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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 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 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 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. 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. 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 $\mathbf{A}_{\text{inf}}$ 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 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 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 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 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 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 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 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 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 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 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, ... Moremath.GRmath.NT20D15, 20E18, 20E22, 20F05, 20F12, 20F14, 20--04, 11R37, 11R11,
11R29, 11Y40 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 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 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 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 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 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 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 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