Carlos Ros Pérez
This work consists of a heuristic study on the distribution of prime numbers in short intervals. We have modelled the occurrence of prime numbers such intervals as a counting experiment. As a result, we have provided an experimental validation and an extension of the Montgomery and Soundararajan conjecture. This is a reduced version of my bachelor`s thesis presented at the University of Valencia.
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A numerical study on the distributions of primes in short intervals of length $h$ over the natural numbers $N$ is presented. Based on Cramér's model in Number Theory, we obtain a heuristic expression applicable when $h \gg \log{N}$ but $h \ll N$, providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.
We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$ ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification.
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $π(x)$. As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than $x$, for the generalized Brun's constants and the first occurrence of a given gap $d$. Also the leading term $\log \log(x)$ in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.
Heuristic Bounded Prime Gaps via a Chaotic Multidimensional Sieve and Random Matrix Theory
本文提出融合混沌扰动与RMT的EMCHS概率筛框架，在GPY与Maynard筛法基础上启发式改进素数间隙界，并用数值证据与部分严格证明加以支撑。
Let N_{a,b}(x) count the number of primes p<=x with p dividing a^k+b^k for some k>=1. It is known that asymptotically N_{a,b}(x) grows like c(a,b)x/log x for some rational number c(a,b) that depends in a rather intricate way on a and b. A simple heuristic formula for N_{a,b}(x) is proposed and it is proved that it is asymptotically exact, i.e. has the same asymptotic behaviour as N_{a,b}(x). Connections with Ramanujan sums and character sums are discussed.
Consider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\rm{GSp}_{2g}(\mathbb{F}_\ell)$ with characteristic polynomial $f\mod\ell$ when $g$ is an odd prime. This infinite product is closely related to a ratio of class numbers. When $g=3$ we conjecture that the product gives the size of an isogeny class of principally polarized abelian threefolds.
We give a more strong heuristic justification of our conjecture on the excess of the odious primes.
An odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. Similarly, an even perfect number $M$ is said to be given in Euclidean form if $M = (2^p - 1)\cdot{2^{p - 1}}$ where $p$ and $2^p - 1$ are primes. In this article, we show how simple considerations surrounding the differences between the underlying properties of the Eulerian and Euclidean forms of perfect numbers give rise to what we will call the Euclid-Euler heuristics for perfect numbers.
We discuss heuristic asymptotic formulae for the number of pairing-friendly abelian varieties over prime fields, generalizing previous work of one of the authors arXiv:math1107.0307
In 2000, Galbraith and McKee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points. We show how their heuristics can be generalized to Jacobians of curves of higher genus. We then elaborate this in genus 2 and study various related issues, such as the probability of cyclicity and the probability of primality of the number of points on the curve itself. Finally, we discuss the asymptotic behavior as the genus tends to infinity.
We present a heuristic asymptotic formula as $x\to \infty$ for the number of isogeny classes of pairing-friendly elliptic curves with fixed embedding degree $k\geq 3$, with fixed discriminant, with rho-value bounded by a fixed $ρ_0$ such that $1<ρ_0<2$, and with prime subgroup order at most $x$.
The Bateman-Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green-Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman-Horn conjecture, its applications, and its origins.
In this work, we add an additional condition to strong pseudo prime test to base 2. Then, we provide theoretical and heuristics evidences showing that the resulting algorithm catches all composite numbers. Our method is based on the structure of singular cubics' Jacobian groups on which we also define an effective addition algorithm.
Sondow et al have studied Ramanujan primes (RPs) and observed numerically that, while half of all primes are RPs asymptotically, one obtains runs of consecutives RPs (resp. non-RPs) which are statistically significantly longer than one would expect if one was tossing an unbiased coin. In this discussion paper we attempt a heuristic explanation of this phenomenon. Our heuristic follows naturally from the Prime Number Theorem, but seems to be only partly satisfactory. It motivates why one should obtain long runs of both RPs and non-RPs, and also longer runs of non-RPs than of RPs. However, it al…
Let $E_0,\ldots,E_n$ be a partition of the set of prime numbers, and define $E_j(x) := \sum_{p \in E_j \atop p \leq x} \frac{1}{p}$. Define $π(x;\mathbf{E},\mathbf{k})$ to be the number of integers $n \leq x$ with $k_j$ prime factors in $E_j$ for each $j$. Basic probabilistic heuristics suggest that $x^{-1}π(x;\mathbf{E},\mathbf{k})$, modelled as the distribution function of a random variable, should satisfy a joint Poisson law with parameter vector $(E_0(x),\ldots,E_n(x))$, as $x \rightarrow \infty$. We prove an asymptotic formula for $π(x;\mathbf{E},\mathbf{k})$ which contradicts these heuri…
We study the asymptotics conjecture of Malle for dihedral groups $D_\ell$ of order $2\ell$, where $\ell$ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen--Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.
The computer data up to $2^{44}\approx 1.76\times 10^{13}$ on the gaps between consecutive twins is presented. The simple derivation of the heuristic formula describing computer results contained in the recent papers by P.F.Kelly and T.Pilling \cite{Kelly1}, \cite{Kelly2} is provided and compared with the ``experimental'' values.
