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On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjectureJun 13 2019We investigate the ``partition function'' integrals $\int_{-1/2}^{1/2} |\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima $\max_{|h| \leq 1/2} |\zeta(1/2 + it + ih)|$, as $T \leq t \leq 2T$ varies. In particular, we prove that ... More

A short note on higher Mordell integralsJun 13 2019In this short note we show that techniques of Bringmann, Kaszian, and Milas hold for computing the higher Mordell integrals associated to a general family of quantum modular forms of depth two and weight one.

Effective algebraic independence of values of E-functionsJun 13 2019E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with polynomial coefficients. They were introduced by Siegel in 1929 to generalize ... More

Explicit degree bounds for right factors of linear differential operatorsJun 13 2019If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominators of very high degree. We give a completely explicit bound for the degrees of the (monic) right factors ... More

On simultaneous rational approximation to a real number and its integral powers, IIJun 13 2019For a positive integer $n$ and a real number $\xi$, let $\lambda_n (\xi)$ denote the supremum of the real numbers $\lambda$ for which there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi^2 ||, \ldots , ||q \xi^n||$ are all ... More

Bessel $δ$-method and subconvexity for $\mathrm{GL}(2)$ $L$-functions in the $t$-aspectJun 13 2019In this paper, we present a new Bessel $\delta$-method. As an application, we give a short proof for the Weyl-type subconvex bound in $t$-aspect for the $L$-function of a holomorphic newform of arbitrary level and nebentypus.

Quadratic points on modular curves with infinite Mordell--Weil groupJun 12 2019Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive ... More

Small-Support Uncertainty Principles on $\mathbb{Z}/p$ over Finite FieldsJun 12 2019We establish an uncertainty principle for functions $f: \mathbb{Z}/p \rightarrow \mathbb{F}_q$ with constant support (where $p \mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: \mathbb{Z}/p \rightarrow \mathbb{F}_q$ for which ... More

On a functional equation for an elliptic dilogarithmJun 12 2019It is known due to S. Bloch that elliptic dilogarithm is subject of big bunch of so-called Steinberg functional equation parametrized by rational functions on an elliptic curve. We show that all of these equations follows from the case of functions of ... More

Divisibility of Andrews' Singular Overpartitions by Powers of 2 and 3Jun 12 2019Andrews introduced the partition function $\overline{C}_{k, i}(n)$, called singular overpartition, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. He also proved ... More

Reciprocal Logarithmic Integrals Expressed as a SeriesJun 12 2019We present a method using contour integration to evaluate definite reciprocal logarithmic integrals and their associated infinite sums. In various cases these generalizations give the values of known mathematical constants such as Catalan's constant and ... More

On pro-$p$ groups with quadratic cohomologyJun 11 2019The main purpose of this article is to study pro-$p$ groups with quadratic $\mathbb{F}_p$-cohomology algebra, i.e. $H^\bullet$-quadratic pro-$p$ groups. Prime examples of such groups are the maximal Galois pro-$p$ groups of fields containing a primitive ... More

An alternative construction of equivariant Tamagawa numbersJun 11 2019We propose a new formulation of the equivariant Tamagawa number conjecture (ETNC) for non-commutative coefficients. We remove Picard groupoids, determinant functors, virtual objects and relative K-groups. Our Tamagawa numbers lie in an idele group instead ... More

On triangulable tensor products of B-pairs and trianguline representationsJun 11 2019We show that if V and V' are two p-adic representations of Gal(Qp^alg/Qp) whose tensor product is trianguline, then V and V' are both potentially trianguline.

On Odd Perfect, MultiPerfect and Harmonic NumbersJun 11 2019This paper introduces a relative of perfect, multiperfect and harmonic numbers. Call `\textit{alpha number}', any positive integer $n$ satisfying $\sigma(n)=(\alpha_1/\alpha_2)n $ with $\alpha_1,\alpha_2$ integers and $\alpha_1\leq \tau (n)$, where $\tau ... More

The $(α, β)-$ramification invariants of a number fieldJun 10 2019Let $L$ be a number field. For a given prime $p$ we define integers $\alpha_{p}^{L}$ and $\beta_{p}^{L}$ with some interesting arithmetic properties. For instance, $\beta_{p}^{L}$ is equal to $1$ whenever $p$ does not ramify in $L$ and $\alpha_{p}^{L}$ ... More

Colored Vertex Models and Iwahori Whittaker FunctionsJun 10 2019We give a recursive method for computing all values of a basis of Iwahori Whittaker functions on a split reductive group $G$ over a nonarchimedean local field $F$ using an action of the Hecke algebra. Then we specialize to $G=GL_r$ where we show that ... More

Arithmetic Chern-Simons theory for arithmetic schemesJun 10 2019In this paper, we generalize the arithmetic Chern-Simons theory to regular flat separated schemes of finite type over rings of integers of number fields by applying the duality theorems for arithmetic schemes.

