Related papers: Equidistributions of Sign Patterns of the Liouvill…

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} \lambda(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for…

Let $\lambda$ and $\mu$ denote the Liouville and M\"obius functions respectively. Hildebrand showed that all eight possible sign patterns for $(\lambda(n), \lambda(n+1), \lambda(n+2))$ occur infinitely often. By using the recent result of…

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…

Denote by $\lambda(n)$ Liouville's function concerning the parity of the number of prime divisors of $n$. Using a theorem of Allouche, Mend\`es France, and Peyri\`ere and many classical results from the theory of the distribution of prime…

In this note we give a short and self-contained proof that, for any $\delta > 0$, $\sum_{x \leq n \leq x+x^\delta} \lambda(n) = o(x^\delta)$ for almost all $x \in [X, 2X]$. We also sketch a proof of a generalization of such a result to…

Summation arithmetic functions of Mertens and Liouville are investigated in the paper. It is proved that the limiting distribution of these functions is the normal. It is also shown that the estimating of standard deviation of these…

Let $\lambda$ be the Liouville function. Assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions (GRH), we show that for every sufficiently large even integer $N$ there are $a,b \geq 1$ such that $$ a+b = N \text{ and }…

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 improve bounds on the degree and sparsity of Boolean functions representing the Legendre symbol as well as on the $N$th linear complexity of the Legendre sequence. We also prove similar results for both the Liouville function for…

We introduce a novel arithmetic function $w(n)$, a generalization of the Liouville function $\lambda(n)$, as the coefficients of a Dirichlet series. By spatially encoding information in a natural way about the distribution of prime factors…

Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers. Kano and Bugeaud have proved, using analytic techniques, that there are normal Liouville…

Let $\lambda$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$ \sum_{n \leq x} \lambda(a_1 n + b_1) \lambda(a_2 n+b_2) = o(x) $$ as $x \to \infty$, for any fixed natural numbers…

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)…

Let $\alpha=0.a_1a_2a_3\ldots$ be an irrational number in base $b>1$, where $0\leq a_i<b$. The number $\alpha \in (0,1)$ is a \textit{normal number} if every block $(a_{n+1}a_{n+2}\ldots a_{n+k})$ of $k$ digits occurs with probability…

In 2008 or earlier, Michel Mend\`es France asked for an instance of a real number $x$ such that both $x$ and $1/x$ are simply normal to a given integer base $b$. We give a positive answer to this question by constructing a number $x$ such…

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$ for all $H \geq X^{\theta}$ with $\theta >…

Using sieves and elementary manipulations, we show that the signs of partial sums of the Liouville function over divisors are in a strong sense equally distributed.

A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency that the digit $1$ appears in the…

Let $\lambda(n)$ be the Liouville function. We study the distribution of \[ \frac{1}{x^{1/2}}\sum_{x\leq n\leq 2x}\lambda(f(n)) \] over random polynomials $f$ of fixed degree $d$ and coefficients bounded in magnitude by $H$. In particular…

In this paper we prove that if $\{\varphi_i(x)=\lambda x+t_i\}$ is an equicontractive iterated function system and $b$ is a positive integer satisfying $\frac{\log b}{\log |\lambda|}\notin\mathbb{Q},$ then almost every $x$ is normal in base…