On Iwasawa $λ$-invariants for abelian number fields and random matrix heuristics
本文研究被 Dirichlet 特征扭曲的 $p$-进 zeta 函数的 $λ$-不变量分布，利用 $p$-进随机矩阵理论给出预测并提供数值证据，同时探讨了 $χ$-正则素数的比例。
An elementary heuristic for Hardy-Littlewood extended Goldbach's conjecture
本文提出一个基于超几何分布和Hoeffding型界的初等组合启发式，用于预测Hardy-Littlewood扩展哥德巴赫猜想。
A Generalized Elliott-Halberstam Conjecture Implying the Twin Prime Hypothesis
本研究提出关乎算术级数素数对分布的移位卷积广义Elliott-Halberstam猜想（GEH-2），给出von Mangoldt函数相关性的分布水平，证明其蕴含孪生素数猜想并讨论相关数论意义。
Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding computational results suggest that the primes are distributed among sequence $(S_n)$ in the same way that they are distributed among positive integers. In other words, taking into account the Prime Number Theorem, this assertion is equivalent to \begin{equation*}\begin{split} &\# \{p:\, p\,\,{\rm is\,\,a\,\, prime\,\, and}\,\, p=S_k \,\,{\rm for\,\,some\,\,} k \,\…
Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the numerical evidence lead to the conjecture that the primes are distributed among sequence $(S_n)$ in the same way that they are distributed among positive integers. More precisely, Conjecture 3.3 in [8] asserts that $π_n\sim \frac{n}{\log n}$ as $n\to \infty$, where $π_n$ denotes the number of primes in the set $\{S_1,S_2,\ldots, S_n\}$. Motivated by this, here we c…
In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes assuming the $abc$-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of free part of abelian group generated by all elements in $X$, and we will use the arithmetic point of view and geometric point of view to give heuristic.
A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples
提出取整连分数（FCT）框架构造Erdős-Straus猜想的三项埃及分数表示，利用移位整数p+i的除数结构而非同余技巧，推导出失败概率的超多项式上界，结合Borel-Cantelli引理启发式证明反例有限，并在10^17、10^52和10^131附近的10^9个素数上验证无反例。
This article discusses deviations from the Cohen-Lenstra heuristics when roots of unity are present. In particular, we propose an explanation for the discrepancy between the observed number of cyclic cubic fields whose 2-class group is $C_2\times C_2$ and the number predicted by the Cohen-Lenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute the invariant when the cubic field is ramified at exactly one prime, up to $10^8$.
We show that the $abc$ conjecture of Masser-Oesterlé-Szpiro for number fields implies that there are infinitely many non-Fibonacci-Wieferich primes. We also provide a new heuristic for the number of such primes beneath a certain value.
Lower bounds on weighted moments of primes in short intervals in number fields
本研究针对数域中短区间素数矩的Montgomery与Soundararajan猜想类似问题，通过调整整数情形下的已有研究方法，建立了加权素数矩的下界，且结果与猜想值相符。
We study the number of ways to decompose a monic polynomial in F_q[t] of degree n as a sum of two monic irreducible polynomials in F_q[t]. Our principal result is an asymptotic formula for the number of such representations in the case when q is large compared to n. In its range of validity, this formula agrees with what is suggested by heuristic arguments from the rational setting. We also present similar results towards an analogue of the twin prime conjecture.
Let g be a non-zero rational number. Let N_{g,t}(x) denote the number of primes p<=x for which the subgroup of the multiplicative group of the finite field having p elements that is generated by g mod p is of residual index t. In Part I, which is published in J. Number Theory 83 (2000), 155-181, a heuristic for N_{g,t}(x) was set up, under the assumption of the Generalized Riemann Hypothesis (GRH), and shown to be asymptotically exact. In this preprint we provide an alternative and rather shorter proof of this result.
Let $K$ be a number field and $r$ an integer. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree two prime ideals of $K$ with trace of Frobenius equal to $r$. Under certain restrictions on $K$, we show that "on average" the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.