A New Type of Abundant NumbersJun 10 2019This article defines a new type of abundant numbers, called largest rho-value (abbreviate LR) numbers, and then shows that Robin hypothesis is true if and only if all LR numbers $>5040$ satisfy Robin inequality.

A geometric Jacquet-Langlands correspondence for paramodular Siegel threefoldsJun 10 2019We study the Picard-Lefschetz formula for the Siegel modular threefold of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology with arbitrary automorphic coefficients. We give some applications to the Langlands ... More

Extension of a Diophantine triple with the property $D(4)$Jun 10 2019In this paper we give an upper bound on the number of extensions of a triple to a quadruple for the Diophantine $m$-tuples with the property $D(4)$ and confirm the conjecture of uniqueness of such extension in some special cases.

Union of Two Arithmetic Progressions with the Same Common Difference Is Not Sum-dominantJun 10 2019Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A = \{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid a_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. We prove the following results ... More

Adelic geometry on arithmetic surfaces II: completed adeles and idelic Arakelov intersection theoryJun 10 2019We work with completed adelic structures on an arithmetic surface and justify that the construction under consideration is compatible with Arakelov geometry. The ring of completed adeles is algebraically and topologically self-dual and fundamental adelic ... More

Explicit methods for the Hasse norm principle and applications to $A_n$ and $S_n$ extensionsJun 09 2019Let $K/k$ be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for $K/k$ and the defect of weak approximation for the norm one torus $R^1_{K/k} \mathbb{G}_m$. ... More

Representation of integers by sparse binary formsJun 09 2019We will give new upper bounds for the number of solutions to the inequalities of the shape $|F(x , y)| \leq h$, where $F(x , y)$ is a sparse binary form, with integer coefficients, and $h$ is a sufficiently small integer in terms of the absolute value ... More

Shintani lifts for Weil representations of unitary groups over finite fieldsJun 09 2019We construct extended Weil representations of unitary groups over finite fields geometrically, and show that they are Shintani lifts for Weil representations.

$\mathbf{A}_{\text{inf}}$ is infinite dimensionalJun 09 2019Given a perfect valuation ring $R$ of characteristic $p$ that is complete with respect to a rank-$1$ nondiscrete valuation, we show that the ring $\mathbf{A}_{\text{inf}}$ of Witt vectors of $R$ has infinite Krull dimension.

The error term in the Sato-Tate theorem of BirchJun 08 2019We establish an error term in the Sato-Tate theorem of Birch. That is, for $p$ prime, $q=p^r$ we show that $\#\{ (a,b) \in \mathbb{F}_q^2 : \theta_{a,b}\in I\} =\mu_{ST}(I)q^2 + O_r(q^{7/4})$ for any interval $I\subseteq[0,\pi]$ where for an elliptic ... More

Difference sets and the primesJun 08 2019We show that if the difference of two elements of a set $A \subseteq [N]$ is never one less than a prime number, then $|A| = O (N \exp (-c (\log N)^{1/3}))$ for some absolute constant $c>0$.

A generalization of Selfridge's questionJun 08 2019Selfridge asked to investigate the pairs $(m,n)$ of natural numbers for which $2^m - 2^n$ divides $x^m - x^n$ for all integers $x.$ This question was answered by different mathematicians by showing that there are only finitely many such pairs. Let $R$ ... More

A Note on the Bateman-Horn ConjectureJun 08 2019We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version of the conjecture ... More

Half-integral weight modular forms and real quadratic $p$-rational fieldsJun 07 2019Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields. For $p=5$ we prove the existence of infinitely many real quadratic $p$-rational fields.

Embeddings of maximal tori in groups of type $F_4$Jun 07 2019We classify maximal tori in groups of type $F_4$ over a local or global field of characteristic different from $2$ and $3$. We prove a local-global principle for embeddings of maximal tori in groups of type $F_4$.

