# Basic zeta functions and some applications in physics

@article{Kirsten2010BasicZF, title={Basic zeta functions and some applications in physics}, author={Klaus Kirsten}, journal={arXiv: High Energy Physics - Theory}, year={2010} }

It is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the Casimir effect and Bose-Einstein condensation. A brief introduction into these areas is given in the respective sections. We will consider exclusively spectral zeta functions, that is zeta functions arising from the eigenvalue spectrum of suitable differential operators. There is a set of technical tools that are at the very heart of understanding analytical properties of essentially every…

## 23 Citations

### Local Zeta Regularization And The Scalar Casimir Effect: A General Approach Based On Integral Kernels

- Physics
- 2017

This is the first one of a series of papers about zeta regularization of the divergences appearing in the vacuum expectation value (VEV) of several local and global observables in quantum field…

### The zeta function for circular graphs

- MathematicsArXiv
- 2013

It is proved that the roots of zeta(n,s) converge for n to infinity to the line Re(s)=1/2 in the sense that for every compact subset K in the complement of this line, and large enough n, no root of the zeta function zeta(-s) is in K.

### Zeta function regularization technique in the electrostatics context for discrete charge distributions

- MathematicsEuropean Journal of Physics
- 2022

Spectral functions, such as the zeta functions, are widely used in quantum field theory to calculate physical quantities. In this work, we compute the electrostatic potential and field due to an…

### Generalized Bernoulli Polynomials and the Casimir Effect in the Einstein Universe

- Mathematics
- 2013

We consider various regularization schemes for calculating the renormalized vacuum energy of a massless scalar field in the n-dimensional Einstein universe. We also study a related problem, namely,…

### The thermodynamic limit of an ideal Bose gas by asymptotic expansions and spectral ζ-functions

- PhysicsJournal of Mathematical Physics
- 2022

We analyze the thermodynamic limit—modeled as the open-trap limit of an isotropic harmonic potential—of an ideal, non-relativistic Bose gas with a special emphasis on the phenomenon of Bose–Einstein…

### Further insights into the mysteries of the values of zeta functions at integers

- Mathematics
- 2021

. We present a remarkably simple and surprisingly natural interpretation of the values of zeta functions at negative integers and zero. Namely we are able to relate these values to areas related to…

### Quantum Mechanics on Laakso Spaces

- Mathematics
- 2010

We first review the spectrum of the Laplacian operator on a general Laakso Space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally,…

### Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

- MathematicsLetters in Mathematical Physics
- 2013

We investigate the existence of the meromorphic extension of the spectral zeta function of a Laplacian on self-similar fractals using the results of Kigami and Lapidus (based on renewal theory) and…

### Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

- Mathematics
- 2013

We investigate the existence of the meromorphic extension of the spectral zeta function of a Laplacian on self-similar fractals using the results of Kigami and Lapidus (based on renewal theory) and…

### Will a physicist prove the Riemann hypothesis?

- MathematicsReports on progress in physics. Physical Society
- 2019

The Riemann Hypothesis is formulated and some physical problems related to this hypothesis are reviewed: the Polya--Hilbert conjecture, the links with Random Matrix Theory, relation with the Lee--Yang theorem on the zeros of the partition function and phase transitions, random walks, billiards etc.

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