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Polyhedral Gauss Sums, and polytopes with symmetry

, 2015
We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n \mathbb Z}$, to ...
Malikiosis, Romanos-Diogenes, Robins, Sinai, Zhang, Yichi +2 more
core +2 more sources

Sister Beiter and Kloosterman: a tale of cyclotomic coefficients and modular inverses [PDF]

, 2012
For a fixed prime $p$, the maximum coefficient (in absolute value) $M(p)$ of the cyclotomic polynomial $\Phi_{pqr}(x)$, where $r$ and $q$ are free primes satisfying $r>q>p$ exists. Sister Beiter conjectured in 1968 that $M(p)\le(p+1)/2$.
Cobeli, Cristian +3 more
core

Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

, 2018
For $\Gamma$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=\Gamma \backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$.
Balkanova, Olga +4 more
core +1 more source

Modular hyperbolas and Beatty sequences

, 2019
Bounds for $\max\{m,\tilde{m}\}$ subject to $m,\tilde{m} \in \mathbb{Z}\cap[1,p)$, $p$ prime, $z$ indivisible by $p$, $m\tilde{m}\equiv z\bmod p$ and $m$ belonging to some fixed Beatty sequence $\{ \lfloor n\alpha+\beta \rfloor : n\in\mathbb{N} \}$ are ...
Technau, Marc
core

Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane. [PDF]