A tale of two omegasJun 07 2019We consider $\omega(n)$ and $\Omega(n)$, which respectively count the number of distinct and total prime factors of $n$. We survey a number of similarities and differences between these two functions, and study the summatory functions $L(x)=\sum_{n\leq ... More

Identifying central endomorphisms of an abelian variety via Frobenius endomorphismsJun 06 2019Assuming the Mumford-Tate conjecture, we show that the center of the endomorphism ring of an abelian variety defined over a number field can be recovered from an appropriate intersection of the fields obtained from its Frobenius endomorphisms. We then ... More

A topological groupoid representing the topos of presheaves on a monoidJun 06 2019Butz and Moerdijk famously showed that every (Grothendieck) topos with enough points is equivalent to the category of sheaves on some topological groupoid. We give an alternative, more algebraic construction in the special case of a topos of presheaves ... More

Eigenvalue equation for the modular graph $C_{a,b,c,d}$Jun 06 2019The modular graph $C_{a,b,c,d}$ on the torus is a three loop planar graph in which two of the vertices have coordination number four, while the others have coordination number two. We obtain an eigenvalue equation satisfied by $C_{a,b,c,d}$ for generic ... More

The Arakelov-Zhang pairing and Julia setsJun 06 2019The Arakelov-Zhang pairing $\langle\psi,\phi\rangle$ is a measure of the "dynamical distance" between two rational maps $\psi$ and $\phi$ defined over a number field $K$. It is defined in terms of local integrals on Berkovich space at each completion ... More

Ternary quadratic forms representing arithmetic progressionsJun 06 2019A positive quadratic form is $(k,\ell)$-universal if it represents all natural numbers $\equiv\ell\pmod k$, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$ such that $k\nmid\ell$ there exists ... More

On $3$-dimensional foliated dynamical systems and Hilbert type reciprocity lawJun 06 2019We show some fundamental results concerning $3$-dimensional foliated dynamical systems (FDS$^3$ for short) introduced by Deninger. Firstly, we give a decomposition theorem for an FDS$^3$, which yields a classification of FDS$^3$'s. Secondly, for each ... More

Low degree points on curvesJun 05 2019In this paper we investigate an arithmetic analogue of the gonality of a smooth projective curve $C$ over a number field $k$: the minimal $e$ such there are infinitely many points $P \in C(\bar{k})$ with $[k(P):k] \leq e$. Developing techniques that make ... More

Estimates for the number of rational points on simple abelian varieties over finite fieldsJun 05 2019Let $A$ be a simple abelian variety of dimension $g$ over the field $\mathbb{F}_q$. The paper provides improvements on the Weil estimates for the size of $A(\mathbb{F}_q)$. For an arbitrary value of $q$ we prove $(\lfloor(\sqrt{q}-1)^2 \rfloor + 1)^g ... More

Rational points and derived equivalenceJun 05 2019We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a pair of hyperkaehler ... More

Variants of Khintchine's theorem in metric Diophantine approximationJun 05 2019Jun 11 2019New results towards the Duffin-Schaeffer conjecture, which is a fundamental unsolved problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function $\psi: \mathbb{N}\to\mathbb{R}$ we denote by ... More

Variants of Khintchine's theorem in metric Diophantine approximationJun 05 2019New results towards the Duffin-Schaeffer conjecture, which is a fundamental unsolved problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function $\psi: \mathbb{N}\to\mathbb{R}$ we denote by ... More

Iterated integrals on products of one variable polylogarithmsJun 05 2019In this paper we show that the iterated integrals on products of one variable polylogarithms from 0 to 1 are actually multiple zeta values if they are convergent. In the divergent case, we define regularized iterated integrals from 0 to 1. By the same ... More

Iterated integrals on products of one variable polylogarithmsJun 05 2019Jun 06 2019In this paper we show that the iterated integrals on products of one variable polylogarithms from 0 to 1 are actually multiple zeta values if they are convergent. In the divergent case, we define regularized iterated integrals from 0 to 1. By the same ... More

Playing a game of billiard with FibonacciJun 05 2019By making use of the greatest common divisor's ($gcd$) properties we can highlight some connections between playing billiard inside a unit square and the Fibonacci sequence as well as the Euclidean algorithm. In particular by defining two maps $\tau$ ... More

Exponential Sums of Witt Towers over AffinoidsJun 05 2019In this paper we construct a Dwork theory for general exponential sums over affinoids in Witt towers. Using this, we compute the degree of the $L$-function, its Hodge polygon and examine when the Hodge and Newton polygons coincide.

