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Register a new userSmall sums of five roots of unity
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Ben Barber
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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.
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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 the orders of these roots of unity in terms of the degrees and the heights of the polynomials $c_i$ and $f_i$.
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 solve the problems involving the Pell equations. Recently, Z.-W Sun studied some trigonometric identities involving this polynomial. In this paper, we generalized their results. As applications of our result, we extend S. Chowlas result on the congruence concerning the fundamental unit of $mathbb{Q}(sqrt{p})$ and give an equivalent form of the extended Ankeny-Artin-Chowla conjecture.
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 partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, $R_n$ has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper half complex plane $mathbb H$, in work of Bringmann, Folsom, Garvan, Kimport, Mahlburg, and Ono. Quantum modular forms, defined by Zagier in 2010, are similar to modular or mock modular forms but are defined on the rationals $mathbb Q$ as opposed to $mathbb H$, and exhibit modular transformations there up to suitably analytic error functions in $mathbb R$; in general, they have been related to diverse areas including number theory, topology, and representation theory. Here, we establish quantum modular properties of $R_n$.
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 $rm{SL}_2(mathbb Z)$, up to nontrivial error terms; however, their domains (the upper half-plane $mathbb H$, and the rationals $mathbb Q$, respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more. In this paper we study the $(n+1)$-variable combinatorial rank generating function $R_n(x_1,x_2,dots,x_n;q)$ for $n$-marked Durfee symbols. These are $n+1$ dimensional multisums for $n>1$, and specialize to the ordinary two-variable partition rank generating function when $n=1$. The mock modular properties of $R_n$ when viewed as a function of $tauinmathbb H$, with $q=e^{2pi i tau}$, for various $n$ and fixed parameters $x_1, x_2, cdots, x_n$, have been studied in a series of papers. Namely, by Bringmann and Ono when $n=1$ and $x_1$ a root of unity; by Bringmann when $n=2$ and $x_1=x_2=1$; by Bringmann, Garvan, and Mahlburg for $ngeq 2$ and $x_1=x_2=dots=x_n=1$; and by the first and third authors for $ngeq 2$ and the $x_j$ suitable roots of unity ($1leq j leq n$). The quantum modular properties of $R_1$ readily follow from existing results. Here, we focus our attention on the case $ngeq 2$, and prove for any $ngeq 2$ that the combinatorial generating function $R_n$ is a quantum modular form when viewed as a function of $x in mathbb Q$, where $q=e^{2pi i x}$, and the $x_j$ are suitable distinct roots of unity.
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 distribution in this case had previously been described, no method to calculate the distribution given the moments was known. We resolve this issue by introducing new invariants associated to the class group, $psi_{L/K}$ and $omega_{L/K},$ and study the distribution of $(G_{L/K}, psi_{L/K}, omega_{L/K})$ using a linear random matrix model. Using this linear model, we calculate the distribution (including our new invariants) in the function field case, and then make local adjustments at the primes lying over $ell$ and $infty$ to make a conjecture in the number field case, which agrees with some numerical experiments.
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