Witten zeta function

If $G$ is a compact semisimple Lie group, the associated Witten zeta function is (the meromorphic continuation of) the series

:$\backslash zeta\_G(s)=\backslash sum\_\backslash rho\backslash frac\{1\}\{(\backslash dim\backslash rho)^s\},$

where the sum is over equivalence classes of irreducible representations of $G$.

In the case where $G$ is connected and simply connected, the correspondence between representations of $G$ and of its Lie algebra, together with the Weyl dimension formula, implies that $\backslash zeta\_G(s)$ can be written as

:$\backslash sum\_\{m\_1,\backslash dots,m\_r>0\}\backslash prod\_\{\backslash alpha\backslash in\; \backslash Phi^+\}\backslash frac\{1\}\{\backslash langle\backslash alpha^\backslash lor,\; m\_1\backslash lambda\_1+\backslash cdots+m\_r\backslash lambda\_r\backslash rangle^\{s\}\},$

where $\backslash Phi^+$ denotes the set of positive roots, $\backslash \{\backslash lambda\_i\backslash \}$ is a set of simple roots and $r$ is the rank.

• $\backslash zeta\_\{SU(2)\}(s)=\backslash zeta(s)$, the Riemann zeta function.
• $\backslash zeta\_\{SU(3)\}(s)=\backslash sum\_\{x=1\}^\{\backslash infty\}\backslash sum\_\{y=1\}^\{\backslash infty\}\backslash frac\{1\}\{(xy(x+y)/2)^s\}.$

If $G$ is simple and simply connected, the abscissa of convergence of $\backslash zeta\_G(s)$ is $r/\backslash kappa$, where $r$ is the rank and $\backslash kappa=|\backslash Phi^\{+\}|$. This is a theorem due to Alex Lubotzky and Michael Larsen.{{Cite journal|last1=Larsen|first1=Michael|last2=Lubotzky|first2=Alexander|year=2008|title=Representation growth of linear groups|journal=Journal of the European Mathematical Society|volume=10|issue=2|pages=351–390|doi=10.4171/JEMS/113|issn=1435-9855|arxiv=math/0607369|s2cid=9322647 }} A new proof is given by Jokke Häsä and Alexander Stasinski

{{cite journal |last1=Häsä |first1=Jokke |last2=Stasinski |first2=Alexander |title=Representation growth of compact linear groups |journal=Transactions of the American Mathematical Society |date=2019 |volume=372 |issue=2 |pages=925–980 |doi=10.1090/tran/7618|doi-access=free |arxiv=1710.09112 }}

which yields a more general result, namely it gives an explicit value (in terms of simple combinatorics) of the abscissa of convergence of any "Mellin zeta function" of the form

$\backslash sum\_\{x\_1,\backslash dots,x\_r=1\}^\{\backslash infty\}\backslash frac\{1\}\{P(x\_1,\backslash dots,x\_r)^s\},$

where $P(x\_1,\backslash dots,x\_r)$ is a product of linear polynomials with non-negative real coefficients.

$\backslash zeta\_\{SU(3)\}$ is absolutely convergent in $\backslash \{\; s\; \backslash in\; \backslash mathbb\{C\},\; \backslash Re(s)>2/3\; \backslash \}$, and it can be extended meromorphicaly in $\backslash mathbb\{C\}$. Its singularities are in $\backslash Bigl\backslash \{\; \backslash frac\{2\}\{3\}\; \backslash Bigr\backslash \}\; \backslash cup\; \backslash Bigl\backslash \{\; \backslash frac\{1\}\{2\}-k,\; k\; \backslash in\; \backslash mathbb\{N\}\; \backslash Bigr\backslash \},$ and all of those singularities are simple poles.{{Cite journal |last=Romik |first=Dan |date=2017 |title=On the number of $n$-dimensional representations of $\operatorname{SU}(3)$, the Bernoulli numbers, and the Witten zeta function |url=http://www.impan.pl/get/doi/10.4064/aa8455-3-2017 |journal=Acta Arithmetica |language=en |volume=180 |issue=2 |pages=111–159 |doi=10.4064/aa8455-3-2017 |issn=0065-1036}} In particular, the values of $\backslash zeta\_\{SU(3)\}(s)$ are well defined at all integers, and have been computed by Kazuhiro Onodera.{{Cite journal |last=Onodera |first=Kazuhiro |date=2014 |title=A functional relation for Tornheim's double zeta functions |url=http://journals.impan.pl/cgi-bin/doi?aa162-4-2 |journal=Acta Arithmetica |language=en |volume=162 |issue=4 |pages=337–354 |doi=10.4064/aa162-4-2 |arxiv=1211.1480 |s2cid=119636956 |issn=0065-1036}}

At $s=0$, we have $\backslash zeta\_\{SU(3)\}(0)\; =\; \backslash frac\{1\}\{3\},$ and $\backslash zeta\_\{SU(3)\}\text{'}(0)=\backslash log(2^\{4/3\}\backslash pi).$

Let $a\; \backslash in\; \backslash mathbb\{N\}^*$ be a positive integer. We have

$\backslash zeta\_\{SU(3)\}(a)=\backslash frac\{2^\{a+2\}\}\{1+(-1)^a\; 2\}\; \backslash sum\_\{k=0\}^\{[a/2]\}\; \{2a-2k-1\; \backslash choose\; a-1\}\; \backslash zeta(2k)\; \backslash zeta(3a-k).$

If a is odd, then $\backslash zeta\_\{SU(3)\}$ has a simple zero at $s=-a,$ and

$\backslash zeta\_\{SU(3)\}\text{'}(-a)=\backslash frac\{2^\{-a+1\}\; (a!)^2\}\{(2a+1)!\}\; \backslash zeta\text{'}(-3a-1)\; +\; 2^\{-a+2\}\; \backslash sum\_\{k=0\}^\{(a-1)/2\}\; \{a\; \backslash choose\; 2k\}\; \backslash zeta(-a-2k)\; \backslash zeta\text{'}(-2a+2k).$

If a is even, then $\backslash zeta\_\{SU(3)\}$ has a zero of order $2$ at $s=-a,$ and

$\backslash zeta\_\{SU(3)\}\text{'}\text{'}(-a)=2^\{-a+2\}\backslash sum\_\{k=0\}^\{a/2\}\; \{a\; \backslash choose\; 2k\}\; \backslash zeta\text{'}(-a-2k)\backslash zeta\text{'}(-2a+2k).$

Category:Zeta and L-functions

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