We prove a necessary and sufficient condition on the list of nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form $u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in \mathbb{Z}[x]$. This condition is independent of $f$. We also show that if this condition holds, then the monic polynomials $f_1,\dots,f_k$ can be chosen to be irreducible in $\mathbb{Z}[x]$.

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.

We prove a result concerning power series $f(z)\in\mathbb{C}[\mkern-3mu[z]\mkern-3mu]$ satisfying a functional equation of the form $$ f(z^d)=\sum_{k=1}^n \frac{A_k(z)}{B_k(z)}f(z)^k, $$ where $A_k(z),B_k(z)\in \mathbb{C}[z]$. In particular, we show that if $f(z)$ satisfies a minimal functional equation of the above form with $n\geqslant 2$, then $f(z)$ is necessarily transcendental. Towards a more complete classification, the case $n=1$ is also considered.

Let $A(n_1,n_2,\dots,n_{m-1})$ be the normalized Fourier coefficients of a Maass cusp form on $\textrm{GL}(m)$. In this paper, we study the cancellation of $A (n_1,n_2,\dots,n_{m-1})$ over Beatty sequences.

Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n-1}z^{n-1} : a_j \in \{0, 1 \ } \big\}$, whose elements are called \emf{zero-one polynomials} and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1, \ldots, n-1 \}$. We also let $\cA_{n,m} = \{ \alf(z) \in \cA_n : \alf(1) = m \}$, whose elements correspond to the ${n \choose m}$ subsets of~$[n]$ of size~$m$, and let $\cB_n = \cA_{n+1} \setminus \cA_n$, whose elements are the zero-one polynomials of degree exactly~$n$. Many researchers have studied norms of polynomials with restricted coefficients. Using $\norm{\alf}_p$ to denote the usual $L_p$ norm of~$\alf$ on the unit circle, one easily sees that $\alf(z) = a_0 + a_1 z + \cdots + a_N z^N \in \bR[z]$ satisfies $\norm{\alf}_2^2 = c_0$ and $\norm{\alf}_4^4 = c_0^2 + 2(c_1^2 + \cdots + c_N^2)$, where $c_k := \sum_{j=0}^{N-k} a_j a_{j+k}$ for $0 \le k \le N$. If $\alf(z) \in \cA_{n,m}$, say $\alf(z) = z^{\beta_1} + \cdots + z^{\beta_m}$ where $\beta_1 < \cdots < \beta_m$, then $c_k$ is the number of times $k$ appears as a difference $\beta_i - \beta_j$. The condition that $\alf \in \cA_{n,m}$ satisfies $c_k \in \{0,1\}$ for $1 \le k \le n-1$ is thus equivalent to the condition that $\{ \beta_1, \ldots, \beta_m \}$ is a \emf{Sidon set} (meaning all differences of pairs of elements are distinct). In this paper, we find the average of~$\|\alf\|_4^4$ over $\alf \in \cA_n$, $\alf \in \cB_n$, and $\alf \in \cA_{n,m}$. We further show that our expression for the average of~$\|\alf\|_4^4$ over~$\cA_{n,m}$ yields a new proof of the known result: if $m = o(n^{1/4})$ and $B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, then almost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that $\lim_{n \to \infty} B(n,m)/\binom{n}{m} = 1$.

The aim of this paper is to study sequences of integers for which the second differences between their squares are constant. We show that there are infinitely many nontrivial monotone sextuples having this property and discuss some related problems.