Exploring transcendentality in superstring amplitudesJun 04 2019It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of maximum transcendentality. In this paper it is argued that there is a natural extension of this definition that applies ... More

On Euler systems for the multiplicative group over general number fieldsJun 04 2019We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning Euler systems for the multiplicative group over arbitrary number fields.

Conditional lower bound for the k-th prime ideal with given Artin symbolJun 04 2019We prove an explicit upper bound for the k-th prime ideal with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions.

Chow's theorem for semi-abelian varieties and bounds for splitting fields of algebraic toriJun 04 2019A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper we generalize Chow's theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every ... More

The density and minimal gap of visible points in some planar quasicrystalsJun 04 2019We give formulas for the density of visible points of several families of planar quasicrystals, which include the Ammann-Beenker point set and vertex sets of some rhombic Penrose tilings. These densities are used in order to calculate the limiting minimal ... More

Spectral average of central values of automorphic L-functions for holomorphic cusp forms on SO_0(m,2) IIJun 04 2019Given a maximal even-integral lattice $\cL$ of signature $(m+, 2-)$ with an odd $m\geq 3$, we consider the holomorphic cusp forms $F$ of weight $l$ on the bounded symmetric domain of type IV of dimension $m$ with respect to the discriminant subgroup of ... More

Inhomogeneous Diophantine Approximation on $M_0$-sets with restricted denominatorsJun 04 2019Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu$ with the property that $ |\widehat{\mu}(t)| \ll (\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence of natural numbers. If $\mathcal{A}$ ... More

Explicit $L^2$ bounds for the Riemann $ζ$ functionJun 03 2019Bounds on the tails of the zeta function $\zeta(s)$, and in particular explicit bounds, are needed for applications, notably for integrals involving $\zeta(s)$ on vertical lines or other paths going to infinity. An explicit version of the traditional ... More

Dynamics of geodesics, and Maass cusp formsJun 03 2019The correspondence principle in physics between quantum mechanics and classical mechanics suggests deep relations between spectral and geometric entities of Riemannian manifolds. We survey---in a way intended to be accessible to a wide audience of mathematicians---a ... More

A constraint for twist equivalence of cusp forms on GL$(n)$Jun 03 2019Jun 08 2019This Note answers, and generalizes, a question of Kaisa Matom\"aki. We show that give two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $\chi$ such that ... More

A constraint for twist equivalence of cusp forms on GL$(n)$Jun 03 2019This note answers, and generalizes, a question of Kaisa Matom\"aki. We show that give two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL_n$ over a number field $F$ of respective conductor $N_1,$ $N_2,$ every character $\chi$ such that ... More

A Tannakian framework for $G$-displays and Rapoport-Zink spacesJun 03 2019We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by B\"ultel and Pappas, and further studied by Lau. We use this framework to define Rapoport-Zink functors associated to triples $(G,\{\mu\},[b])$, where $G$ ... More

Standard Lattices of Compatibly Embedded Finite FieldsJun 03 2019Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field $\mathbb{F}_p$ at once. They can also be used to represent the algebraic closure $\bar{\mathbb{F}}_p$, and to represent ... More

Quantitative non-divergence and Diophantine approximation on manifoldsJun 03 2019The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational points lying ... More

Primitive divisors of sequences associated to elliptic curvesJun 03 2019Let $\{nP+Q\}_{n\geq0}$ be a sequence of points on an elliptic curve defined over a number field $K$. In this paper, we study the denominators of the $x$-coordinates of this sequence. We prove that, if $Q$ is a torsion point of prime order, then for $n$ ... More

Arithmetic topology in Ihara theory II: Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbolsJun 03 2019We introduce mod $l$ Milnor invariants of a Galois element associated to Ihara's Galois representation on the pro-$l$ fundamental group of a punctured projective line ($l$ being a prime number), as arithmetic analogues of Milnor invariants of a pure braid. ... More

A note on the Erdő-Straus ConjectureJun 03 2019In this paper I make a fundamental assertion about the Erd\H{o}s-Straus conjecture. Suppose that for some prime $p$ there exists $x,y,z \in \mathbb{N}$ with $x \leq y \leq z$ so that $$ \frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}. $$ If $p \nmid ... More

Polynomial analogue of the Kempner functionJun 03 2019In the integer case, the Kempner function of a positive integer $n$ is defined to be the smallest positive integer $k$ such that $n$ divides the factorial $k!$. In this paper, we first define a natural order for polynomials in $\F_q[t]$ over a finite ... More

On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoidsJun 02 2019We study semigroup C*-algebras of semigroups associated with number fields and initial data arising naturally from class field theory. Using K-theoretic invariants, we investigate how much information about the initial number-theoretic data is encoded ... More

Congruences modulo powers of 11 for some eta-quotientsJun 02 2019The partition function $ p_{[1^c11^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{11}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{11 n})^d}.\] In this paper, we prove infinite families of congruences for ... More

A note on linear forms in two logarithms: the argument of an algebraic powerJun 02 2019In this note, we shall give an improved lower bound for the argument of a power of a given algebraic number which has absolute value one but is not a root of unity.

