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The Schinzel Hypothesis for PolynomialsFeb 21 2019The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is replaced by ... More Sums of linear transformations in higher dimensionsFeb 20 2019In this paper, we prove the following two results. Let $d$ be a natural number and $q,s$ be co-prime integers such that $1 \leq |s| < |q|$. Then there exists a constant $\delta > 0$ depending only on $q,s$ and $d$ such that for any finite subset $A$ of ... More A note on divisorial correspondences of semi-abelian varietiesFeb 20 2019Let S be a locally noetherian scheme and consider two extensions G_1 and G_2 of abelian S-schemes by S-tori. In this note we prove that the fppf-sheaf Corr _S(G_1,G_2) of divisorial correspondences between G_1 and G_2 is representable. Moreover, using ... More Many solutions to the $S$-unit equation $a+1=c$Feb 20 2019We show that there are arbitrarily large sets $S$ of $s$ primes for which the number of solutions to $a+1=c$ where all prime factors of $ac$ lie in $S$ has $\gg \exp( s^{1/4}/\log s)$ solutions. Sums of integers and sums of their squaresFeb 19 2019Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares ... More Sums of integers and sums of their squaresFeb 19 2019Feb 20 2019Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares ... More Weighted greatest common divisors and weighted heightsFeb 18 2019We introduce the weighted greatest common divisor of a tuple of integers and explore some of it basic properties. Furthermore, for a set of heights $\mathfrak w=(q_0, \ldots , q_n)$, we use the concept of the weighted greatest common divisor to define ... More $p$-divisible groups and relative crystalline representationsFeb 18 2019Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W(k)\langle t_1^{\pm 1}, \ldots, t_m^{\pm 1}\rangle$ ... More Geometric local epsilon factorsFeb 18 2019Inspired by the work of Laumon on $\varepsilon$-factors and by Deligne's $1974$ letter to Serre, we give an explicit cohomological definition of $\varepsilon$-factors for $\ell$-adic Galois representations over henselian discrete valuation fields of positive ... More Orthogonal polynomial expansions for the Riemann xi functionFeb 17 2019We study infinite series expansions for the Riemann xi function $\Xi(t)$ in three specific families of orthogonal polynomials: (1) the Hermite polynomials; (2) the symmetric Meixner-Pollaczek polynomials $P_n^{(3/4)}(x;\pi/2)$; and (3) the continuous ... More On two lattice points problems about the parabolaFeb 16 2019We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the square root cancellation ... More Kloosterman sums with twice-differentiable functionsFeb 15 2019We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \lfloor f(n)\rfloor+ y \lfloor f(n)\rfloor^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, ... More Jacobi Sums and Correlations of Sidelnikov SequencesFeb 15 2019We consider the problem of determining the cross-correlation values of the sequences in the families comprised of constant multiples of $M$-ary Sidelnikov sequences over $\mathbb{F}_q$, where $q$ is a power of an odd prime $p$. We show that the cross-correlation ... More On Siegel eigenvarieties at Saito-Kurokawa pointsFeb 15 2019We study the geometry of the Siegel eigenvariety $\mathcal{E}_{\Delta}$ of paramodular tame level $\Delta$ associated to a squarefree $N\in \mathbb{N}_{+}$ at certain points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal eigenform of $\mathrm{S}_{2k-2}(\Gamma_0(N))$ ... More On semilinear sets and asymptotically approximate groupsFeb 15 2019Let $G$ be any group and $A$ be an arbitrary subset of $G$ (not necessarily symmetric and not necessarily containing the identity). The $h$-fold product set of $A$ is defined as $$A^{h} :=\lbrace a_{1}.a_{2}...a_{h} : a_{1},\ldots,a_n \in A \rbrace.$$ ... More On the mod-$p$ distribution of discriminants of $G$-extensionsFeb 15 2019This paper was motivated by a recent paper by Krumm and Pollack investigating modulo-$p$ behaviour of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers ... More On an extension of the Landau-Gonek formulaFeb 14 2019We prove an extension of the Landau-Gonek formula. As an application we recover unconditionally some of the consequences of a pair correlation estimate that previously was known under the Riemann hypothesis. As one corollary we prove that at least two-thirds ... More From partitions to Hodge numbers of Hilbert Schemes of SurfacesFeb 14 2019We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, ... More From partitions to Hodge numbers of Hilbert Schemes of SurfacesFeb 14 2019Feb 15 