Prime numbers can be considered as the building blocks of natural numbers. For example, 12 can be factored in a unique way as 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.
By the fundamental theorem of arithmetic, every positive integer admits a unique prime decomposition where denotes the i-th prime number and a natural number.
This can be used in the framework of second quantized field theory to define Fock states
which means that particles are in state , …, particles are in state . Thus, the states can be labeled by positive integers due to the unicity of prime decomposition. The elementary particles associated to the prime numbers are called primons.
Alternatively, the creation operators can be associated to the prime numbers such that
where 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 .
We define the energy of an individual primon as a logarithm
The energy of a non-interacting multi-particle state is given by the sum of the energies of the constituents
Note that the ground state corresponds to .
By definition, the canonical partition function is
where corresponds to the inverse temperature in physics. A quick computation shows
where is the famous Riemann zeta function. For this reason, such a system is called a Riemann gas. The zeta function is convergent for . It is divergent at , which is similar to the Hagedorn temperature. The Euler product formula allows to rewrite the partition function as
In number theory, a function is called an arithmetic function. Such a function is said to be multiplicative if and whenever and are coprime.
The partition function is a particular case of Dirichlet series
where is a multiplicative arithmetic function. In the case of the Riemann gas, the function is for all .
The previous Riemann gas is a bosonic model. In the case of a fermionic model, we have to take into account the constraint . In order to select the admissible states, we define the arithmetic function
We can check that this function is multiplicative :
- is trivial
- if and are coprime then their prime decompositions do not have common factors, thus .
The partition function is given by the Dirichlet series
Note that works like a sieve to filter the fermionic constraint. In order to establish a relationship with the zeta function, we compute
The constraint of the fermionic model can be extended to where is a parameter. This leads to define the arithmetic function
It is multiplicative like and the partition function is computed in a similar way
We recover the fermionic model when and the bosonic model when .
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 of two partition functions.
The Dirichlet convolution on arithmetic functions is similar to the convolution in Fourier theory
where means that divides . In particular, the Dirichlet series preserves the multiplicative structure as follows
Hence, the Dirichlet convolution provides an interpretation of the product .
Moreover, the arithmetic functions form the Dirichlet ring with pointwise addition and Dirichlet convolution as multiplication . The multiplicative identity is given by if and otherwise.
MIXING WITH INTERACTIONS
The Dirichlet convolution can be extended with a kernel in order to consider arithmetic interactions in the summation
This can be used to define the unitary Dirichlet convolution restricted to unitary divisors. Recall that is a unitary divisor of if and and are coprime. It is encoded by the kernel if and otherwise.
It is then possible to compute the partition function of two bosonic Riemann gases. The trick consists of using the equality , which implies
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).
We justify the equality .
The left hand side counts the number of unitary divisors of
Suppose that has prime decomposition . Any divisor has prime decomposition with . If for some then and have the common factor and cannot be coprime. Thus, unitary divisors require or . The number of unitary divisors is then equal to
where is the number of distinct prime factors of .
The right hand side counts the number of square free divisors
Square free divisors require or . The number of square free divisors is then also equal to
- Bakas and Bowick, Curiosities of arithmetic gases, 1991
- Julia, Thermodynamic limit in number theory, 1993
- Schumayer and Hutchinson, Physics of the Riemann Hypothesis, 2011