相关论文: A general criterion for the P\'{o}lya-Carlson dich…

Let $\sum a_nx^n\in\bar{\mathbb{Q}}[[x]]$ be the power series representation of a rational function and let $f:\ \{0,1,\ldots\}\rightarrow \bar{\mathbb{Q}}$ be a so-called almost quasi-polynomial. Under a necessary stability condition, we…

We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for automorphisms of non-finitely generated torsion abelian groups and for endomorphisms of groups $\mathbb Z_p^d,$…

In this paper we continue to study the Reidemeister zeta function. We prove P\'olya -- Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta function for a large class of automorphisms…

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions.…

In this paper, we give a new proof of a result due to Bezivin that a D-finite Mahler function is necessarily rational. This also gives a new proof of the rational-transcendental dichotomy of Mahler functions due to Nishioka. Using our…

For a $\mathbb{Z}^d$-action $\alpha$ by commuting homeomorphisms of a compact metric space, Lind introduced a dynamical zeta function that generalizes the dynamical zeta function of a single transformation. In this article, we investigate…

We present results and background rationale in support of a Polya--Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of dynamical zeta functions of compact group automorphisms.

We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely…

In this paper we extend the Zeta function regularization technique, which gives a meaningful solution to divergent power series, in order to assign finite values to divergent integral of certain transcendental functions $f(x)$. The…

This work develops an analytic framework for the study of the $\zeta$-function associated with general sequences of complex numbers. We show that a contour integral representation, commonly used when studying spectral $\zeta$-functions…

Let $\sigma$ denote an endomorphism of a smooth algebraic group $G$ over the algebraic closure of a finite field, and assume all iterates of $\sigma$ have finitely many fixed points. Steinberg gave a formula for the number of fixed points…

Let $\Gamma$ be a group and $r_n(\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}$. When…

Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $\sigma_a(F)=\sum_{\substack{H\leq F}}|H|^a$ where $H$ ranges over the subgroups of $F$. We infer from this formula that the $p$-component of the…

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(\alpha;z)=\sum_{n\geq 2}(\log n)^{\alpha}(\eta_n+{\rm i} \theta_n)/n^z$, properly scaled and normalized, where…

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the…

Let $K$ be a compact set with connected complement on the half-plane Re$(s)>0$, and let $f$ be a continuous function on $K$ which is analytic in its interior. We prove that for any parameter $0<\alpha<1, \alpha \neq \frac 1 2$ then $f(s)$…

We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is…

The zeta-function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta-function of a surface is…

We introduce a one-parameter family of series associated to the Riemann $\zeta$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the…

Let $F(z)$ be a $k$-regular series in $\mathbb{Z}[[z]]$ and $b$ be an integer with $b\ge2$. Bell, Bugeaud and Coons [BelBC] proved that $F(\frac{1}{b})$ is either rational or transcendental. In [Mi], we introduce a generalized $k$-regular…