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Multiple zeta values and iterated log-sine integralsApr 22 2019We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among ... More Improved algorithms for left factorial residuesApr 19 2019We present improved algorithms for computing the left factorial residues $!p=0!+1!+...+(p-1)! \!\mod p$. We use these algorithms for the calculation of the residues $!p\!\mod p$, for all primes $p$ up to $2^{40}$. Our results confirm that Kurepa's left ... More An Elementary Approach to a Curious Identity from RamanujanApr 19 2019In his notebooks, Ramanujan wrote the following identity: \begin{equation} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 ... More On singular moduli that are S-unitsApr 18 2019Recently Yu. Bilu, P. Habegger and L. K\"uhne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that ... More A $q$-analogue of Wilson's congruenceApr 18 2019Let ${\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\ldots,n\}$. Let $${\mathfrak f}_n(q):=\sum_{\sigma\in{\mathcal C}_{n+1}}q^{{\mathrm maj}\,\sigma} $$ be a $q$-analogue of the factorial $n!$, where ${\mathrm maj}$ denotes ... More Perfect State Transfer on Weighted Graphs of the Johnson SchemeApr 18 2019We characterize perfect state transfer on real-weighted graphs of the Johnson scheme $\mathcal{J}(n,k)$. Given $\mathcal{J}(n,k)=\{A_1, A_2, \cdots, A_k\}$ and $A(X) = w_0A_0 + \cdots + w_m A_m$, we show, using classical number theory results, that $X$ ... More Critical $L$-values for some quadratic twists of Gross curvesApr 18 2019Let $K=\Bbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\beta)}$ denote its quadratic twist, with $\beta=\sqrt{-q}$. The curve $E$ is defined over the Hilbert class field $H$ of $K$. ... More A proof of the Hasse-Weil inequality for genus 2 à la ManinApr 18 2019We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956. Approaching Cusick's conjecture on the sum-of-digits functionApr 18 2019Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2. \] We prove ... More Poissonian pair correlation on manifolds via the heat kernelApr 17 2019We define a notion of Poissonian pair correlation (PPC) for Riemannian manifolds without boundary and prove that PPC implies uniform distribution in this setting. This extends earlier work by Grepstad and Larcher, Aistleitner, Lachmann, and Pausinger, ... More Quotients of numerical semigroups generated by two numbersApr 17 2019In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, denoted <a,b> d. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction expansions and Ostrowski-like ... More Subsets of $\mathbb{F}^*_p$ with only small products or ratiosApr 17 2019Let $p$ be a fixed prime. We estimate the number of elements of a set $A \subseteq \mathbb{F}^*_p$ for which $$ s_1s_2 \equiv a \pmod{p} \quad \mbox{for some}\quad a \in [-X,X] \quad \mbox{for all}\quad s_1,s_2 \in A. $$ We also consider variations and ... More Hausdorff and packing dimension of Diophantine setsApr 17 2019Using the variational principle in parametric geometry of numbers, we compute the Hausdorff and packing dimension of Diophantine sets related to exponents of Diophantine approximation, and their intersections. In particular, we extend a result of Jarn\'ik ... More The Siegel variance formula for quadratic formsApr 17 2019We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m\times m}$ and $B_{n\times n}$, where $m\geq n$. By using the oscillator representation, we give a formula for this variance sum in terms of ... More The module of vector-valued modular forms is Cohen-MacaulayApr 17 2019Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M(\rho)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M(H)$ denote ... More The product of parts or "norm" of a partitionApr 16 2019In this article we study the norm of an integer partition, which is defined as the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of current research ... More Quotients of Hurwitz PrimesApr 16 2019Quotient sets have attracted the attention of mathematicians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gaussian primes is also dense in the complex plane. Sittinger ... More Distribution of determinant of sum of matricesApr 16 2019Let $\mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $\det S$ for certain types of subsets $S$ in the ring $M_2(\mathbb F_q)$ of $2\times 2$ matrices with entries in $\mathbb F_q$. For $i\in \mathbb{F}_q$, let $D_i$ ... More Gamma factors for the Asai cube representationApr 16 2019We prove an equality between the gamma factors for the Asai cube representation of ${\rm R}_{E/F}{\rm GL}_2$ defined by the Weil$-$Deligne representations and the local zeta integrals of Ikeda and