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Metrical irrationality results related to values of the Riemann $ζ$-functionFeb 12 2018We introduce a one-parameter family of series associated to the Riemann $\zeta$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the parameter, where almost ... More

Deligne's conjecture for automorphic motives over CM-fields, Part I: factorizationFeb 08 2018This is the first of two papers devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. The present ... More

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

Normal elements of completed group algebras over ${\rm SL}_3(\mathbb{Z}_p) $Feb 08 2018Let $p$ be a prime integer and $\mathbb{Z}_p$ be the ring of $p$-adic integers. By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group ${\rm SL}_3(\mathbb{Z}_p)$ is a unit. ... 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

Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representationsFeb 07 2018A fundamental difficulty in the study of automorphic representations, representations of $p$-adic groups and the Langlands program is to handle the non-generic case. In this work we develop a complete local and global theory of tensor product $L$-functions ... 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

Pair correlation of sequences $(\lbrace a_n α\rbrace)_{n \in \mathbb{N}}$ with maximal order of additive energyFeb 07 2018We show for sequences $\left(a_{n}\right)_{n \in \mathbb{N}}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb{N}}$ does not have Poissonian pair correlations ... 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

Counting rational points and lower bounds for Galois orbitsFeb 06 2018In this article we present a new method to obtain polynomial lower bounds for Galois orbits of torsion points of one dimensional group varieties.

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

Markov spectrum near Freiman's isolated points in $M\setminus L$Feb 06 2018Freiman proved in 1968 that the Lagrange and Markov spectra do not coincide by exhibiting a countable infinite collection $\mathcal{F}$ of isolated points of the Markov spectrum which do not belong the Lagrange spectrum. In this paper, we describe the ... 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

Congruences for the Coefficients of the Powers of the Euler ProductFeb 05 2018Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we ... 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

Linear relations among asymptotic frequencies in continued fractionsFeb 05 2018Let $H(m,d)$ denote the asymptotic frequency of the natural numbers $k\equiv d \mod m$ in the continued fraction expansions of almost all numbers $x\in[0,1)$. For a fixed number $m\ge 4$, we study $\mathbb Q$-linear relations among the numbers $H(m,d)$, ... More

Prime and Möbius correlations for very short intervals in $\mathbb{F}_q[x]$Feb 04 2018We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as cancellation in sums ... 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

The finiteness of the genus of a finite-dimensional division algebra, and some generalizationsFeb 01 2018We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the ... 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

The first simultaneous sign change and non-vanishing of Hecke eigenvalues of newformsJan 31 2018Let $f$ and $g$ be two distinct newforms with weights $k_1, k_2 \ge 2$ and square-free levels $N_1, N_2 \ge 1$ respectively and which are normalized Hecke eigenforms. Further, for $n\ge 1$, let $ a_f(n)$ and $a_g(n)$ be the $n$-th Fourier-coefficients ... More

Sums with the Mobius function twisted by characters with powerful moduliJan 31 2018In their recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, ... More

On continued fraction expansions of quadratic irrationals in positive characteristicJan 30 2018Let $P$ be a prime polynomial in the variable $Y$ over a finite field and let $f$ be a quadratic irrational in the field of formal Laurant series in the variable $Y^{-1}$. We study the asymptotic properties of the degrees of the coefficients of the continued ... 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

Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$Jan 30 2018Feb 05 2018We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on $\mathrm{GL}_2$ over a number field which is not totally real, improving the one obtained by Marshall. The main ... 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

Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for $ζ(2m+1)$Jan 28 2018A comprehensive study of the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}\exp{(-an^{N}x)}}{1-\exp{(-n^{N}x)}}, 0<a\leq 1,\ x>0$, $N\in\mathbb{N}$ and $h\in\mathbb{Z}$, is undertaken. Two of the general transformations of ... 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

Reductions of modular Galois representations of Slope (2,3)Jan 26 2018We compute the semisimplifications of the mod-$p$ reductions of $2$-dimensional crystalline representations of $\Gal(\Qb_p/\Q_p)$ of slope $(2,3)$ and arbitrary weight, building on work of Bhattacharya-Ghate

An improved upper bound for the size of the multiplicative 3-Sidon setsJan 26 2018We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots,n\}$ ... 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 \[w(x)=x\exp\big(-(\sqrt{2}+o(1))\sqrt{\log x\log\log x}\big).\]

Connectedness of The Moduli Space of Artin-Schreier Curves of Fixed GenusJan 25 2018We study the moduli space $\mathcal{AS}_{g}$ of Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of positive characteristic $p$. The moduli space is partitioned by irreducible strata, where each stratum parameterizes Artin-Schreier ... More

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

Frobenius linear translators giving rise to new infinite classes of permutations and bent functionsJan 25 2018We show the existence of many infinite classes of permutations over finite fields and bent functions by extending the notion of linear translators, introduced by Kyureghyan [12]. We call these translators Frobenius translators since the derivatives of ... More

Automorphisms with eigenvalues in $S^1$ of a ${\mathbb Z}$-lattice with cyclic finite monodromyJan 24 2018For any finite set $M\subset {\mathbb Z}_{\geq 1}$ of positive integers, there is up to isomorphism a unique ${\mathbb Z}$-lattice $H_M$ with a cyclic automorphism $h_M:H_M\to H_M$ whose eigenvalues are the unit roots with orders in $M$ and have multiplicity ... More

