# Number Theory

## New submissions

[ total of 25 entries: 1-25 ]
[ showing up to 2000 entries per page: fewer | more ]

### New submissions for Tue, 20 Mar 18

[1]
Title: The density of visible points in the Ammann-Beenker point set
Subjects: Number Theory (math.NT)

The relative density of visible points of the integer lattice $\mathbb{Z}^d$ is known to be $1/\zeta(d)$ for $d\geq 2$, where $\zeta$ is Riemann's zeta function. In this paper we prove that the relative density of visible points in the Ammann-Beenker point set is given by $2(\sqrt{2}-1)/\zeta_K(2)$, where $\zeta_K$ is Dedekind's zeta function over $K=\mathbb{Q}(\sqrt{2})$.

[2]
Title: The arithmetic derivative and Leibniz-additive functions
Subjects: Number Theory (math.NT)

An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$f(mn)=f(m)h_f(n)+f(n)h_f(m)$$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative $D$; namely, $D$ is Leibniz-additive with $h_D(n)=n$. In this paper, we study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function $f$ is totally determined by the values of $f$ and $h_f$ at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions.

[3]
Title: On class groups of random number fields
Subjects: Number Theory (math.NT)

The aim of the present paper is to add, in several ways, to our understanding of the Cohen-Lenstra-Martinet heuristics on class groups of random number fields. Firstly, we point out several difficulties with the original formulation, and offer possible corrections. Secondly, we recast the heuristics in terms of Arakelov class groups of number fields. Thirdly, we propose a rigorously formulated Cohen-Lenstra-Martinet conjecture.

[4]
Title: Orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$
Comments: Banach Center Publ. (to appear). arXiv admin note: text overlap with arXiv:1611.07840, arXiv:1611.08181
Subjects: Number Theory (math.NT)

This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calculations extend those given by Zagier and Kramarz \cite{ZK} and by Watkins \cite{Wat}. Our main observations concern the asymptotic formula for the frequency of orders of Tate-Shafarevich groups. In the last section we propose a similar asymptotic formula for the class numbers of real quadratic fields.

[5]
Title: Bounded error uniformity of the linear flow on the torus
Authors: Bence Borda
Subjects: Number Theory (math.NT)

A linear flow on the torus $\mathbb{R}^d / \mathbb{Z}^d$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $\mathbb{Q}$. In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne--Ehrenfest type theorem for the mod $1$ discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.

[6]
Title: A Positive Proportion of Hasse Principle Failures in a Family of Châtelet Surfaces
Authors: Nick Rome
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)

We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop asymptotic formulae for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bret\`{e}che and Browning.

### Cross-lists for Tue, 20 Mar 18

[7]  arXiv:1803.06394 (cross-list from math.CO) [pdf, ps, other]
Title: Combinatorial proofs of two Euler type identities due to Andrews
Subjects: Combinatorics (math.CO); Number Theory (math.NT)

We prove combinatorially some identities related to Euler's partition identity (the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts). They were conjectured by Beck and proved by Andrews via generating functions.
Let $a(n)$ be the number of partitions of $n$ such that the set of even parts has exactly one element, $b(n)$ be the difference between the number of parts in all odd partitions of $n$ and the number of parts in all distinct partitions of $n$, and $c(n)$ be the number of partitions of $n$ in which exactly one part is repeated. Then, $a(n)=b(n)=c(n)$. The identity $a(n)=c(n)$ was proved combinatorially (in greater generality) by Fu and Tang. We prove combinatorially that $a(n)=b(n)$ and $b(n)=c(n)$. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. Let $c_1(n)$ be the number of partitions of $n$ such that there is exactly one part occurring three times while all other parts occur only once and let $b_1(n)$ to be the difference between the total number of parts in the partitions of $n$ into distinct parts and the total number of different parts in the partitions of $n$ into odd parts. We prove combinatorially that $c_1(n)=b_1(n)$. In addition to these results by Andrews, we prove combinatorially that $b_1(n)=a_1(n)$, where $a_1(n)$ counts partitions of $n$ such that the set of even parts has exactly one element and satisfying some additional conditions. Moreover, we offer an analog of these results for the number of partitions of $n$ with exactly one part occurring two times while all other parts occur only once.