2019We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, ... More The digit exchanges in the beta expansion of algebraic numbersFeb 14 2019In this article, we investigate the $\beta$-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where $\beta$ is a Pisot or Salem number. Moreover, we define a new class of algebraic ... More Extended Congruences for Harmonic NumbersFeb 14 2019We derive $p$-adic expansions for the generalized Harmonic numbers $H^{(j)}_{p-1}$ and $H^{(j)}_{\frac{p-1}{2}}$ involving the Bernoulli numbers $B_j$ and the the base-2 Fermat quotient $q_p$. While most of our results are not new, we obtain them elementarily, ... More Searching for modular companionsFeb 14 2019In this note, we report on the results of a computer search performed to find possible modular companions to certain $q$-series identities and conjectures. For the search, we use conditions arising from the asymptotics of Nahm sums. We focus on two sets ... More An instance where the major and minor arc integrals meetFeb 13 2019We apply the circle method to obtain an asymptotic formula for the number of integral points on a certain sliced cubic hypersurface related to the Segre cubic. Unusually, the major and minor arc integrals in this application are both positive and of the ... More Log-decay $F$-isocrystals on higher dimensional varietiesFeb 13 2019Let $k$ be a perfect field of positive characteristic and let $X$ be a smooth irreducible quasi-compact scheme over $k$. The Drinfeld-Kedlaya theorem states that for an irreducible $F$-isocrystal on $X$, the gap between consecutive generic slopes is bounded ... More Some properties of coefficients of cyclotomic polynomialsFeb 12 2019This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with $t\geq 3$ odd, ... More Matrix scaling, explicit Sinkhorn limits, and arithmeticFeb 12 2019The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the Sinkhorn limits of certain ... More A partial theta function Borwein conjectureFeb 12 2019We present an infinite family of Borwein type $+ - - $ conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument. Local model of Hilbert-Siegel moduli schemes in $Γ_1(p)$-levelFeb 12 2019We construct a local model for Hilbert-Siegel moduli schemes with $\Gamma_1(p)$-level bad reduction over $\text{Spec }\mathbb{Z}_{q}$, where $p$ is a prime unramified in the totally real field and $q$ is the residue cardinality over $p$. Our main tool ... More Degenerate central factorial numbers of the second kindFeb 12 2019In this paper, we introduce the degenerate central factorial polynomials and numbers of the second kind which are degenerate versions of the central factorial polynomials and numbers of the second kind. We derive some properties and identities for those ... More Euler product asymptotics on Dirichlet L-functionsFeb 12 2019We derive the asymptotic behaviour of partial Euler products for Dirichlet $L$-functions $L(s, \chi)$ in the critical strip upon assuming only the Generalised Riemann Hypothesis (GRH). Capturing the behaviour of the partial Euler products on the critical ... More Explicit bounds for small prime nonresiduesFeb 12 2019Let $\chi$ be a Dirichlet character modulo a prime~$p$. We give explicit upper bounds on $q_1<q_2<\dots<q_n$, the $n$ smallest prime nonresidues of $\chi$. More precisely, given $n_0$ and $p_0$ there exists an absolute constant $C=C(n_0,p_0)>0$ such that ... More Abelian Varieties and Finitely Generated Galois GroupsFeb 11 2019This paper surveys the methods that have been used to attack the conjecture, still open, that an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank. The minimal cone of an algebraic Laurent seriesFeb 11 2019For a given Laurent series that is algebraic over the field of power series in several indeterminates over a characteristic zero field, we show that the convex hull of its support is essentially a polyhedral rational cone. One of the main tools for proving ... More Gaussian Generalized Tetranacci NumbersFeb 11 2019In this paper, we define Gaussian generalized Tetranacci numbers and as special cases, we investigate Gaussian Tetranacci and Gaussian Tetranacci-Lucas numbers with their properties. Halász's theorem for Beurling generalized numbersFeb 11 2019We show that Hal\'{a}sz's theorem holds for Beurling numbers under the following two mild hypotheses on the generalized number system: existence of a positive density for the generalized integers and a Chebyshev upper bound for the generalized primes. ... More Congruences on sums of $q$-binomial coefficientsFeb 11 2019We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by Guo. We also ... More Weighted prime geodesic theoremsFeb 11 