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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

Class field theory, Diophantine analysis and the asymptotic Fermat's Last TheoremFeb 20 2019Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine ... More

Differential Forms on Hyperelliptic Curves with Semistable ReductionFeb 20 2019Let $C$ be a hyperelliptic curve over a local field $K$ with odd residue characteristic, defined by some affine Weierstrass equation $y^2=f(x)$. We assume that $C$ has semistable reduction and denote by $\mathcal{X} \rightarrow \textrm{Spec}\, \mathcal{O}_K$ ... 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.

Jensen polynomials for the Riemann zeta function and other sequencesFeb 19 2019In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function $\zeta(s)$ at its point of symmetry. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an asymptotic ... More

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

Une nouvelle démonstration du théorème d'André sur les $E$-fonctions au sens largeFeb 19 2019We give a new proof of a theorem of Andr\'e (2014) stating that every polynomial relation over $\overline{\mathbb{Q}}$ between values of a family of $E$-functions in the broad sense $(f_1, \dots, f_n)$ comes from a polynomial relation over $\overline{\mathbb{Q}}(z)$ ... More

On the theory of higher rank Euler, Kolyvagin and Stark systems, III: applicationsFeb 19 2019In an earlier article we proved the existence of a canonical Kolyvagin derivative homomorphism between the modules of Euler and Kolyvagin systems (in any given rank) that are associated to $p$-adic representations over number fields. We now explain how ... More

Uniform Yomdin-Gromov parametrizations and points of bounded height in valued fieldsFeb 18 2019We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of F_q[t]-points of bounded degrees of algebraic ... 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

Reduced norms of division algebras over complete discrete valuation fields of local-global typeFeb 18 2019Let $F$ be a complete discrete valuation field whose residue field $k$ is a global field of positive characteristic $p$. Let $D$ be a central division $F$-algebra of $p$-power degree. We prove that the subgroup of $F^*$ consisting of reduced norms of ... More

Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximationFeb 18 2019The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss regarding equidistribution ... 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

Congruences involving the $U_{\ell}$ operator for weakly holomorphic modular formsFeb 18 2019Let $\lambda$ be an integer, and $f(z)=\sum_{n\gg-\infty} a(n)q^n$ be a weakly holomorphic modular form of weight $\lambda+\frac 12$ on $\Gamma_0(4)$ with integral coefficients. Let $\ell\geq 5$ be a prime. Assume that the constant term $a(0)$ is not ... More

On Petersson norms of generic cusp forms and special values of adjoint $L$-functions for ${\rm GSp}_4$Feb 18 2019We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${\rm GSp}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations. Our formula ... 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

Dynamics of continued fractions and distribution of modular symbolsFeb 17 2019We formulate a thermodynamical approach to the study of distribution of modular symbols, motivated by the work of Baladi-Vall\'ee. We introduce the modular partitions of continued fractions and observe that the statistics for modular symbols follow from ... More

A simple proof of a congruence for a series involving the little $q$-Jacobi polynomialsFeb 16 2019We give a simple and a more explicit proof of a mod $4$ congruence for a series involving the little $q$-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.

An effective Ratner equidistribution theorem for multiplicative Diophantine approximation on planar linesFeb 16 2019In this paper, we prove an effective asymptotic equidistribution result for one-parameter unipotent orbits in $\mathrm{SL}(3, \mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$. This enables us to establish a strengthening of the Littlewood conjecture, valid for ... More

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varietiesFeb 16 2019We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group ... 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

Counting points on hyperelliptic curves of type $y^2=x^{2g+1} + ax^{g+1} + bx$Feb 15 2019In this work, we investigate hyperelliptic curves of type $C: y^2 = x^{2g+1} + ax^{g+1} + bx$ over the finite field $\mathbb{F}_q, q = p^n, p > 2$. For the case of $g = 3$ and $4$ we propose algorithms to compute the number of points on the Jacobian of ... 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

Determinants with Bernoulli polynomials and the restricted partition functionFeb 14 2019Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We study two natural determinants of order $rD$ with Bernoulli polynomials and we present connections with ... 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

A Classification of order $4$ class groups of $\mathbb{Q}{(\sqrt{n^2+1})}$Feb 14 2019We classify all order 4 class groups of the family of real quadratic fields $\mathbb{Q}{(\sqrt{n^2+1})}$. The main tools used are special values of Dedekind zeta functions attached to these fields and generalized Dedekind sum.

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

Universal optimality of the $E_8$ and Leech lattices and interpolation formulasFeb 13 2019We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely ... More

On a certain local identity for Lapid-Mao's conjecture and formal degree conjecture : even unitary group caseFeb 13 2019Lapid and Mao formulated a conjecture on an explicit formula of Whittaker Fourier coefficients of automorphic forms on quasi-split classical groups and metaplectic groups as an analogue of Ichino-Ikeda conjecture. They also showed that this conjecture ... More

Fibonacci Sequences And Real Quadratic p-Rational FieldsFeb 13 2019We study the p-rationality of real quadratic fields in terms of generalized Fibonacci numbers and their periods modulo positive integers.

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

A Conditional Explicit Result for the Prime Number Theorem in Short IntervalsFeb 13 2019We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

Discorrelation between primes in short intervals and polynomial phasesFeb 13 2019Let $H = N^{\theta}$. We obtain estimates for the exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} \Lambda(n) e(\alpha n^k), \] which are valid for all $k \geq 1$ and $\theta > 2/3$. As a consequence of this, we deduce a short ... 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

Special values of generalized multiple Hurwitz zeta function at non-positive integersFeb 12 2019In this paper, we provide an alternative method to calculate the values of generalized multiple Hurwitz zeta function at non-positive integers by means of \emph{Raabe}'s formula and the \textit{Bernoulli} numbers.