The strategy of pattern recognition via Artin transfers applied to finite towers of 2-class fieldsJun 02 2019The isomorphism type of the Galois group of the 2-class field tower of quadratic number fields having a 2-class group with abelian type invariants (4,4) is determined by means of information on the transfer of 2-classes to unramified abelian 2-extensions, ... More

A formula on Stirling numbers of the second kind and its application to the unstable $K$-theory of stunted complex projective spacesJun 02 2019A formula on Stirling numbers of the second kind $S(n, k)$ is proved. As a corollary, for odd $n$ and even $k$, it is shown that $k!S(n, k)$ is a positive multiple of the greatest common divisor of $j!S(n, j)$ for $k+1\leq j\leq n$. Also, as an application ... More

On circular distributions and a conjecture of ColemanJun 01 2019We investigate a conjecture of Robert Coleman concerning the module of circular distributions.

Generalized Heegner cycles on Mumford curvesJun 01 2019We study generalised Heegner cycles, originally introduced by Bertolini-Darmon-Prasanna for modular curves, in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic $p$-adic ... More

Generalising the Wallis ProductMay 31 2019In 1655, John Wallis whilst at the University of Oxford discovered the famous and beautiful formula for pi, now known as Wallis' Product. Since then, several analogous formulae have been discovered generalising the original. One more modern proof of the ... More

A Universal HKR TheoremMay 31 2019In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a \emph{filtered circle} interpolating between the usual topological circle and a formal version of it. By mapping to schemes we ... More

Hankel Continued fractions and Hankel determinants of the Euler numbersMay 31 2019The Euler numbers occur in the Taylor expansion of $\tan(x)+\sec(x)$. Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. ... More

A bijective proof and generalization of Siladić's TheoremMay 31 2019In a recent paper, Dousse introduced a refinement of Siladi\'c's theorem on partitions, where parts occur in two primary and three secondary colors. Her proof used the method of weighted words and $q$-difference equations. The purpose of this paper is ... More

Equidistribution and the shrinking target problem for sequences of polynomialsMay 31 2019Jun 06 2019Let $(f_n)_{n=1}^{\infty}$ be a sequence of polynomials and $\alpha>1$. In this paper we study the distribution of the sequence $(f_n(\alpha))_{n=1}^{\infty}$ modulo one. We give sufficient conditions for a sequence $(f_n)_{n=1}^{\infty}$ to ensure that ... More

Equidistribution and the shrinking target problem for sequences of polynomialsMay 31 2019Let $(f_n)_{n=1}^{\infty}$ be a sequence of polynomials and $\alpha>1$. In this paper we study the distribution of the sequence $(f_n(\alpha))_{n=1}^{\infty}$ modulo one. We give sufficient conditions for a sequence $(f_n)_{n=1}^{\infty}$ to ensure that ... More

Arithmetic Chern-Simons theory with real placesMay 31 2019The goal of this paper is two folds: we generalize the arithmetic Chern-Simons theory over totally imaginary number fields to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern-Simons invariant with coefficient ... More

Characterizing the mod-$\ell$ local Langlands correspondence by nilpotent gamma factorsMay 31 2019Let $F$ be a $p$-adic field and choose an algebraic closure $\mathbb{F}_{\ell}$, with $\ell$ different from $p$. We define ``nilpotent lifts'' of irreducible generic $\mathbb{F}_{\ell}$-representations of $GL_n(F)$, which take coefficients in the ring ... More

Tetrahedron trinomial coefficient transformMay 31 2019We introduce the tetrahedron trinomial coefficient transform which takes a Pascal-like arithmetical triangle to a sequence. We define a Pascal-like infinite tetrahedron H, and prove that the application of the tetrahedron trinomial transform to one face ... More

A structure theorem for finite fieldsMay 31 2019We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev polynomials ... More

