Between the conjectures of P\'{o}lya and Tur\'{a}n

This paper is concerned with the constancy in the sign of $L(X, \alpha) = \sum_{1}^{X} \frac{\lambda(n)}{n^{\alpha}}$, where $\lambda(n)$ the Liouville function. The non-positivity of $L(X, 0)$ is the P\'{o}lya conjecture, and the non-negativity of $L(X, 1)$ is the Tur\'{a}n conjecture --- both of which are false. By constructing an auxiliary function, evidence is provided that $L(X, \frac{1}{2})$ is the best contender for constancy in sign. The core of this paper is the conjecture that $L(X, \frac{1}{2}) \leq 0$ for all $X\geq 17$: this has been verified for $X\leq 300,001$.Comment: 5 page

A cell complex in number theory

Let De_n be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic (the Mertens function) is closely related to deep properties of the prime number system. In this paper we study the asymptotic behavior of the individual Betti numbers and of their sum. We show that De_n has the homotopy type of a wedge of spheres, and that as n tends to infinity: \sum \be_k(\De_n) = \frac{2n}{\pi^2} + O(n^{\theta}),\;\; \mbox{for all} \theta > \frac{17}{54}. We also study a CW complex tDe_n that extends the previous simplicial complex. In tDe_n all numbers up to n correspond to cells and its Euler characteristic is the summatory Liouville function. This cell complex is shown to be homotopy equivalent to a wedge of spheres, and as n tends to infinity: \sum \be_k(\tDe_n) = \frac{n}{3} + O(n^{\theta}),\;\; \mbox{for all} \theta > \frac{22}{27}.Comment: 16 page

Sign Changes of the Liouville function on quadratics

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) \end{equation} for any polynomial $f(x)$ with integer coefficients which is not of form $bg(x)^2$. } \vspace{1mm} \noindent The prime number theorem is equivalent to \eqref{a.1} when $f(x)=x$. Chowla's conjecture is proved for linear functions but for the degree greater than 1, the conjecture seems to be extremely hard and still remains wide open. One can consider a weaker form of Chowla's conjecture, namely, \vspace{1mm} \noindent {\bf Conjecture 1 (Cassaigne, et al).} {\em If $f(x) \in \Z [x]$ and is not in the form of $bg^2(x)$ for some $g(x)\in \Z[x]$, then $\lambda (f(n))$ changes sign infinitely often.} Clearly, Chowla's conjecture implies Conjecture 1. Although it is weaker, Conjecture 1 is still wide open for polynomials of degree $>1$. In this article, we study Conjecture 1 for the quadratic polynomials. One of our main theorems is {\bf Theorem 1.} {\em Let $f(x) = ax^2+bx +c$ with $a>0$ and $l$ be a positive integer such that $al$ is not a perfect square. Then if the equation $f(n)=lm^2$ has one solution $(n_0,m_0) \in \Z^2$, then it has infinitely many positive solutions $(n,m) \in \N^2$.} As a direct consequence of Theorem 1, we prove some partial results of Conjecture 1 for quadratic polynomials are also proved by using Theorem 1

Distribution of neighboring values of the Liouville and M\"obius functions

Let $\lambda(n)$ and $\mu(n)$ denote the Liouville function and the M\"obius function, respectively. In this study, relationships between the values of $\lambda(n)$ and $\lambda(n+h)$ up to $n\leq10^8$ for $1\leq h\leq1,000$ are explored. Chowla's conjecture predicts that the conditional expectation of $\lambda(n+h)$ given $\lambda(n)=1$ for $1\leq n\leq X$ converges to the conditional expectation of $\lambda(n+h)$ given $\lambda(n)=-1$ for $1\leq n\leq X$ as $X\rightarrow\infty$. However, for finite $X$, these conditional expectations are different. The observed difference, together with the significant difference in $\chi^2$ tests of independence, reveals hidden additive properties among the values of the Liouville function. Similarly, such additive structures for $\mu(n)$ for square-free $n$'s are identified. These findings pave the way for developing possible, and hopefully efficient, additive algorithms for these functions. The potential existence of fast, additive algorithms for $\lambda(n)$ and $\mu(n)$ may eventually provide scientific evidence supporting the belief that prime factorization of large integers should not be too difficult. For $1\leq h\leq1,000$, the study also tested the convergence speeds of Chowla's conjecture and found no relation on $h$