# Bose-Einstein Condensation for an Exponential Density of States Function and Lerch Zeta Function

## TL;DR

This paper explores a novel connection between Bose-Einstein condensation with exponential density of states and Lerch zeta functions, proposing a potential approach to the Riemann hypothesis through quantum statistical methods.

## Contribution

It introduces a new class of Lerch zeta functions derived from BEC with exponential density of states and suggests a quantum framework to approach the Riemann hypothesis.

## Key findings

- Derivation of Lerch zeta functions from BEC models.
- Proposed method to relate BEC classical limit to zeros of the Riemann zeta function.
- Introduction of creation-annihilation operators for BEC on complex Hilbert space.

## Abstract

I show how Bose-Einstein condensation (BEC) in a non interacting bosonic system with exponential density of states function yields to a new class of Lerch zeta functions. By looking on the critical temperature, I suggest that a possible strategy to prove the "Riemann hypothesis" problem. In a theorem and a lemma I suggested that the classical limit $\hbar\to 0$ of BEC can be used as a tool to find zeros of real part of the Riemann zeta function with complex argument. It reduces the Riemann hypothesis to a softer form. Furthermore I propose a pair of creation-annihilation operators for BEC phenomena. This set of creation-annihilation operators is defined on a complex Hilbert space. They build a set up to interpret this type of BEC as a creation-annihilation phenomenon for a virtual hypothetical particle.