Related papers: On t-adic Littlewood conjecture for generalised Th…

We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to…

The Lambert W function gives the solutions of a simple exponential polynomial. The generalized Lambert W function was defined by Mez\"{o} and Baricz, and has found applications in delay differential equations and physics. In this article we…

As an analogue of Mahler's classification for real numbers, Bundschuh introduced a classification for Laurent series over a finite field, divided into $A,S,T,U$-numbers.It is known that each of $A,S,U$-numbers is nonempty.On the other hand,…

We study the function field analogue of Shanks bias. For Liouville function $\lambda(f)$, we compare the number of monic polynomials $f$ with $\lambda(f) \chi_m(f) = 1$ and $\lambda(f) \chi_m(f) = -1$ for a nontrivial quadratic character…

We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we…

Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a…

Let $F$ be an entire function represented by absolutely convergent for all $z\in\mathbb{C}$ Dirichlet series of the form $ F(z) = \sum\nolimits_{n=0}^{+\infty} a_{n}e^{z\lambda_{n}},$\ where a sequence $(\lambda_n)$ such that…

In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi(m)/(k+1)$ residue classes in the…

A group of infinite products over low-order rational polynomials evaluated at the sequence of prime numbers is loosely called the Hardy-Littlewood constants. In this manuscript we look at them as factors embedded in a super-product over…

The aim of this work is to present a possible adaptation of the Manin-Mumford conjecture to the $T-$modules, a mathematical object which has been introduced in the 1980's by G. Anderson as the natural analogue of the abelian varieties in…

For a sequence $\gamma=(\gamma_n)_{n\ge 1}$, define \[ L_\gamma(z):=\sum_{n\ge 1}\gamma_n\frac{z^n}{1-z^n} =\sum_{n\ge 1}\Bigl(\sum_{d\mid n}\gamma_d\Bigr)z^n. \] We prove a short rigidity theorem: if $\gamma$ is eventually linearly…

Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of…

We prove a Littlewood-type theorem on random analytic functions for not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space $H^2(\dd)$ by a Gaussian process whose covariance matrix $K$…

We prove that every completely monotone function defined on a right-unbounded open interval admits a Newton series expansion at every point of that interval. This result can be viewed as an analog of Bernstein's little theorem for…

Let $\mathbb{F}_{q}$ denote the finite field of order $q$ (a power of a prime $p$). We study the $p$-adic valuations for zeros of $L$-functions associated with exponential sums of the following family of Laurent polynomials…

Any power series with unit constant term can be factored into an infinite product of the form $\prod_{n\geq 1} (1-q^n)^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice…

For a simple Lie algebra $\mathfrak g$ we define a system of linear ODEs with polynomial coefficients, which we call the topological equation of $\mathfrak g$-type. The dimension of the space of solutions regular at infinity is equal to the…

We classify the coefficients $(a_1,...,a_n) \in \mathbb{F}_q[t]^n$ that can appear in a linear relation $\sum_{i=1}^n a_i \gamma_i =0$ among Galois conjugates $\gamma_i \in \overline{\mathbb{F}_q(t)}$. We call such an $n$-tuple a Smyth…

This paper contains the results of my PhD-thesis. I will show the K- and L-theoretic Farrell-Jones conjecture (FJC) for the general linear groups over the rationals and over the rational functions over a finite field. This especially…

This paper studies the Unramified Fontaine-Mazur Conjecture for $ p $-adic Galois representations and its generalizations. We prove some basic cases of the conjecture and provide some useful criterions for verifying it. In addition, we…