Related papers: Diophantine problems related to the Omega constant

Problems formulated in terms of logarithmic or exponential equations often use the Lambert $W$ function in their solutions. Expansions, approximations and bounds on $W$ have been derived in an effort to gain a better understanding of the…

We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

Let $d(n)$ be the number of divisors of $n$, let $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. Several…

We give closed-form expressions for the Dirichlet beta function at even positive integers and for the Dirichlet lambda function at odd positive integers, based on the function J(s) defined via convergent integral. We also show fundamental…

We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely…

In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where…

We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…

We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…

In this paper, I propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem -\Delta u = f(x,u) in Omega, u restricted to the boundary of Omega is 0. Positive answers to these problems would…

We give some comments on W.M. Schmidt's theorem on Diophantine approximations with positive integers and our recent results on the topic.

The recent negative answer to Hilbert's tenth problem over rings of integers relies on a theorem that for every extension of number fields $L/K$, if there is an abelian variety $A$ over $K$ such that $0 < \operatorname{rank} A(K) =…

Little is known about the zeros of the Digamma function. Establishing some Weierstrassian infinite product representations for a given regularization of the Digamma function we find interesting sums of its zeros. In addition, we study the…

Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|\zeta(1/2+it)|$ on the critical line, and zeros…

We establish several new $\Omega$-theorems for logarithmic derivatives of the Riemann zeta function and Dirichlet $L$-functions. In particular, this improves on earlier work of Landau (1911), Bohr-Landau (1913), and recent work of Lamzouri.

We refine a result of the last two Authors of [8] on a Diophantine approximation problem with two primes and a $k$-th power of a prime which was only proved to hold for $1<k<4/3$. We improve the $k$-range to $1<k\le 3$ by combining Harman's…

We investigate some Diophantine approximation constants related to the simultaneous approximation of $(\zeta,\zeta^{2},\ldots,\zeta^{k})$ for Liouville numbers $\zeta$. For a certain class of Liouville numbers including the famous…

We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of…

The presence of a cosmological constant, Lambda, in an action with higher powers of the curvature can produce rapidly oscillating metrics. We develop a perturbative approach for generating periodic solutions to the non-linear field…

Let k\geq 2 and consider the Diophantine inequality |x_1^k-\alp_2 x_2^k-\alp_3 x_3^k| <\tet. Our goal is to find non-trivial solutions in the variables x_i, 1\leq i\leq 3, all of size about P, assuming that \tet is sufficiently large. We…