Related papers: On a Goldbach-type problem for the Liouville funct…

We consider the Hardy-H\'enon system $-\Delta u =|x|^a v^p$, $-\Delta v =|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the…

We study the pointwise convergence and the $\Gamma$-convergence of a family of non-local, non-convex functionals $\Lambda_\delta$ in $L^p(\Omega)$ for $p>1$. We show that the limits are multiples of $\int_{\Omega} |\nabla u|^p$. This is a…

Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2.…

We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, are bounded from above everywhere. It follows that subharmonic functions of…

We establish a Liouville-type theorem for nonnegative weak supersolutions to $\mathcal{L}_K u = u^q$ in $\mathbb{R}^n$, where $\mathcal{L}_K$ is a translation-invariant integro-differential operator of order $2s$ with $s \in (0,1)$. The…

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or} \ \textrm{div}(|y|^aF(x,y))\;.$$ Under suitable regularity assumptions for the…

We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel-Olivier test for real-valued functions. Analogous results for cosine series…

Carnevale and Voll conjectured that j (--1) j $\lambda$ 1 j $\lambda$ 2 j = 0 when $\lambda$ 1 and $\lambda$ 2 are two distinct integers. We check the conjecture when either $\lambda$ 2 or $\lambda$ 1 -- $\lambda$ 2 is small. We investigate…

A common problem in analytic number theory is to bound the sum of an arithmetic function over a set of integers. Nair and Tenenbaum found a very general bound that applies to short sums of a multivariable arithmetic function over polynomial…

Let $\Lambda$ be the von Mangoldt function and $(r_G(n) = \sum_{m_1 + m_2 = n} \Lambda(m_1) \Lambda(m_2))$ be the counting function for the Goldbach numbers. Let $N \geq 2$ be an integer. We prove that $$\begin{align} &\sum_{n \le N} r_G(n)…

In this paper we show that, if an increasing sequence $\Lambda=(\lambda_k)_{k\in\mathbb{Z}}$ has gaps going to infinity $\lambda_{k+1}-\lambda_k\to +\infty$ when $k\to\pm\infty$, then for every $T>0$ and every sequence…

The Chowla conjecture asserts that the values of the Liouville function form a normal sequence of plus and minus ones. Reinterpreted in the language of ergodic theory it asserts that the Liouville function is generic for the Bernoulli…

We consider Liouville-type theorems for the following H\'{e}non-Lane-Emden system \hfill -\Delta u&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture…

In 2023, the first author and Vandehey proved that the largest $k$ for which the string of equalities $\lambda(n+1)=\lambda(n+2)=\cdots=\lambda(n+k)$ holds for some $n\leq x$, where $\lambda$ is the Carmichael $\lambda$ function, is bounded…

We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for…

We prove some Liouville-type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary, thereby confirming some cases of Wang's conjecture (J. Geom. Anal. 31,…

For every non-hyper-FC-central countable amenable group and every $k\geq 2$, we provide a sequence of symmetric, fully supported probability measures such that their convex combination is non-Liouville (that is it admits a non-constant…

Let $\Lambda(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\varepsilon>0$ the asymptotic formula $$ \sum_{n\le x} \Lambda\Big(\Big[\frac{x}{n}\Big]\Big) =…

In the present paper, we adopt a pretentious approach and prove a strongly uniform estimate for the sums of the von Mangoldt function $\Lambda$ on arithmetic progressions. This estimate is analogous to an estimate that Linnik established in…

In this paper, we can show that \begin{align*} S_{\Lambda}(x)=\sum_{1\leq n\leq x}\Lambda \left(\left[\frac{x}{n}\right]\right)= \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n(n+1)}x +O\left(x^{7/15+1/195+\varepsilon}\right), \end{align*} where…