Piatetski-Shapiro$-$Rallis, where $E$ is an \'etale cubic ... More p-adic equidistribution of CM pointsApr 16 2019Let $X$ be a modular curve and consider a sequence of Galois orbits of CM points in $X$, whose $p$-conductors tend to infinity. Its equidistribution properties in $X({\bf C})$ and in the reductions of $X$ modulo primes different from $p$ are well understood. ... More On the Ramsey number of the Brauer configurationApr 16 2019We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions monochromatic with their common difference. Such a result has been obtained independently and in much greater generality by Sanders. ... More Dimension bound for doubly badly approximable affine formsApr 16 2019We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $\epsilon$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension ... More Modeling rational numbers by Cantor seriesApr 15 2019In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case of an arbitrary ... More An algorithm for determining torsion growth of elliptic curvesApr 15 2019We present a fast algorithm that takes as input an elliptic curve defined over $\mathbb Q$ and an integer $d$ and returns all the number fields $K$ of degree $d'$ dividing $d$ such that $E(K)_{tors}$ contains $E(F)_{tors}$ as a proper subgroup, for all ... More Ranks of overpartitions: asymptotics and inequalitiesApr 15 2019In this paper we compute asymptotics for $ \overline{N}(a,c,n), $ the number of overpartitions of $ n $ with rank congruent to $ a $ modulo $ c. $ As an application we prove, among others, some inequalities conjectured by Ji, Zhang and Zhao (2018), and ... More Universal norms and Greenberg conjectureApr 15 2019We investigate the group of universal norms attached to the cyclotomic Z {\ell}-tower of a totally real number field in connection with Grenberg's conjecture on Iwasawa invariants of such a field. An elliptic analogue of Fukuhara's trigonometeric identitiesApr 15 2019We obtain new elliptic function identities, which are an elliptic analogue of Fukuhara's trigonometric identities. We show that the coefficients of Laurent expansions at $z=0$ of our elliptic identities give rise to some reciprocity laws for elliptic ... More Integral points on twisted Markoff surfacesApr 15 2019We study the integral Hasse principle for affine varieties of the form ax^2+y^2+z^2-xyz=m ,using Brauer-Manin obstruction, and we produce examples whose Brauer groups include 4-torsion elements .We will construct their explicit representatives and compute ... More Value distribution of derivatives in polynomial dynamicsApr 15 2019For every $m\in\mathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in $\mathbb{C}\setminus\{0\}$ under the $m$-th order derivatives of the iterates of a polynomials $f\in \mathbb{C}[z]$ ... More On the largest element in D(n)-quadruplesApr 13 2019Let $n$ be a nonzero integer. A set of nonzero integers $\{a_1,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$ is called a $D(n)$-$m$-tuple. In this paper, we consider the question, for given integer $n$ which is not a ... More The Massey vanishing conjecture for number fieldsApr 13 2019A conjecture of Min\'a\v{c} and T\^an predicts that for any n>2, any prime p and any field k, the Massey product of n Galois cohomology classes in H^1(k,Z/pZ) must vanish if it is defined. We establish this conjecture when k is a number field. Elekes-Rónyai Theorem revisitedApr 12 2019In this paper it is proven that for any $f\in\mathbb{R}(x_1,x_2)$ and $A_1,A_2$ nonempty finite subsets of $\mathbb{R}$ such that $|A_1|=|A_2|$ and $f$ is defined in $A_1\times A_2$, we have that \begin{equation*} |f(A_1,A_2)|=\Omega\left(|A_1|^{\frac{4}{3}}\right) ... More On some determinants involving cyclotomic unitsApr 12 2019For each odd prime $p$, let $\zeta_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the determinant of the matrix ... More On sum-product basesApr 11 2019Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we consider by probabilistic arguments the existence of thin sets $A,A'$ of integers such that $AA+A=\mathbb{N}_0$ and $A'A'+A'A'=\mathbb{N}_0$. Generic cuspidal representations of $U(2,1)$Apr 11 2019Let $F$ be any non-Archimedean local field with a Galois involution $\sigma$ and $F_0$ be the fixed field for the action of $\sigma$. When the residue characteristic of $F_0$ is odd, using the explicit construction of cuspidal representations of classical ... More Generic cuspidal representations of $U(2,1)$Apr 11 2019Apr 12 2019Let $F$ be any non-Archimedean local field with a Galois involution $\sigma$ and $F_0$ be the fixed field for the action of $\sigma$. When the residue characteristic of $F_0$ is odd, using the explicit construction of cuspidal representations