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Zeta Function Expression of Spin Partition Functions on Thermal AdS_{3}

## Abstract

**:**

## 1. Introduction

## 2. The Action of $SL(2,\mathbb{C})$ on ${\mathbb{H}}^{3}$, Thermal $Ad{S}_{3}$, and the Patterson–Selberg Zeta Function

## 3. The Main Result and Some Examples

**Theorem 1.**

## 4. A Gangolli–Patterson-Type Formula in the Presence of Spin

**Theorem 2.**

## Conflicts of Interest

## Errata

- The statement ${X}_{a}\stackrel{\mathrm{def}}{=}{\Gamma}_{a}\setminus {\mathbb{H}}^{2}\stackrel{\mathrm{def}}{=}\left\{\left[\begin{array}{cc}1& na\\ 0& 1\end{array}\right]\right|n\in \mathbb{Z}\}$ in Equation (40) should read ${X}_{a}\stackrel{\mathrm{def}}{=}{\Gamma}_{a}\setminus {\mathbb{H}}^{2},{\Gamma}_{a}\stackrel{\mathrm{def}}{=}\left\{\left[\begin{array}{cc}1& na\\ 0& 1\end{array}\right]\right|n\in \mathbb{Z}\}$.
- In Equation (58), a parenthesis should be added: ${I}_{s-\frac{1}{2}}\left(2\pi \right|n|\frac{{y}_{2}}{a}$ should read ${I}_{s-\frac{1}{2}}\left(2\pi \right|n|\frac{{y}_{2}}{a})$.
- In the third sentence following Equation (63), the phrase “extended to deformation (14)” should read “extended to the deformation (14)”. Thus, in the article, “the” should be added.
- In the sentence that concludes Section 4, the phrase “single zero $s=\frac{1}{2}$ of z” should read “single non-trivial zero $s=\frac{1}{2}$ of z”, since clearly, the gamma function used in Definition (72) has trivial zeros at $s=-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\cdots $.
- The phrase “By the second Formula in (46)” that precede Equation (82) should read (simply) “By Formula (47)”.
- The phrase “Fourier transform of ${\widehat{f}}_{t}\left(x\right)$” that follows Equation (86) should read “Fourier transform of ${f}_{t}\left(x\right)$”.
- In Equation (89), we should have parentheses: ${K}_{ir}\frac{2\pi ny}{a}$ should read ${K}_{ir}\left(\frac{2\pi ny}{a}\right)$.
- In Equation (50), we need another parenthesis: ${G}_{a}({x}_{1}+{n}_{1}a,{y}_{1}),({x}_{2}+{n}_{2}a,y);s)$ should read ${G}_{a}(({x}_{1}+{n}_{1}a,{y}_{1}),({x}_{2}+{n}_{2}a,y);s)$.

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L.Williams, F. Zeta Function Expression of Spin Partition Functions on Thermal *AdS*_{3}. *Mathematics* **2015**, *3*, 653-665.
https://doi.org/10.3390/math3030653

**AMA Style**

L.Williams F. Zeta Function Expression of Spin Partition Functions on Thermal *AdS*_{3}. *Mathematics*. 2015; 3(3):653-665.
https://doi.org/10.3390/math3030653

**Chicago/Turabian Style**

L.Williams, Floyd. 2015. "Zeta Function Expression of Spin Partition Functions on Thermal *AdS*_{3}" *Mathematics* 3, no. 3: 653-665.
https://doi.org/10.3390/math3030653