[8]  arXiv:1803.06700 (cross-list from math.LO) [pdf, ps, other]
Title: Categoricity of Shimura Varieties
Subjects: Logic (math.LO); Algebraic Geometry (math.AG); Number Theory (math.NT)

We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity.

[9]  arXiv:1803.06859 (cross-list from math.DS) [pdf, ps, other]
Title: Approximation of non-archimedean Lyapunov exponents and applications over global fields
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)

Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show:
- a quantified version of Silverman's and Ingram's recent comparison between the critical height and any ample height on the moduli space $\mathcal{M}_d(\bar{\mathbb{Q}})$,
- two improvements of McMullen's finiteness of the mutiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period,
- a characterization of non-affine isotrivial rational maps defined over the function field $\mathbb{C}(X)$ of a normal projective variety $X$ in terms of the growth of the degree of the multipliers.

### Replacements for Tue, 20 Mar 18

[10]  arXiv:1512.04234 (replaced) [pdf, ps, other]
Title: Two applications of the spectrum of numbers
Subjects: Number Theory (math.NT); Formal Languages and Automata Theory (cs.FL)
[11]  arXiv:1606.00300 (replaced) [pdf, ps, other]
Title: Del Pezzo surfaces over finite fields and their Frobenius traces
Comments: 25 pages. Fixed various typos and improved exposition
Subjects: Number Theory (math.NT)
[12]  arXiv:1705.08159 (replaced) [pdf, ps, other]
Title: Diagonal forms of higher degree over function fields of $p$-adic curves
Authors: Susanne Pumpluen
Comments: Some small corrections/changes have been done with respect to the first version
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
[13]  arXiv:1706.05589 (replaced) [pdf, ps, other]
Title: Higher congruences between newforms and Eisenstein series of squarefree level
Authors: Catherine Hsu
Subjects: Number Theory (math.NT)
[14]  arXiv:1709.05780 (replaced) [pdf, ps, other]
Title: On the indivisibility of derived Kato's Euler systems and the main conjecture for modular forms
Comments: Some errors and typos are corrected. Two non-ordinary examples are added at the end
Subjects: Number Theory (math.NT)
[15]  arXiv:1709.10362 (replaced) [pdf, ps, other]
Title: Some analytic aspects of automorphic forms on GL(2) of minimal type
Comments: 28 pages. To appear in Comm. Math. Helv
Subjects: Number Theory (math.NT)
[16]  arXiv:1710.06135 (replaced) [pdf, ps, other]
Title: The depth structure of motivic multiple zeta values
Authors: Jiangtao Li
Subjects: Number Theory (math.NT)
[17]  arXiv:1802.00085 (replaced) [pdf, other]
Title: Explicit bounds for primes in arithmetic progressions
Comments: 67 pages. A typo detected in the initial version of Lemma 2.18 was repaired; its effects propagated through the paper, resulting in a change to many of the constants in our theorems. Results of computations, and the code used for those computations, can be found at: this http URL
Subjects: Number Theory (math.NT)
[18]  arXiv:1802.08185 (replaced) [pdf, ps, other]
Title: On quaternion algebra over the composite of quadratic number fields and over some dihedral fields
Comments: This is a preliminary form of the article. Categories: number theory; rings and algebras
Subjects: Number Theory (math.NT)
[19]  arXiv:1802.10012 (replaced) [pdf, ps, other]
Title: Integral points on generalised affine Châtelet surfaces
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
[20]  arXiv:1803.05236 (replaced) [pdf, other]
Title: Some negative results related to Poissonian pair correlation problems
Subjects: Number Theory (math.NT)
[21]  arXiv:1603.08360 (replaced) [pdf, other]
Title: Lyapunov spectrum of Markov and Euclid trees
Comments: Slightly improved version with more details added to the proof of Theorem 3
Journal-ref: Nonlinearity, Volume 30, Number 12 (2017), 4428-53
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
[22]  arXiv:1604.01938 (replaced) [pdf, ps, other]
Title: On the Noether number of $p$-groups
Subjects: Group Theory (math.GR); Commutative Algebra (math.AC); Number Theory (math.NT); Representation Theory (math.RT)
[23]  arXiv:1611.06470 (replaced) [pdf, ps, other]
Title: Bounded orbits of Diagonalizable Flows on finite volume quotients of products of $SL_2(\mathbb{R})$