2019Prime geodesic theorems for weighted infinite graphs and weighted building quotients are given. The growth rates are expressed in terms of the spectral data of suitable translation operators inspired by a paper of Bass. On the Diophantine equations f (x) = g(y)Feb 11 2019The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for which the Diophantine ... More Number theoretical properties of Romik's dynamical systemFeb 10 2019We study a dynamical system that was originally defined by Romik in 2008 using an old theorem of Berggren concerning Pythagorean triples. Romik's system is closely related to the Farey map on the unit interval which generates an additive continued fraction ... More Über die Winkel zwischen UnterräumeFeb 10 2019We prove a metric statement about approximation of a $n$-dimensional linear subspace $A$ in $\mathbb{R}^d$ by $n$-dimensional rational subspaces. We consider the problem of finding a rational subspace $B$ of bounded height $H=H(B)$ for which the angle ... More On a problem of Pillai with Fibonacci numbers and powers of $3$Feb 09 2019Consider the sequence $ \{F_{n}\}_{n\geq 0} $ of Fibonacci numbers defined by $ F_0=0 $, $ F_1 =1$ and $ F_{n+2}=F_{n+1}+ F_{n} $ for all $ n\geq 0 $. In this paper, we find all integers $ c $ having at least two representations as a difference between ... More Dynamically Defined Sequences with Small DiscrepancyFeb 08 2019We study the problem of constructing sequences $(x_n)_{n=1}^{\infty}$ on $[0,1]$ in such a way that $$ D_N^* = \sup_{0 \leq x \leq 1} \left| \frac{ \left\{1 \leq i \leq N: x_i \leq x \right\}}{N} - x \right|$$ is uniformly small. A result of Schmidt shows ... More Counting multi-quadratic number fields of bounded discriminantFeb 08 2019We prove an asymptotic formula for the number of multi-quadratic number fields of bounded discriminant with a power-saving error term. Furthermore, we explicitly calculate the leading coefficient and extend our result to totally real multi-quadratic number ... More On Locally GCD Equivalent Number FieldsFeb 08 2019Local GCD Equivalence is a relation between extensions of number fields which is weaker than the classical arithmetic equivalence. It was originally studiedd by Lochter, using an equivalent relation called Weak Kronecker Equivalence. Among the many results ... More On the behavior of the logarithm of the Riemann zeta-functionFeb 08 2019The purpose of the present paper is to reveal the relation between the behavior of the logarithm of the Riemann zeta-function $\log{\zeta(s)}$ and the distribution of zeros of the Riemann zeta-function. We already know some examples for the relation by ... More Lattices from tight frames and vertex transitive graphsFeb 07 2019We show that real tight frames that generate lattices must be rational. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. We use this observation to describe a construction of strongly eutactic ... More A short introduction to Monstrous MoonshineFeb 07 2019This paper is an introduction to the Monstrous Moonshine correspondence aiming at an undergraduate level. We review first the classification of finite simple groups and some properties of the monster $\mathbb{M}$, and then the theory of classical modular ... More A short introduction to Monstrous MoonshineFeb 07 2019Feb 18 2019This paper is an introduction to the Monstrous Moonshine correspondence aiming at an undergraduate level. We review first the classification of finite simple groups and some properties of the monster $\mathbb{M}$, and then the theory of classical modular ... More Good reduction of K3 Surfaces in equicharacteristic pFeb 07 2019We show that for smooth and proper varieties over local fields with no non-trivial vector fields, good reduction descends over purely inseparable extensions. We use this to extend the Neron-Ogg-Shafarevich criterion for K3 surfaces to the equicharacteristic ... More Subfactors and Hecke groupsFeb 07 2019We study a relation between the Hecke groups and the index of subfactors in a von Neumann algebra. Such a problem was raised by V. F. R. Jones. We solve the problem using the notion of a cluster C*-algebra. On the difference between Zumkeller numbersFeb 06 2019Feb 09 2019In this paper, we prove that for every $\ell \in \mathbb{N}$ there are infinitely many $(a,b)$ that both $a$ and $b$ are Zumkeller numbers and $b-a= \ell$ Diophantine approximation on curvesFeb 06 2019Let $g$ be a dimension function. The Generalised Baker-Schmidt Problem (1970) concerns the $g$-dimensional Hausdorff measure ($\HH^g$-measure) of the set of $\psi$-approximable points on non-degenerate manifolds. The problem relates the `size' of the ... More