:\mathbb{N}^{*}\rightarrow\mathbb{R}$ is called an arithmetic function. Such a function is said to be multiplicative if $f\left(1\right)=1$ and $f\left(ab\right)=f\left(a\right)f\left(b\right)$ whenever $a$ and $b$ are coprime.

The partition function is a particular case of Dirichlet series

$\displaystyle Z_{f}\left(s\right)=\sum_{n=1}^{\infty}\frac{f\left(n\right)}{n^{s}}$

where $f$ is a multiplicative arithmetic function. In the case of the Riemann gas, the function is $f\left(n\right)=1$ for all $n$.

FERMIONIC MODEL

The previous Riemann gas is a bosonic model. In the case of a fermionic model, we have to take into account the constraint $r_{i}\in\left\{ 0,1\right\}$. In order to select the admissible states, we define the arithmetic function

$\mu_{2}\left(p_{1}^{r_{1}}\ldots p_{k}^{r_{k}}\right)=1\textrm{ if all }r_{i}\in\left\{ 0,1\right\} \textrm{ and }0\textrm{ otherwise}$

We can check that this function is multiplicative :

• $\mu_{2}\left(1\right)=1$ is trivial
• if $a$ and $b$ are coprime then their prime decompositions do not have common factors, thus $\mu_{2}\left(ab\right)=\mu_{2}\left(a\right)\mu_{2}\left(b\right)$.

The partition function is given by the Dirichlet series

$\displaystyle Z_{\mu_{2}}\left(s\right)=\sum_{n=1}^{\infty}\frac{\mu_{2}\left(n\right)}{n^{s}}$

Note that $\mu_{2}$ works like a sieve to filter the fermionic constraint. In order to establish a relationship with the zeta function, we compute

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}=\prod_{p\textrm{ prime}}\sum_{r=0}^{\infty}\frac{1}{p^{r}}$

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{p\textrm{ prime}}\sum_{r=0}^{\infty}\frac{1}{p^{rs}}$

$\displaystyle Z_{\mu_{2}}\left(s\right)=\prod_{p\textrm{ prime}}\sum_{r=0}^{1}\frac{1}{p^{rs}}=\prod_{p\textrm{ prime}}\left(1+p^{-s}\right)=\prod_{p\textrm{ prime}}\frac{1-p^{-2s}}{1-p^{-s}}$

$\displaystyle Z_{\mu_{2}}\left(s\right)=\frac{\zeta\left(s\right)}{\zeta\left(2s\right)}$

PARAFERMIONIC MODEL

The constraint $r_{i}\in\left\{ 0,1\right\}$ of the fermionic model can be extended to $r_{i}\in\left\{ 0,\ldots,\kappa - 1\right\}$ where $\kappa$ is a parameter. This leads to define the arithmetic function

$\displaystyle \mu_{\kappa}\left(p_{1}^{r_{1}}\ldots p_{k}^{r_{k}}\right)=1\textrm{ if all }r_{i}\in\left\{ 0,1,\ldots,\kappa - 1\right\} \textrm{ and }0\textrm{ otherwise}$

It is multiplicative like $\mu_{2}$ and the partition function is computed in a similar way

$\displaystyle Z_{\mu_{\kappa}}\left(s\right)=\prod_{p\textrm{ prime}}\sum_{r=0}^{\kappa-1}\frac{1}{p^{rs}}=\prod_{p\textrm{ prime}}\frac{1-p^{-\kappa s}}{1-p^{-s}}$

$\displaystyle Z_{\mu_{\kappa}}\left(s\right)=\frac{\zeta\left(s\right)}{\zeta\left(\kappa s\right)}$

We recover the fermionic model when $\kappa=2$ and the bosonic model when $\kappa\rightarrow\infty$.

MIXING WITHOUT INTERACTIONS

In physics, the partition function for a system composed of subsystems is the product of the partition functions for the individual sub-systems. Thus, we need to find an interpretation of the product $Z_{f}\left(s\right)Z_{g}\left(s\right)$ of two partition functions.

