Let $\Gamma$ be a group and $r_n\left(\Gamma \right)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function ${𝒵}_{\Gamma }\left(s\right)={\sum }_{n=1}^{\infty }{r}_n\left(\Gamma \right)n^{-s}$. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then ${𝒵}_{\Gamma }\left(s\right)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place...

For nonnegative integers a, b, c and positive integer n, let N(a,b,c;n) denote the number of representations of n by the form ${\sum }_{i=1}^a(x{²}_i+x_i{y}_i+y{²}_i)+2{\sum }_{j=1}^b(u{²}_j+u_j{v}_j+v{²}_j)+4{\sum }_{k=1}^c(r{²}_k+r_k{s}_k+s{²}_k)$. Explicit formulas for N(a,b,c;n) for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for N(2,1,0;n), N(2,0,1;n), N(1,2,0;n), N(1,0,2;n) and N(1,1,1;n) by employing the (p, k)-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.

Let $G\cong C_{n_1}\oplus ...\oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1·...·w_n$ is a sequence of integers, all but at most one relatively prime to $\left|G\right|$, and $S$ is a sequence over $G$ with $\left|S\right|\ge \left|W\right|+\left|G\right|-1\ge \left|G\right|+1$, the maximum multiplicity of $S$ at most $\left|W\right|$, and $\sigma \left(W\right)\equiv 0mod\left|G\right|$, then there exists a nontrivial subgroup $H$ such that every element $g\in H$ can be represented as a weighted subsequence sum of the form $g=\underset{i=1}{\sum ^n}{w}_i{s}_i$, with $s_1·...·s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples...

Let $\Gamma$ be a set of binary quadratic forms of the same discriminant, $\Delta$ a set of arithmetical progressions and $m$ a positive integer. We investigate the representability of prime powers $p^m$ lying in some progression from $\Delta$ by some form from $\Gamma$.

Soient $A_p=\{0,\cdots ,p-1\}$ et $Z\subseteq A_p^{\mathbb{N}}\times A_p^{\mathbb{N}}$ un sous-système. $Z$ est une représentation en base $p$ d’une fonction $f$ du tore si pour tout point $x$ du tore, ses développements en base $p$ sont liés par le couplage $Z$ aux développements en base $p$ de $f\left(x\right)$. On prouve que si $f$ est représentable en base $p$ alors $f\left(x\right)=(ux+\frac{m}{p-1})\text{mod}1$, où $u\in \mathbb{Z}\text{et}m\in A_p$. Réciproquement, toutes les fonctions de ce type sont représentables en base $p$ par un transducteur. On montre finalement que les fonctions du tore qui peuvent être représentées par automate cellulaire sont exclusivement les multiplications...