Reduction of certain crystalline representations and local constancy in the weight spaceJan 23 2018We study the mod $p$ reduction of crystalline local Galois representations of dimension 2 under certain conditions on its weight and slope. Berger showed that for a fixed non-zero trace of the Frobenius, the reduction process is locally constant for varying ... 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

Fractional Powers of the Generating Function for the Partition FunctionJan 22 2018Jan 30 2018Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$. In this article, ... More

Stable lattices in modular Galois representations and Hida deformationJan 22 2018In this paper, we discuss the variation of the number of the isomorphic class of stable lattices when the weight and the level varies in Hida deformation by using the Kubota-Leopoldt $p$-adic $L$-function. As a corollary, we give a sufficient condition ... 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

A method for construction of rational points over elliptic curves II: Points over solvable extensionsJan 18 2018I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over a rational ... 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

Level Reciprocity in the twisted second moment of Rankin-Selberg L-functionsJan 18 2018We prove an exact formula for the second moment of Rankin-Selberg $L$-functions $L(1/2,f \times g)$ twisted by $\lambda_f(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$. The formula is a reciprocity ... 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

On partitions into squares of distinct integers whose reciprocals sum to 1Jan 18 2018In 1963, Graham proved that all integers greater than 77 (but not 77 itself) can be partitioned into distinct positive integers whose reciprocals sum to 1. He further conjectured that for any sufficiently large integer, it can be partitioned into squares ... More

Sums of Kloosterman sums over primes in an arithmetic progressionJan 17 2018For $q$ prime, $X \geq 1$ and coprime $u,v \in \mathbb{N}$ we estimate the sums \begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*} where $\text{Kl}_2(p;q)$ denotes ... 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

Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjectureJan 15 2018Let F be a totally real field of degree d and let p be an odd prime which is totally split in F. We define and study one-dimensional partial eigenvarieties interpolating Hilbert modular forms over F with weight varying only at a single place v above p. ... 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

Algebraic differential formulas for the shuffle, stuffle and duality relations of iterated integralsJan 09 2018Feb 02 2018In this paper we prove certain algebraic identities, which correspond to differentiations of the shuffle relation, the stuffle relation, and the relations which arise from M\"obius transformations of iterated integrals. These formulas provide fundamental ... More

On a $q$-analogue of the number of representations of an integer as a sum of two squaresJan 09 2018Kassel and Reutenauer \cite{kassel2016fourier} introduced a $q$-analogue of the number of representations of an integer as a sum of two squares. We establish some connections between the prime factorization of $n$ and the coefficients of this $q$-analogue. ... 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

Algebraic curves $A^{\circ l}(x)-U(y)=0$ and arithmetic of orbits of rational functionsJan 06 2018We give a description of pairs of complex rational functions $A$ and $U$ of degree at least two such that for every $d\geq 1$ the algebraic curve $A^{\circ d}(x)-U(y)=0$ has a factor of genus zero or one. In particular, we show that if $A$ is not a "generalized ... 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

Symmetry breaking for representations of rank one orthogonal groups IIDec 30 2017For a pair $(G,G')=(O(n+1,1), O(n,1))$ of reductive groups, we investigate intertwining operators (symmetry breaking operators) between principal series representations $I_\delta(V,\lambda)$ of $G$, and $J_\varepsilon(W,\nu)$ of the subgroup $G'$. The ... More

Betti numbers for numerical semigroup ringsDec 30 2017We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.

Characterizing moonshine functions by vertex-operator-algebraic conditionsDec 29 2017Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the Monstrous Moonshine Conjecture implies any finite order automorphism of $V$ has graded trace given by a completely replicable function, ... 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

Completeness of the list of spinor regular ternary quadratic formsNov 15 2017Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers ... 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

Diophantine triples in linear recurrence sequences of Pisot typeNov 10 2017The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness ... 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.

Improved Complexity Bounds for Counting Points on Hyperelliptic CurvesOct 10 2017We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by ... More

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

Factorization of Dyck words and the distribution of the divisors of an integerSep 13 2017In [CaballeroHooleyDelta], we associated a Dyck word $\langle\! \langle n \rangle\! \rangle_{\lambda}$ to any pair $(n, \lambda)$ consisting of an integer $n \geq 1$ and a real number $\lambda > 1$. The goal of the present paper is to show a relationship ... More

A relationship between the ideals of $\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right]$ and the Fibonacci numbersSep 13 2017Let $C_n(q)$ be the number of ideals of codimension $n$ of $\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right]$, where $\mathbb{F}_q$ is the finite field with $q$ elements. Kassel and Reutenauer [KasselReutenauer2015A] proved that $C_n(q)$ is a polynomial ... 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

On the Diophantine equation $(x+1)^{k}+(x+2)^{k}+...+(2x)^{k}=y^n$Sep 01 2017In this work, we give upper bounds for $n$ on the title equation. Our results depend on assertions describing the precise exponents of $2$ and $3$ appearing in the prime factorization of $T_{k}(x)=(x+1)^{k}+(x+2)^{k}+...+(2x)^{k}$. Further, on combining ... More

Distribution modulo one and denominators of the Bernoulli polynomialsAug 23 2017Let $\{\cdot\}$ denote the fractional part and $n \geq 1$ be a fixed integer. In this short note, we show for any prime $p$ the one-to-one correspondence $$\sum_{\nu \geq 1} \left\{\frac{n}{p^\nu}\right\} > 1 \quad \iff \quad p \mid \mathrm{denom}( B_n(x) ... 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