Related papers: Exponential sums of the M\"{o}bius function and fl…

The aim of this paper is twofold. Firstly, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the…

We explore a class of meromorphic functions on elliptic curves, termed \emph{elliptic orthogonal a-polynomials} ($a$-EOPs), which extend the classical notion of orthogonal polynomials to compact Riemann surfaces of genus one. Building on…

Generalization of the Euler polynomials ${{A}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{m}^{n}}{{x}^{m}}}$ are the polynomials ${{\alpha }_{n}}\left( x \right)={{\left( 1-x…

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

Motivated by the variations of Sarnak's conjecture due to El Abdalaoui, Kulaga-Przymus, Lemanczyk, De La Rue and by the observation that the Mobius function is a good weight (with limit zero) for the polynomial pointwise ergodic theorem in…

Let $f,g_1,\dots,g_m$ be polynomials with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Denote by $\text{diag}(g)$ the diagonal matrix with coefficients $g=(g_1,\dots,g_m)$ and denote by $\nabla g$ the Jacobian of $g$. Let…

Paul Erd\H{o}s posed a problem on the asymptotic estimation of decomposing 1 into a sum of infinitely many unit fractions in \cite{Erd80}. We point out that this problem can be solved in the same way as the finite case, as shown in…

Let f be a sum of exponentials of the form exp(2 pi i N x), where the N are distinct integers. We call f an idempotent trigonometric polynomial (because the convolution of f with itself is f) or, simply, an idempotent. We show that for…

We give an overview of recent progress around a problem introduced by Elekes and R\'onyai. The prototype problem is to show that a polynomial $f\in \mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset \mathbb{R}^2$,…

Let $f$ be a polynomial of degree $d>1$ in $n$ variables over $\mathbb{Z}$. Let $f_d$ be the homogeneous part of degree $d$ of $f$ and $s$ be the dimension of the critical locus of $f_d$. In this paper, we prove Igusa's conjecture for…

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…

Recently the author presented a new approach to solving the coefficient problems for various classes of holomorphic functions $f(z) = \sum\limits_0^\infty c_n z^n$, not necessarily univalent. This approach is based on lifting the given…

Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi…

Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $\mu$ be the M\"obius function and $\mu_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=\mu^2g$ and $f=\mu_2^2g$ have unbounded partial sums. Our proofs…

In this paper matrix orthogonal polynomials in the real line are described in terms of a Riemann--Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The…

Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of…

We show that the $L^1$ norm of an exponential sum of length $X$ and with coefficients equal to the Liouville or M\"{o}bius function is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville…

This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the…

Landau examined the partial sums of the M\"obius function and the Liouville function for a number field $K$. First we shall try again the same problem by using a new Perron's formula due to Liu and Ye. Next we consider the equivalent…