The Dirichlet convolution on arithmetic functions is similar to the convolution in Fourier theory

$\displaystyle \left(f\star g\right)\left(n\right)=\sum_{d|n}f\left(d\right)g\left(\frac{n}{d}\right)$

where $d|n$ means that $d$ divides $n$. In particular, the Dirichlet series preserves the multiplicative structure as follows

$\displaystyle Z_{f\star g}\left(s\right)=Z_{f}\left(s\right)Z_{g}\left(s\right)$

Hence, the Dirichlet convolution provides an interpretation of the product $Z_{f}\left(s\right)Z_{g}\left(s\right)$.

Moreover, the arithmetic functions form the Dirichlet ring with pointwise addition $\left(f+g\right)\left(n\right)=f\left(n\right)+g\left(n\right)$ and Dirichlet convolution as multiplication $\left(f\star g\right)\left(n\right)$. The multiplicative identity is given by $I\left(n\right)=1$ if $n=1$ and $I\left(n\right)=0$ otherwise.

MIXING WITH INTERACTIONS

The Dirichlet convolution can be extended with a kernel $K\left(n,d\right)$ in order to consider arithmetic interactions in the summation

$\displaystyle \left(f\star_{K}g\right)\left(n\right)=\sum_{d|n}K\left(n,d\right)f\left(d\right)g\left(\frac{n}{d}\right)$

This can be used to define the unitary Dirichlet convolution restricted to unitary divisors. Recall that $d$ is a unitary divisor of $n$ if $d|n$ and $d$ and $\frac{n}{d}$ are coprime. It is encoded by the kernel $U\left(n,d\right)=1$ if $\left(d,\frac{n}{d}\right)=1$ and $U\left(n,d\right)=0$ otherwise.

$\displaystyle \left(f\star_{U}g\right)\left(n\right)=\sum_{d|n,\left(d,n/d\right)=1}f\left(d\right)g\left(\frac{n}{d}\right)$

It is then possible to compute the partition function $Z_{1\star_{U}1}\left(s\right)$ of two bosonic Riemann gases. The trick consists of using the equality $1\star_{U}1=\mu_{2}\star1$, which implies

$\displaystyle Z_{1\star_{U}1}\left(s\right)=Z_{\mu_{2}\star1}\left(s\right)=Z_{\mu_{2}}\left(s\right)Z_{1}\left(s\right)=\frac{\zeta\left(s\right)^{2}}{\zeta\left(2s\right)}$

Thus, the unitary interacting mixing of two bosonic gases is equivalent to a non-interacting mixing between a bosonic gas and a fermionic gas (Bakas and Bowick).

Proof

We justify the equality $1\star_{U}1=\mu_{2}\star1$.

The left hand side counts the number of unitary divisors of $n$

$\displaystyle \left(1\star_{U}1\right)\left(n\right)=\sum_{d|n,\left(d,n/d\right)=1}1$

Suppose that $n$ has prime decomposition $n=p_{1}^{r_{1}}\ldots p_{k}^{r_{k}}$. Any divisor $d$ has prime decomposition $d=p_{1}^{r_{1}'}\ldots p_{k}^{r_{k}'}$ with $0\leq r_{i}'\leq r_{i}$. If $0 for some $i$ then $d$ and $\frac{n}{d}$ have the common factor $p_{i}$ and cannot be coprime. Thus, unitary divisors require $r_{i}'=0$ or $r_{i}'=r_{i}$. The number of unitary divisors is then equal to

$\displaystyle \left(1\star_{U}1\right)\left(n\right)=2^{q\left(n\right)}$

where $q\left(n\right)$ is the number of distinct prime factors of $n$.

The right hand side counts the number of square free divisors

$\displaystyle \left(\mu_{2}\star1\right)\left(n\right)=\sum_{d|n}\mu_{2}\left(d\right)$

Square free divisors require $r_{i}'=0$ or $r_{i}'=1$. The number of square free divisors is then also equal to

$\displaystyle\left(\mu_{2}\star1\right)\left(n\right)=2^{q\left(n\right)}$

REFERENCES

1. Bakas and Bowick, Curiosities of arithmetic gases, 1991
2. Julia, Thermodynamic limit in number theory, 1993
3. Schumayer and Hutchinson, Physics of the Riemann Hypothesis, 2011