Henri Cohen
In this paper we give a survey of recent methods for the asymptotic and exact enumeration of number fields with given Galois group of the Galois closure. In particular, the case of fields of degree up to 4 is now almost completely solved, both in theory and in practice. The same methods also allow construction of the corresponding complete tables of number fields with discriminant up to a given bound.
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We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these approximations are at least as good as the logarithmic integral approximation. As a corollary, we provide some reformulations of the Riemann hypothesis in terms of the prime counting function and the harmonic numbers.
We 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 fields.
Each number field has an associated finite abelian group, the class group, that records certain properties of arithmetic within the ring of integers of the field. The class group is well-studied, yet also still mysterious. A central conjecture of Brumer and Silverman states that for each prime $\ell$, every number field has the property that its class group has very few elements of order $\ell$, where "very few" is measured relative to the absolute discriminant of the field. This paper surveys recent progress toward this conjecture, and outlines its close connections to counting prime numbers,…
We give an upper bound for the number elliptic Carmichael numbers $n \le x$ that have recently been introduced by J. H. Silverman. We also discuss several possible ways for further improvements.
In this note, we prove a power-saving remainder term for the function counting $S_3$-sextic number fields. We also give a prediction on the second main term. We also present numerical data on counting functions for $S_3$-sextic number fields. Our data indicates that our prediction is likely to be correct, and it also suggests the existence of additional lower order terms which we have not yet been able to explain.
A link between error terms when counting smooth and rough numbers
研究光滑与粗糙数计数函数误差项之间的关系，并利用该关系得到de Bruijn近似Λ在光滑数计数中的误差上界。
Counting ideals in abelian number fields
本文在阿贝尔数域中改进了计数整数理想的方法，特别是在范数受限的情况下，提升了对互质理想对计数的结果。
The goal is to obtain an asymptotic formula for the number of quadratic extensions with bounded discriminant of a some quadratic number field with odd class number. This extends an already known result for Q.
A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$.
We study an asymptotic formula for counting integral points over an equation defined by a non-degenerated indefinite integral ternary quadratic form $f$ representing a non-zero integer $a$ such that $-a\cdot det(f)$ is square over a number field. In particular, we prove that the finite part of this asymptotic formula is given by the product of local density times $1-p^{-1}$ over all finite primes $p$ over $\Bbb Z$.
On the counting function of square-full numbers
在黎曼假设及ζ(s)零点为单零点的前提下，研究全平方数的分布，改进了其计数函数的误差项并补充了额外主项。
In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin's form: $$ π(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev functions as in conventional analytic methods, this new approach uses contour-integration method to analyze Riemann's prime counting function $J(x)$, which only differs from $π(x)$ by $\mathcal O(\sqrt x/\log x)$.
We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston.
In this paper, a new formula for π^(2)(N) is formulated, it is a function that counts the number of semi-primes not exceeding a given number N. A semi-prime is a natural number that is the product of precisely two prime numbers, the two primes in the product may equal each other. Semi-prime numbers are also a case of almost primes. Since a formula for this is already known, a new identity that uses the prime counting function is created by equating the two functions.
We prove results concerning the specialisation of torsion line bundles on a variety $V$ defined over $\mathbb{Q}$ to ideal classes of number fields. This gives a new general technique for constructing and counting number fields with large class group.
Counting Frobenius Pseudoprimes
本研究将Frobenius伪素数计数从一、二次情形推广到任意次数，给出Grantham Frobenius素性检验的伪证数计数公式，以及Frobenius伪素数平均数量的条件渐近上下界。
In this paper we establish a number of new estimates concerning the prime counting function π(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in small intervals.
Counting quintic fields with genus number one
本文研究了五次数域的亏格数，计算了亏格数为1的比例与平均亏格数，并证明正比例的五次数域具有任意大的亏格数，这些结果同样适用于$S_5$-五次数域。
Counting Eisenstein polynomials satisfying a condition from genus theory
本文给出了满足代数数域亏格数研究中产生的附加条件的奇素数次首一 Eisenstein 多项式数量的渐近公式。
Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other are prime number indexed basis entities taken from the discrete or continuous Fourier basis.
We prove a bound on the number of primes with a given splitting behaviour in a given field extension. This bound generalises the Brun-Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg's Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to $x$.
Let $k$ be a number field. For $\mathcal{H}\rightarrow \infty$, we give an asymptotic formula for the number of algebraic integers of absolute Weil height bounded by $\mathcal{H}$ and fixed degree over $k$.
Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which a…
The theorem below gives another way of computing the distribution prime counting function without using recursion and the values of Prime numbers
Counting biquadratic number fields with quaternionic and dihedral extensions
本文研究了可嵌入四元数或二面体扩张的双二次数域的计数问题，通过用 Hilbert 符号表达逆 Galois 问题的可解性，并应用 Heath-Brown 的方法界定关联二次特征标的和，建立了有界判别式下此类数域数量的渐近公式。
Let $f(n)$ count the number of subsets of $\{1,...,n\}$ without an element dividing another. In this paper I show that $f(n)$ grows like the $n$-th power of some real number, in the sense that $\lim_{n\rightarrow \infty}f(n)^{1/n}$ exists. This confirms a conjecture of Cameron and Erdös, proposed in a paper where they studied a number of similar problems, including the well known "Cameron-Erdös os Conjecture" on counting sum-free subsets.
