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Motivated by questions in number theory, Myerson asked how small the sum of 5 complex nth roots of unity can be. We obtain a uniform bound of O(n^{-4/3}) by perturbing the vertices of a regular pentagon, improving to O(n^{-7/3}) infinitely often. The corresponding configurations were suggested by examining exact minimum values computed for n <= 221000. These minima can be explained at least in part by selection of the best example from multiple families of competing configurations related to close rational approximations.
Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2ldots$. We estimate
Let $p>3$ be a prime. Gauss first introduced the polynomial $S_p(x)=prod_{c}(x-zeta_p^c),$ where $0<c<p$ and $c$ varies over all quadratic residues modulo $p$ and $zeta_p=e^{2pi i/p}$. Later Dirichlet investigated this polynomial and used this to sol
In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary two-variable partitio
Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of $rm{SL}_2(mathbb Z)$,
Let $L/K$ be a quadratic extension of global fields. We study Cohen-Lenstra heuristics for the $ell$-part of the relative class group $G_{L/K} := textrm{Cl}(L/K)$ when $K$ contains $ell^n$th roots of unity. While the moments of a conjectural distribu