of classical ... More New Identities for Padovan NumbersApr 11 2019In \cite{Choi-Jo}, the $am+b$ ($0\leq b<a$) subscripted Tribonacci numbers are studied. This work is devoted to study a new generalization of Fibonacci numbers called Padovan numbers. In particular, the $am+b$ subscripted Padovan numbers will be expressed ... More A Framework for Modular Properties of False Theta FunctionsApr 10 2019False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives ... More On the Random Wave Conjecture for Dihedral Maaß FormsApr 10 2019We prove two results on arithmetic quantum chaos for dihedral Maass forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level $1$ forms, these ... More Equivariant splitting of the Hodge--de Rham exact sequenceApr 10 2019Let $X$ be an algebraic curve with an action of a finite group $G$ over a field $k$. We show that if the Hodge-de Rham short exact sequence of $X$ splits $G$-equivariantly then the action of $G$ on $X$ is weakly ramified. In particular, this generalizes ... More Density results for specialization sets of Galois coversApr 10 2019We provide evidence for this conclusion: given a finite Galois cover $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We show that this ... More Dynamically affine maps in positive characteristicApr 09 2019We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function ... More Borel's stable range for the cohomology of arithmetic groupsApr 09 2019In this note, we remark on the range in Borel's theorem on the stable cohomology of the arithmetic groups Sp(2n,Z) and SO(n,n;Z). This improves the range stated in Borel's original papers, an improvement that was known to Borel. Our main task is a technical ... More The structure and number of Erdős covering systemsApr 09 2019Introduced by Erd\H{o}s in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. Many beautiful questions and conjectures about covering systems have been posed over the past several ... More A K-theoretic Selberg trace formulaApr 09 2019Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of operators on the Hilbert space L^2(G/H) associated to compactly supported smooth functions ... More Quadratic Chabauty for (bi)elliptic curves and Kim's conjectureApr 09 2019We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets $\mathcal{X}(\mathbb{Z}_p)_2$ containing ... More Determinant groups of hermitian lattices over local fieldsApr 09 2019We describe the determinants of the automorphism groups of hermitian lattices over local fields. Using a result of G. Shimura, this yields an explicit method to compute the special genera in a given genus of hermitian lattices over a number field. Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant θ_3Apr 09 2019Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant \theta_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, ... More Hausdorff dimension of the large values of Weyl sumsApr 09 2019The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d n^{d})) \right| ... More A note on multiplicative automatic sequencesApr 08 2019We prove that any $q$-automatic completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character. This answers a question of J. P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin ... More A genus formula for the positive étale wild kernelApr 08 2019Let $F$ be a number field and let $i\geq 2$ be an integer. In this paper, we study the positive \'{e}tale wild kernel $\mathrm{WK}^{\mbox{\'{e}t},+}_{2i-2}F$, which is the twisted analogue of the $2$-primary part of the narrow class group. If $E/F$ is ... More Group Operation on Nodal CurvesApr 08 2019In this work, we present an efficient method for computing in the Generalized Jacobian of special singular curves. The efficiency of the operation is due to representation of an element in the Jacobian group by a single polynomial. SanD primes and numbersApr 07 2019We define S(um)anD(ifference) numbers as ordered pairs $(m,\, m+\Delta)$ such that the digital-sum $DS(m(m+\Delta))=\Delta.$ We consider both the decimal and the binary case. If both $m$ and $m+\Delta$ are prime numbers, we refer to SanD {\em primes}. ... More From Ramanujan Graphs to Ramanujan ComplexesApr 06 2019Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments are surveyed. ... More Integral points on Hilbert moduli schemesApr 06 2019We use the method of Faltings (Arakelov, Par\v{s}in, Szpiro) in order to explicitly study integral points on a class of varieties over $\mathbb Z$ called Hilbert moduli schemes. For instance, integral models of Hilbert modular varieties are classical ... More A p-adic analogue of Siegel's Theorem on sums of squaresApr 06 2019Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and ... More