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Search: MSC category 11N60 ( Distribution functions associated with additive and positive multiplicative functions )

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1. CMB 2011 (vol 56 pp. 251)

Borwein, Peter; Choi, Stephen K. K.; Ganguli, Himadri
Sign Changes of the Liouville Function on Quadratics
Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that \begin{equation*} \label{a.1} \sum_{n\le x} \lambda (f(n)) =o(x)\tag{$*$} \end{equation*} for any polynomial $f(x)$ with integer coefficients which is not of form $bg(x)^2$. When $f(x)=x$, $(*)$ is equivalent to the prime number theorem. Chowla's conjecture has been proved for linear functions, but for degree greater than 1, the conjecture seems to be extremely hard and remains wide open. One can consider a weaker form of Chowla's conjecture. Conjecture 1. [Cassaigne et al.] If $f(x) \in \mathbb{Z} [x]$ and is not in the form of $bg^2(x)$ for some $g(x)\in \mathbb{Z}[x]$, then $\lambda (f(n))$ changes sign infinitely often. Clearly, Chowla's conjecture implies Conjecture 1. Although weaker, Conjecture 1 is still wide open for polynomials of degree $\gt 1$. In this article, we study Conjecture 1 for quadratic polynomials. One of our main theorems is the following. Theorem 1 Let $f(x) = ax^2+bx +c $ with $a\gt 0$ and $l$ be a positive integer such that $al$ is not a perfect square. If the equation $f(n)=lm^2 $ has one solution $(n_0,m_0) \in \mathbb{Z}^2$, then it has infinitely many positive solutions $(n,m) \in \mathbb{N}^2$. As a direct consequence of Theorem 1, we prove the following. Theorem 2 Let $f(x)=ax^2+bx+c$ with $a \in \mathbb{N}$ and $b,c \in \mathbb{Z}$. Let \[ A_0=\Bigl[\frac{|b|+(|D|+1)/2}{2a}\Bigr]+1. \] Then either the binary sequence $\{ \lambda (f(n)) \}_{n=A_0}^\infty$ is a constant sequence or it changes sign infinitely often. Some partial results of Conjecture 1 for quadratic polynomials are also proved using Theorem 1.

Keywords:Liouville function, Chowla's conjecture, prime number theorem, binary sequences, changes sign infinitely often, quadratic polynomials, Pell equation
Categories:11N60, 11B83, 11D09

2. CMB 2004 (vol 47 pp. 589)

Liu, Yu-Ru
A Generalization of the Erdös-Kac Theorem and its Applications
We axiomatize the main properties of the classical Erd\"os-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

Categories:11N60, 11N80

3. CMB 1999 (vol 42 pp. 68)

Gittenberger, Bernhard; Thuswaldner, Jörg M.
The Moments of the Sum-Of-Digits Function in Number Fields
We consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange's method we obtain the main term and smaller order terms which contain periodic fluctuations.

Categories:11A63, 11N60

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