Related papers: Diophantine problems related to the Omega constant
A motivated q-extension of the values of the Riemann zeta function at positive integers is presented. Several irrationality and transcendence results as well as new general problems for these q-zeta values are stated.
Based on a Problem and its solution published on the pages of SIAM Review, we give an interesting integral representation for the Lambert $W$ function in this short note. In particular, our result yields a new integral representation for…
In this paper we give a survey of what is currently known about Diophantine exponents of lattices and propose several problems.
We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.
We obtain new omega results for the error terms in two classical lattice point problems. These results are likely to be the best possible.
We study some problems in metric Diophantine approximation over local fields of positive characteristic.
In this paper we introduce the $p$-adic analogue of the Lambert $W$ function, and study its main properties.
The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series which involves a product of Riemann zeta-functions of a special form.
This paper collects polynomial Diophantine equations that are simple to state but apparently difficult to solve.
We discuss several open problems in Diophantine approximation. Among them there are famous Littlewood's and Zaremba's conjectures as well as some new and not so famous problems.
We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.
These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…
The number of lattice points in $d$-dimensional hyperbolic or elliptic shells $\{m : a<Q[m]<b\}$, which are restricted to rescaled and growing domains $r\;\Omega$, is approximated by the volume. An effective error bound of order…
Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as…
In this paper we aim to prove two inequalities involving the classical approximation constants $w_{n}^{\prime}(\zeta),\hat{w}_{n}^{\prime}(\zeta)$ that stem from the simultaneous approximation problem $|\zeta^{j}x-y_{j}|$, $1\leq j\leq n$,…
We describe the spectrum of ordinary Diophantine exponents for $d$-dimensional lattices. The result reduces the problem to two-dimensional case and uses argument of metric theory.
Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…
We discuss a problem initially thought for the Mathematical Olympiad but which has several interpretations. The recurrence sequences involved in this problem may be generalized to recurrence sequences related to a much larger set of…
This paper investigates the generalized convexity properties of the Lambert $W$ function, defined as the solution to $W(z)e^{W(z)}=z$. Focusing on $H_{p,q}$-convexity and concavity with respect to H\"older means, we derive necessary and…