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.

Multiple Stieltjes constants and Laurent type expansion of the multiple zeta functions at integer pointsFeb 12 2019In this article, we study the local behaviour of the multiple zeta functions at integer points and write down a Laurent type expansion of the multiple zeta functions around these points. Such an expansion involves a convergent power series whose coefficients ... More

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

Irrationality and transcendence of continued fractions with algebraic integersFeb 12 2019We extend a result of Han\v{c}l, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{\alpha_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their ... 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

Modular graph functions and odd cuspidal functions - Fourier and Poincaré seriesFeb 11 2019Modular graph functions are $SL(2,{\mathbb Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $\tau$. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain ... 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.

Overconvergent Hilbert modular forms via perfectoid modular varietiesFeb 11 2019We give a new construction of overconvergent $p$-adic Hilbert modular forms for an arbitrary totally real field $F$ based on the theory of perfectoid Hilbert modular varieties at infinite level. We show that our definition varies in $p$-adic weight families, ... More

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

On asymptotic properties of the generalized Dirichlet $L$-functionsFeb 11 2019Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$ with $\sigma>1$ ... 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

An improvement of the duality formalism of the rational etale siteFeb 10 2019We improve the arithmetic duality formalism of the rational etale site. This improvement allows us to avoid some exotic approximation arguments on local fields with ind-rational base, thus simplifying the proofs of the previously established duality theorems ... More

Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integersFeb 10 2019We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta\in[1,2]$, there is a unique KMS$_\beta$ state, and we prove that it ... 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

An Euler phi function for the Eisenstein integers and some applicationsFeb 09 2019The Euler phi function on a given integer $n$ yields the number of positive integers less than $n$ that are relatively prime to $n$. Equivalently, it gives the order of the group of units in the quotient ring $\mathbb{Z}/(n)$. We generalize the Euler ... More

Decoupling for certain quadratic surfaces of low co-dimensionsFeb 09 2019We prove sharp $\ell^{p}L^{p}$ decoupling inequalities for $2$ quadratic forms in $4$ variables. We also recover several previous results (arXiv:1409.1634, arXiv:1501.07224, arXiv:1609.02022, arXiv:1609.04107) in a unified way.

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

A finiteness result for $p$-adic families of Bianchi modular formsFeb 08 2019We study $p$-adic families of cohomological automorphic forms for ${\mathrm{GL}}(2)$ over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. ... 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

Zeros of the Lerch zeta-function and of its derivative for equal parametersFeb 08 2019A. Speiser proved that the Riemann hypothesis is equivalent to the absence of non-real zeros of the derivative of the Riemann zeta-function left of the critical line. His result has been extended by N. Levinson and H.L. Montgomery to the statement that ... More

Zeros of the Lerch zeta-function and of its derivative for equal parametersFeb 08 2019Feb 11 2019A. Speiser proved that the Riemann hypothesis is equivalent to the absence of non-real zeros of the derivative of the Riemann zeta-function left of the critical line. His result has been extended by N. Levinson and H.L. Montgomery to the statement that ... More

New insight into results of Ostrowski and Lang on sums of remainders using Farey sequencesFeb 08 2019The sums $S(x,t)$ of the centered remainders $kt-\lfloor kt\rfloor - 1/2$ over $k \leq x$ and corresponding Dirichlet series were studied by A. Ostrowski, E. Hecke, H. Behnke and S. Lang for fixed real irrational numbers $t$. Their work was originally ... 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

Inequalities for weighted sums of Mertens functionsFeb 08 2019In this article we derive some polynomial inequalities for Mertens functions.

An approach to harmonic analysis on non-locally compact groups II: an invariant measure on groups of ordered typeFeb 08 2019We consider a class of non-locally compact groups on which one may define a left-invariant, finitely additive measure taking values in some finitely generated extension of the field $\mathbb{R}$ of real numbers. In particular, we recover previously studied ... More

Measure and Integration on $GL_2$ over a Two-Dimensional Local FieldFeb 08 2019We define a translation-invariant measure and integral on $GL_2$ over a two-dimensional local field $F$ by combining elements of the classical $GL_2$ theory and the theory developed by Fesenko for the field $F$ itself. We give several alternate expressions ... More

Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic GeneralisationsFeb 08 2019We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by quartic polynomials, ... 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

Gamma factors for genuine principal series of covering groups (with an appendix by Caihua Luo)Feb 07 2019We consider the local coefficients matrix associated with intertwining operators of a genuine principal series of covering groups, and investigate some of its arithmetic invariants. In particular, it is shown that the determinant of such a matrix in the ... 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 density of sumsets and product setsFeb 07 2019In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.

A classical functional generalization of the first Barnes lemmaFeb 06 2019We give a brief account and a simpler proof of a contour integral formula for the Gauss hypergeometric function. Such formula is alternative to Barnes's integral formula and generalizes the first Barnes Lemma.

On an Irreducibility Criterion of Anca I. Bonciocat and Nicolae C. BonciocatFeb 06 2019Feb 19 2019There is an error in the main result. We provide an easy generalization to the irreducibility criterion in the title.

On an Irreducibility Criterion of Anca I. Bonciocat and Nicolae C. BonciocatFeb 06 2019Feb 12 2019We provide an easy generalization to the irreducibility criterion in the title.

On the difference between Zeumkeller numbersFeb 06 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 $b-a= \ell$

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

On the Hom Version of the Grothendieck Conjecture for Hyperbolic Polycurves of Dimension 2Feb 06 2019In this paper, we study the Hom version of the Grothendieck Conjecture for hyperbolic polycurves of dimension 2. We group theoretically characterize dominant morphisms from regular varieties to hyperbolic polycurves of dimension 2 in some sense. Also, ... More

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of momentsFeb 06 2019It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength ... More