Basic properties of a generalized third order sequence of numbersMay 30 2019Jun 04 2019We study the properties of the third order sequence $(w_n)=\left(w_n(a,b,c; r, s,t)\right)$ defined by the recurrence relation $w_n = rw_{n - 1} + sw_{n - 2} + tw_{n - 3}\, (n \ge 3)$ with $w_0 = a,\,w_1 = b,\,w_2=c$, where $a$, $b$, $c$, $r$, $s$ and ... More

Basic properties of a generalized third order sequence of numbersMay 30 2019Jun 11 2019We study the properties of the third order sequence $(w_n)=\left(w_n(a,b,c; r, s,t)\right)$ defined by the recurrence relation $w_n = rw_{n - 1} + sw_{n - 2} + tw_{n - 3}\, (n \ge 3)$ with $w_0 = a,\,w_1 = b,\,w_2=c$, where $a$, $b$, $c$, $r$, $s$ and ... More

Representing Ordinal Numbers with Arithmetically Interesting Sets of Real NumbersMay 30 2019For a real number $x$ and set of natural numbers $A$, define $x \ast A := \{ x a \bmod 1: a\in A\}\subseteq [0,1).$ We consider relationships between $x$, $A$, and the order-type of $x\ast A$. For example, for every irrational $x$ and order-type $\alpha$, ... More

Non-linear additive twist of Fourier coefficients of $GL(3)$ Maass formsMay 30 2019Let $\lambda_{\pi}(1,n)$ be the Fourier coefficients of the Hecke-Maass cusp form $\pi$ for $SL(3,\mathbb{Z})$. The aim of this article is to get a non trivial bound on the sum which is non-linear additive twist of the coefficients $\lambda_{\pi}(1,n)$. ... More

Classification of Strongly Positive Representations of Even General Unitary GroupsMay 30 2019We explicitly construct the structure of Jacquet modules of parabolically induced representations of even unitary groups and even general unitary groups over a $p$-adic field $F$ of characteristic different than two. As an application, we obtain a classification ... More

Degenerate principal series in the general caseMay 30 2019Let $G_n$ denote either the group $SO(2n+1, F)$, $Sp(2n, F)$, or $GSpin(2n+1, F)$ over a non-archimedean local field of characteristic different than two. We determine all composition factors of degenerate principal series of $G_n$, using methods based ... More

Prime powers dividing products of consecutive integer values of $x^{2^n}+1$May 30 2019Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ ... More

Prime powers dividing products of consecutive integer values of $x^{2^n}+1$May 30 2019May 31 2019Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ ... More

Hypergeometric rational approximations to $ζ(4)$May 29 2019We give a new hypergeometric construction of rational approximations to $\zeta(4)$, which absorbs the earlier one from 2003 based on Bailey's ${}_9F_8$ hypergeometric integrals. With the novel ingredients we are able to get a better control of arithmetic ... More

A theorem of Bombieri-Vinogradov type with few exceptional moduliMay 29 2019If a set S of pairwise coprime moduli q, less than x^(9/40), is considered, one obtains the expected behavior for primes up to x in arithmetic progressions mod q, except for a subset of S whose cardinality is bounded by a power of log x.

On the density at integer points of a system comprising an inhomogeneous quadratic form and a linear formMay 29 2019We prove an analogue of the Oppenheim conjecture for a system comprising an inhomogeneous quadratic form and a linear form in $3$ variables using dynamics on the space of affine lattices.

On mixed joint discrete universality for a class of zeta-functions IIIMay 29 2019We present the most general at this moment results on the discrete mixed joint value-distribution and the universality property for the class of Matsumoto zeta-functions and periodic Hurwitz zeta-functions under certain linear independence condition on ... More

Higher Correlations and the Alternative HypothesisMay 28 2019The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between nontrivial zeros is supposed ... More

Interpolation over ZZ and torsion in class groupsMay 28 2019We prove an interpolation result over the integers, and use this to give a new proof that class groups of rings of integers are torsion.

Practical numbers among the binomial coefficientsMay 28 2019A "practical number" is a positive integer $n$ such that every positive integer less than $n$ can be written as a sum of distinct divisors of $n$. We prove that most of the binomial coefficients are practical numbers. Precisely, letting $f(n)$ denote ... More

On multiplicative automatic sequencesMay 28 2019We show that any automatic multiplicative sequence either coincides with a Dirichlet character or is identically zero when restricted to integers not divisible by small primes. This answers a question of Bell, Bruin and Coons. A similar result was obtained ... More