arXiv subjects

Tom Moshaiov

Publications and source records attributed to Tom Moshaiov.

3 recordsLinked to original sources

arXiv2026

Polynomial Expressions for Symmetric Group Characters on Cycles

In \cite{[CZ]}, Cohen and Zemel showed that for a partition $λ\vdash k$, the dimension of the irreducible representation of $S_{n}$ corresponding to the partition $(n-k,λ) \vdash n$ is a polynomial of degree $k$ in $n$, whose coefficients in the binomial basis count standard Young tableaux of shape $λ$ with special restrictions. In this paper, we generalize their results on the representation's dimension to character values on arbitrary cycles.

math.CO↗

arXiv2023

An elementary proof of the rationality of $ζ(2n)/π^{2n}$

In $1735$ Euler \cite{1} proved that for each positive integer $k$, the series $ζ(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rational multiple of $π^{2k}$. Many demonstrations of this fact are now known, and Euler's discovery is traditionally proven using non-elementary techniques, such as Fourier series or the calculus of residues \cite{2}. We give an elementary proof, similar to Cauchy's \cite{3} proof of the identity $ζ(2) = π^2/6$, only extended recursively for all values $ζ(2k)$. Our main formula $$ζ(2k)=-\dfrac{(-π^2)^{k}}{4^{2k}-4^{k}}\left[\dfrac{4^{k}k}{(2k)!}+{\displaystyle \sum_{\ell=1}^{k-1}}(4^{2\ell}-4^{\ell})\dfrac{4^{k-\ell}}{(2k-2\ell)!}\dfrac{ζ(2\ell)}{(-π^2)^{\ell}}\right] \phantom{spa}k = 1,2,3,\dots$$ may be derived from previously known formulae \cite{4}. Remarkably, Apostol \cite{5} discovered a proof similar to ours, yet arrived at a different formula, relating $ζ(2k)$ to the Bernoulli numbers, à la Euler.

math.GM↗

arXiv2022

An alternative proof for the irreducibility of the p-th cyclotomic polynomial

Let $p$ be a prime number. As a standard application of the irreducibility criterion of Eisenstein, it is well known that the $p$-th cyclotomic polynomial $Φ_p(t)=1+t+\dots+t^{p-1}$ is the minimal polynomial of $e^{2πi/p}$ over $\mathbb{Q}$. This note provides an alternative proof, utilizing determinants to prove a lemma due to Kronecker.

math.HO↗