Related papers: On the Erd\"{o}s flat polynomials problem, Chowla …

We establish a generalization of Littlewood's criterion on $L^\alpha$-flatness by proving that there is no $L^\alpha$-flat polynomials, $\alpha>0$, within the class of analytic polynomials on the unit circle of the form $…

It is shown that Erd\"{o}s--Littlewood's polynomials are not $L^\alpha$-flat when $\alpha > 2$ is an even integer (and hence for any $\alpha \geq 4$). This provides a partial solution to an old problem posed by Littlewood. Consequently, we…

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 present a simple proof on the existence of $L^1$-flat analytic polynomials with coefficients $0,1$ on the circle and on the real line and we give an example of a conservative ergodic map and flow whose unitary operators admits a simple…

Polynomials with coefficients in $\{-1,1\}$ are called Littlewood polynomials. Using special properties of the Rudin-Shapiro polynomials and classical results in approximation theory such as Jackson's Theorem, de la Vall\'ee Poussin sums,…

We generalize a result of Matom\"aki, Radziwi{\l}{\l}, and Tao, by proving an averaged version of a conjecture of Chowla and a conjecture of Elliott regarding correlations of the Liouville function, or more general bounded multiplicative…

We exhibit a class of Littlewood polynomials that are not $L^\alpha$-flat for any $\alpha \geq 0$. Indeed, it is shown that the sequence of Littlewood polynomials is not $L^\alpha$-flat, $\alpha \geq 0$, when the frequency of $-1$ is not in…

We exhibit a sequence of flat polynomials with coefficients $0,1$. We thus get that there exist a sequences of Newman polynomials that are $L^\alpha$-flat, $0 \leq \alpha <2$. This settles an old question of Littlewood. In the opposite…

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…

Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro…

Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that \vspace{1mm} \noindent {\bf Conjecture (Chowla).} {\em \begin{equation} \label{a.1} \sum_{n\le x} \lambda (f(n)) =o(x)…

Assuming the existence of Siegel zeros, we prove that there exists an increasing sequence of positive integers for which Chowla's Conjecture on $k$-point correlations of the Liouville function holds. This extends work of Germ\'an and…

We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)^mu=|x|^av^p,\; (-\Delta)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions…

We establish Liouville type theorems for elliptic systems with various classes of non-linearities on $\mathbb{R}^N$. We show among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the…

Let $\lambda$ denote the Liouville function. We show that the logarithmic mean of $\lambda(\lfloor \alpha_1n\rfloor)\lambda(\lfloor \alpha_2n\rfloor)$ is $0$ whenever $\alpha_1,\alpha_2$ are positive reals with $\alpha_1/\alpha_2$…

Let $\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-S\'ark\"ozy asks to show that if $P(x)\in \mathbb{Z}[x]$, then the sequence $\lambda(P(n))$ changes sign infinitely often,…

We define a random Liouville function (\lambda_Q) which depends on a random set (Q) of primes and prove that (A_Q = \{n \in \mathbb{N} | \lambda_Q(n) = -1 \}) is normal almost everywhere. This fact enables us to generate a family of normal…

Given a smooth function $K(x)$ satisfying a polynomially cone condition and $x\cdot\nabla K\leq 0$, we prove that there is no solution $u\in C^\infty(\mathbb{R}^2)$ of the equation $$-\Delta u=K(x)e^{2u}\quad \mathrm{on}\;\mathbb{R}^2$$…

We present an elementary problem on analytic polynomials with coefficients $\pm 1$ or in $\{0,\pm 1 \}$ which implies Riemann hypothesis. It is turns out that this problem is a particular case of the weak form of a flat polynomials problem…

In the first part of this article, we complete the program announced in the preliminary note [8] by proving a conjecture presented in [9] that states the equivalence of contractibility and p_{1}-stability for generalized spaces of formal…