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Some remarks on the non-real roots of polynomialsFeb 08 2018Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of real roots ... More Gaussian binomial coefficients with negative argumentsFeb 08 2018Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in ... More There is no Khintchine threshold for metric pair correlationsFeb 07 2018Let $\mathcal{A}\left(\alpha\right)$ denote the sequence $\left(\alpha a_{n}\right)_{n}$, where $\alpha\in\left[0,1\right]$ and where $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the ... More On the gaps between consecutive primesFeb 07 2018Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$ for some small ... More Cyclotomic torsion points in elliptic schemesFeb 07 2018An elliptic curve defined over a number field possesses only a finite number of torsion points defined over the cyclotomic closure of its field of definition. In analogy to the relative version of the Manin-Mumford conjecture stated by Masser and Zannier, ... More On the polynomial Szemerédi's theorem in finite fieldsFeb 06 2018Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $\gamma>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial polynomial progression ... More On growth of the number of determinants with restricted entriesFeb 06 2018Let $A$ be a finite subset of a field $\mathbb{F}$ and $D_n(A)$ be a set of all matrices with entries in $A$, namely $$ D_n(A)=\{D\in \mathbb{F}\ |\ \exists a_{ij}\in A, 1 \le i,j \le n, \det\bigl((a_{ij})\bigr)=D\}, $$ where the symbol $(a_{ij})$ defines ... More Local Energy Optimality of Periodic SetsFeb 06 2018We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic ... More Local to global principle for the moduli space of K3 surfacesFeb 06 2018Recently S. Patrikis, J.F. Voloch and Y. Zarhin have proven, assuming several well known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, ... More Examples of $q$-series for shifted Ramanujan seriesFeb 06 2018Feb 07 2018We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_0$, we call the resulting series: "Ramanujan series with the shift $x_0$". Then, we relate these shifted series to some $q$-series. Finally, we indicate a possible way ... More Roots of Polynomials and The Derangement ProblemFeb 06 2018We present a new killing-a-fly-with-a-sledgehammer proof of one of the oldest results in probability which says that the probability that a random permutation on $n$ elements has no fixed points tends to $e^{-1}$ as $n$ tends to infinity. Our proof stems ... More $q$-Analouges of two Ramanujan-type forumlas for $1/π$Feb 06 2018We give $q$-analouges of the following two Ramanujan-type forumlas for $1/\pi$: \begin{align*} \sum_{k=0}^\infty \frac{(6k+1)(\frac{1}{2})_k^3}{k!^3 4^k} =\frac{4}{\pi}, \quad\text{and}\quad \sum_{k=0}^\infty (-1)^k(6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 ... More New conjecture related to a conjecture of McIntoshFeb 06 2018We introduce a new conjecture on products of two distinct primes that would provide a partial answer to a conjecture of McIntosh. Also, $\binom{2p-1}{p-1}-1$ is written in terms of a polynomial in prime $p$ over the integers and we discuss one way this ... More A propos d'une version faible du problème inverse de GaloisFeb 06 2018The aim of this paper is to construct fields $k$ which fulfill the Weak Inverse Galois Problem (stating that, for every finite group $G$, there exists a finite separable extension $L/k$ such that ${\rm{Aut}}(L/k)=G$), but which do not fulfill the usual ... More On residue maps for affine curvesFeb 05 2018We establish several compatibility results between residue maps in \'etale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and ... More Cuspidal cohomology of stacks of shtukasFeb 05 2018Let $G$ be a connected split reductive group over a finite field ${\mathbb F}_q$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalisation of the space of automorphic forms over a function field. In this paper, we construct a constant term ... More Averages of Hecke eigenvalues over thin sequencesFeb 02 2018Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer ... More Explicit bounds for primes in arithmetic progressionsJan 31 2018We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the ... More Analysis of the Continued Logarithm AlgorithmJan 30 2018Feb 01 2018The Continued Logarithm Algorithm - CL for short- introduced by Gosper in 1978 computes the gcd of two integers; it seems very efficient, as it only performs shifts and subtractions. Shallit has studied its worst-case complexity in 2016 and showed it ... More Conway river and Arnold sailJan 30 2018We establish a simple relation between two geometric constructions in number theory: the Conway river of a real indefinite binary quadratic form and the Arnold sail of the corresponding pair of lines. A note on expansion in prime fieldsJan 29 2018Let $\beta,\epsilon \in (0,1]$, and $k \geq \exp(122 \max\{1/\beta,1/\epsilon\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^{\beta}$, then there exists a sum of the form $$S = a_{1}B \pm \ldots \pm a_{k}B, ... More The $q$-unit circleJan 27 2018We define the unit circle for global function fields. We demonstrate that this unit circle (endearingly termed the \emph{$q$-unit circle}, after the finite field $\mathbb{F}_q$ of $q$ elements) enjoys all of the properties akin to the classical unit circle: ... More Highest perfect power of a product of integers less than $x$Jan 26 2018For $x\geq 3$, we define $w(x)$ as the highest integer $w$ for which there exist integers $m, y\geq 1$ and $1\leq n_1<\dots<n_m\leq x$ such that $n_1\cdots n_m=y^w$. We show that \[w(x)=x\exp\big(-(\sqrt{2}+o(1))\sqrt{\log x\log\log x}\big).\] Subconvex bounds on GL(3) via degeneration to frequency zeroJan 25 2018Feb 04 2018For a fixed cusp form $\pi$ on $\operatorname{PGL}_3(\mathbb{Z})$ and a varying Dirichlet character $\chi$ of prime conductor $q$, we prove that the subconvex bound \[ L(\pi \otimes \chi, \tfrac{1}{2}) \ll q^{3/4 - \delta} \] holds for any $\delta < 1/36$. ... More Stable gonality is computableJan 23 2018Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the minimum number ... More A remark on uniform boundedness for Brauer groupsJan 22 2018The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate ... More Counting subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$Jan 22 2018Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring $\Bbb{Z}_m ... More Cohomology of $p$-adic Stein spacesJan 20 2018We compute the $p$-adic \'etale and the pro-\'etale cohomologies of the Drinfeld half-space of any dimension. The main input is a new comparison theorem for the $p$-adic pro-\'etale cohomology of $p$-adic Stein spaces. On certain multiples of Littlewood and Newman polynomialsJan 19 2018Polynomials with all the coefficients in $\{ 0,1\}$ and constant term 1 are called Newman polynomials, whereas polynomials with all the coefficients in $\{ -1,1\}$ are called Littlewood polynomials. By exploiting an algorithm developed earlier, we determine ... More A note on multivariable $(\varphi,Γ)$-modulesJan 19 2018Let $F/{\mathbb Q}_p$ be a finite field extension, let $k$ be a field of characteristic $p$. Fix a Lubin Tate group $\Phi$ for $F$ and let $\Gamma\times\cdots\times\Gamma$ with $\Gamma={\mathcal O}_F^{\times}$ act on $k[[t_1,\ldots,t_n]][\prod_it_i^{-1}]$ ... More Difference sets and power residuesJan 19 2018Jan 31 2018Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb Z_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb Z_p})^n$ be an arbitrary subset such that $$ \{ \mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq ... More Least primitive root and simultaneous power-non residuesJan 18 2018Jan 20 2018Let $p$ be a prime and let $g(p)$ be the least primitive root modulo $p$. We prove that for any $\epsilon>0$ and $p$ large enough the bound \begin{align} g(p)\ll p^{\frac{1}{4\sqrt{e}}+\epsilon} \nonumber \end{align} holds for most prime $p$ such that ... More Where do odd perfect numbers live?Jan 18 2018The existence of a perfect odd number is an old open problem of number theory. An Euler's theorem states that if an odd integer $ n $ is perfect, then $ n $ is written as $ n = p ^ rm ^ 2 $, where $ r, m $ are odd numbers, $ p $ is a prime number of the ... More Real-analytic Eisenstein series via the Poincaré bundleJan 17 2018A classical construction of Katz gives a purely algebraic construction of real-analytic Eisenstein series using the Gau\ss--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$-adic properties ... More A Multi-layer Recursive Residue Number SystemJan 15 2018We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by means of an RNS ... More Tribonacci numbers and primes of the form $p=x^2+11y^2$Jan 14 2018In this paper we show that for any prime number $p$ not equal to $11$ or $19$, the Tribonacci number $T_{p-1}$ is divisible by $p$ if and only if $p$ is of the form $x^2+11y^2$. We first use class field theory on the Galois closure of the number field ... More Heuristics in direction of a p-adic Brauer--Siegel theoremJan 12 2018Jan 24 2018Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K andlet T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We conjecture the existence ... More Diophantine inequalities for generic ternary diagonal formsJan 11 2018Let k\geq 2 and consider the Diophantine inequality |x_1^k-\alp_2 x_2^k-\alp_3 x_3^k| <\tet. Our goal is to find non-trivial solutions in the variables x_i, 1\leq i\leq 3, all of size about P, assuming that \tet is sufficiently large. We study this problem ... More Arithmetic surfaces and adelic quotient groupsJan 08 2018We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact sequence. We relate ... More The reciprocal sum of primitive nondeficient numbersJan 05 2018We investigate the reciprocal sum of primitive nondeficient numbers, or pnds. In 1934, Erd\H{o}s