In this note we describe a new method of counting the number of unordered factorizations of a natural number by means of a generating function and a recurrence relation arising from it, which improves an earlier result in this direction.
We define the notion of a weighted discriminant and corresponding counting function for number fields, and what it means for these counting functions to have a mass formula for a set of primes. We extend a result of Kedlaya to show that any proper counting function for a finite group $Γ$ has a mass formula for the set of primes not dividing $|Γ|$. We also prove that if $Γ$ is an $\ell$-group for some prime $\ell$, then there are only finitely many weighted discriminant counting functions for $Γ$-extensions of $\Q$ that have a mass formula for all primes. Finally, we enumerate all such counting…
For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.
Counting Polynomial-type Exceptional Units on Algebraic Varieties over Number Fields
本文研究了代数数域上代数簇上的多项式型例外单位的计数问题，通过中国剩余定理和Hensel提升技术，推导出在良好降解条件下精确计数公式，并利用Lang-Weil不等式建立渐进行为估计，特别是对二次度数以下的簇改进了误差项。
Power Savings for Counting (Twisted) Abelian Extensions of Number Fields
本文研究了数域阿贝尔扩张的计数问题，证明了误差项具有显著的幂次节省，并在假设广义Lindelöf假设下达到平方根幂次节省，同时揭示了部分未知的低阶项结构。
Counting abelian number fields with restricted ramification type
本文研究了按任意高度函数计数生成元受限的阿贝尔数域，求出显式前导常数，将其解释为$BG$上的Batyrev-Manin猜想版本，并证明了其在局部条件下的等分布性。
Let $r, v, n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations $$ n=1^{r}\cdot |k_{1}|^{v}+2^{r}\cdot |k_{2}|^{v}+3^{r}\cdot |k_{3}|^{v}+\ldots, $$ for ${\bf k}=(k_1,k_2,k_3,\dots)\in\mathbb{Z}^{\infty}$. For each $(r,v)\in\mathbb{N}\times\{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established.
Counting bounded elements of a number field
本文研究数域中赋值有界元素的计数问题，估计元素数量与极大线性无关元素数量，并结合群论、分歧理论与数的几何推导出理想格逐次极小的新界。
We count the number of isomorphism classes of degree $d$-twists of some polarized abelian varieties over finite fields of odd prime dimension. This can be seen as a higher dimensional analogue of the counting problem for elliptic curves case.
We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.
An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a co…
Weighted enumeration of number-fields and counting points that take bounded squarefree values along certain polynomials using Pseudo and Sudo maximal orders
利用Ekedhal筛法计数与判别式多项式密切相关的多项式的有界无平方因子值，并得到判别式有界数域数量的加权下界，其limsup远大于当前已知界。
Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a morphism between weighted projective spaces, allowing us to count rational points whose image under this morphism has bounded height. This method provides a more general and simpler proof for a result of the first-named author and Najman on counting elliptic curves with prescribed level structures over number fields. We further include some examples of applicati…
The main topic of this contribution is the problem of counting square-free numbers not exceeding $n$. Before this work we were able to do it in time (Comparing to the Big-O notation, Soft-O ($\softO$) ignores logarithmic factors) $\softO(\sqrt{n})$. Here, the algorithm with time complexity $\softO(n^{2/5})$ and with memory complexity $\softO(n^{1/5})$ is presented. Additionally, a parallel version is shown, which achieves full scalability. As of now the highest computed value was for $n=10^{17}$. Using our implementation we were able to calculate the value for $n=10^{36}$ on a cluster.
We obtain an upper bound on the number of extended irreducible q-ary Goppa codes of degree $r$ and length $q^n+1$, where $q=p^t$ and $n$ and $r>2$ are prime numbers.
A Keith number is a positive integer N with the decimal representation a_1a_2...a_n such that n>=2 and N appears in the sequence (K_m) given by the recurrence K_1=a_1,...,K_n=a_n and K_m=K_{m-1}+K_{m-2}+...+K_{m-n} for m>n. We prove that there are only finitely many Keith numbers using only one decimal digit (i.e., a_1=a_2=...=a_n), and that the set of Keith numbers is of asymptotic density zero.
A lattice point counting approach for the study of the number of self-avoiding walks on $\mathbb{Z}^{d}$
该研究通过将正方形格自回避行走计数简化为多维域整点计数，得到其长度为$n$的渐近估计，并为任意$d\geq 2$维$\mathbb{Z}^{d}$格自回避行走数量性质研究提供统一框架。
A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successfully applied to several hard special cases of Manin's conjecture over the field Q of rational numbers. Combining this method with techniques developed by Schanuel, we give a proof of Manin's conjecture over arbitrary number fields for the singular cubic surface S given by the equation w^3 = x y z.
In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and we apply our result to give an easy proof of a result which says that dense subsets A and B of {1,2,...,x} always produce asymptotically the expected number of x^r - smooth sums a+b, where a in A and b in B. Recall that a number n is said to be y-smooth if all its prime divisors are at most y.
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