showed that the reciprocal sum of pnds converges, which he used to prove that abundant numbers have a natural density. However no one has investigated the ... More Counting rational points on quadric surfacesJan 03 2018We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical forms ... More Polynomial functions as splinesDec 25 2017For a map $f:V\to H$ from a vector space over a finite field $k$ to a finite abelian group we define a function $f_m$ on the set $C_m(V)$ of $m$-cubes. We say that $f$ is of degree $<m$ if $f_m\equiv 0$. Note that in the case that $H=k$ and $char (k) ... More The Hilbert's-Tenth-Problem OperatorDec 23 2017For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$, which is naturally ... More Linear independence of powers of singular moduli of degree 3Dec 19 2017We show that two distinct singular moduli $j(\tau),j(\tau')$, such that for some positive integers $m, n$ the numbers $1,j(\tau)^m$ and $j(\tau')^n$ are linearly dependent over $\mathbb{Q}$ generate the same number field of degree at most $2$. This completes ... More Imaginary Multiquadratic Fields of Class Number Dividing $2^m$Dec 19 2017This paper gives a method to find all imaginary multiquadratic fields of class number dividing $2^{m},$ provided the list of all imaginary quadratic fields of class number dividing $2^{m+1}$ is known. We give a bound on the degree of such fields. As an ... More Branches on division algebrasDec 11 2017We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra B containing a given suborder, for certain important families of such suborders, including rings of integers of division subalgebras of B or ... More Exceptional digit frequencies and expansions in non-integer basesNov 28 2017In this paper we study the set of digit frequencies that are realised by elements of the set of $\beta$-expansions. The main result of this paper demonstrates that as $\beta$ approaches $1,$ the set of digit frequencies that occur amongst the set of $\beta$-expansions ... More Shift radix systems with general parametersNov 27 2017There are two dimensional expanding SRS, which have some periodic orbits. The aim of the present note is to describe as good as possible such unusual points. We give all regions, to which points belong obvious cycles, like $(1), (-1), (1,-1), (1,0), (-1,0)$. ... More On the proximity of large primesNov 15 2017By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of primes the ... More Quadratic Irrationals, Generating Functions, and Lévy ConstantsOct 24 2017We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. Consequently we can compute the L\'evy constant of any quadratic ... More Constrained ternary integersOct 23 2017An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the ... More Visible Points On Exponential CurvesOct 16 2017We provide two new bounds on the number of visible points on exponential curves modulo a prime for all choices of primes. We also provide one new bound on the number of visible points on exponential curves modulo a prime for almost all primes. On a function introduced by Erdös and NicolasSep 27 2017Erd\"os and Nicolas [erdos1976methodes] introduced an arithmetical function $F(n)$ related to divisors of $n$ in short intervals $\left] \frac{t}{2}, t\right]$. The aim of this note is to prove that $F(n)$ is the largest coefficient of polynomial $P_n(q)$ ... More Divisors on overlapped intervals and multiplicative functionsSep 27 2017Consider the real numbers $$ \ell_{n,k} = \ln\left( \tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n} \right) $$ and the intervals $\mathcal{L}_{n,k} = \left]\ell_{n,k}-\ln 3,\ell_{n,k}\right]$. For all $n \geq 1$, define $$ \frac{L_n(q)}{q^{n-1}} ... More Systems of forms in many variablesSep 26 2017We consider systems $\vec{F}(\vec{x})$ of $R$ homogeneous forms of the same degree $d$ in $n$ variables with integral coefficients. If $n\geq d2^dR+R$ and the coefficients of $\vec{F}$ lie in an explicit Zariski open set, we give a nonsingular Hasse principle ... More Middle divisors and $σ$-palindromic Dyck wordsSep 13 2017Sep 23 2017Given a real number $\lambda > 1$, we say that $d|n$ is a $\lambda$-middle divisor of $n$ if $$ \sqrt{\frac{n}{\lambda}} < d \leq \sqrt{\lambda n}. $$ We will prove that there are integers having an arbitrarily large number of $\lambda$-middle divisors. ... More On prime numbers of the form $2^n \pm k$Sep 13 2017Consider the set $\mathcal{K}$ of integers $k$ for which there are infinitely many primes $p$ such that $p+k$ is a power of $2$. The aim of this paper is to show a relationship between $\mathcal{K}$ and the limits points of some set rational numbers related ... More Number systems over ordersAug 16 2017Let $\mathbb{K}$ be a number field of degree $k$ and let $\mathcal{O}$ be an order in $\mathbb{K}$. A generalized number system over $\mathcal{O}$ (GNS for short) is a pair $(p,\mathcal{D})$ where $p \in \mathcal{O}[x]$ is monic and $\mathcal{D}\subset\mathcal{O}$ ... More