For a Galois extension $K/\mathbb{F}_q(t)$ of Galois group $Γ$ with $\gcd(q,|Γ|)=1$, we define an invariant $ω_K$, and show that it determines the Weil pairing of the curve corresponding to $K$ and it descends to the prime-to-$|Γ|$-torsion part of the lifting invariants of Hurwitz schemes introduced by Ellenberg--Venkatesh--Westerland and Wood. By keeping track of the image of $ω_K$, we compute, as $K$ varies and $q\to \infty$, the average number of surjections from the Galois group of maximal unramified extension of $K$ to $H$, for any $Γ$-group $H$ whose order is prime to $q|Γ|$. Motivated b…
For degrees $3$ to $6$, we first give numerical results on polynomials which take many prime values on an interval of consecutive values of the variable. In particular, we have improved Ruby's record for the "$n$ out of $n$" case, for $n\ =\ 58$, by using a polynomial of degree $6$. In the theoretical part of this paper, we describe a heuristic probabilistic model in the "$n$ out of $n$" case: exactly $n$ (different) prime values on an interval of $n$ consecutive values of the variable. We find that the heuristic value of the probability of the event "$n$ out of $n$" for a generic polynomial i…
If the prime numbers are pseudo-randomly distributed, then analogy with quantum systems suggests that counting primes might be modeled by a non-homogeneous Poisson process. Consequently, postulating underlying gamma statistics, more-or-less standard heuristic arguments borrowed from quantum mechanics in the context of functional integration allows to derive analytic expressions of several average counting functions associated with prime numbers. The expressions are certain sums of incomplete gamma functions that are closely related to logarithmic-type integral functions --- which in turn are w…
Brizolis asked the question: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? The first author previously extended this question to ask about not only fixed points but also two-cycles, and gave heuristics (building on work of Zhang, Cobeli, Zaharescu, Campbell, and Pomerance) for estimating the number of such pairs given certain conditions on g and h. In this paper we extend these heuristics and prove results for some of them, building again on the aforementioned work. We also make some new conjectures and prove some average versions of …
Discrepancies in the distribution of Gaussian primes
本文通过分析无限多个L函数的解析性质，研究高斯素数分布中的不均性，并提出奇数平方的特殊性质。
Fibonacci primes, primes of the form $2^n-k$ and beyond
本文研究指数增长线性递推序列中的素数分布，通过改进多项式素数计数启发式提出素数项数量的渐近猜想，并与有限计算数据比较。
A dynamic sieve method is designed according to the basic sieve method. It mainly refers to the symbolic dynamics theory. By this method, we could connect the prime system with familiar 'Logistic Mapping'. An interesting discovery is that the pattern of primes could be depicted by a series of orbits of this mapping. Some heuristic proofs for open problems like twin primes are obtained through this relation. This research gives a new viewpoint for the distribution of prime numbers.
A heuristic for ray class groups of quadratic number fields
本文提出实二次域射线类群在固定模数下局部平均行为的启发式模型，引入Arakelov射线类群并假设其随机分布，推导出Cohen-Lenstra启发式及基本单位等分布结果，与Varma的3-挠子群平均结果一致。
Smooth Subsum Search: A heuristic for practical integer factorization
本文提出一种基于可被因子基素数整除的和来表示平滑性候选的整数分解启发式方法，其Python实现比最快的自初始化二次筛法快5至10倍。
Combining the Hardy-Littlewood k-tuple conjecture with a heuristic application of extreme-value statistics, we propose a family of estimator formulas for predicting maximal gaps between prime k-tuples. Computations show that the estimator a(log(x/a)-b) satisfactorily predicts the maximal gaps below x, where a is the expected average gap between the same type of k-tuples, a=O(log^k x). Heuristics suggest that maximal gaps between prime k-tuples near x are approximately a*log(x/a), and thus have the order O(log^{k+1}x). The distribution of maximal gaps around the trend curve a*log(x/a) is close …
On the Moments of the Number of Representations as Sums of Two Prime Squares
本文研究整数表为两个素数平方之和的表示数的各阶矩，利用 Selberg 筛法获得无条件上界，并在 Green-Tao 定理推广的条件下证明下界，且从第五阶矩起得到预期数量级。
Estimating the Number of Primes In Unusual Domains
本文研究非自然数域（如同余类、高斯整数及二次扩张）中素数的分布，提出启发式猜想并给出经验证据，同时探讨相关困难。
We propose a new model for counting $\mathbf{GL}_2(\mathbb{F}_p)$-number fields having the same local properties as the splitting field of the mod $p$-Galois representation associated with an elliptic curve over the rational numbers. We explain how this new model and Bhargava's local-to-global heuristics for counting $\mathbf{GL}_2(\mathbb{F}_p)$-number fields both shed light on the problem of estimating the number of elliptic curves over the rational numbers of squarefree conductor $N < X.$ The new model predicts the existence of significantly more $\mathbf{GL}_2(\mathbb{F}_p)$-number fields …
Let $L/K$ be a quadratic extension of global fields. We study Cohen-Lenstra heuristics for the $\ell$-part of the relative class group $G_{L/K} := \textrm{Cl}(L/K)$ when $K$ contains $\ell^n$th roots of unity. While the moments of a conjectural distribution in this case had previously been described, no method to calculate the distribution given the moments was known. We resolve this issue by introducing new invariants associated to the class group, $ψ_{L/K}$ and $ω_{L/K},$ and study the distribution of $(G_{L/K}, ψ_{L/K}, ω_{L/K})$ using a linear random matrix model. Using this linear model, …
For any prime p and any positive integer n, let B(p,n) denote the n-th layer of the cyclotomic Z_p-extension over the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular B(p,n), we provide evidence for the following conjecture: For all primes p and positive integers n, the ideal class group of B(p,n) is trivial.
We study the problem of writing Gaussian primes as the sum of two squares, both of which are interesting arithmetically, in particular, when one is the square of a prime and the other the square of an almost-prime.
We survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.
A classical problem in analytic number theory is to study the distribution of $αp$ modulo 1, where $α$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p+2$ is an almost-prime (the existence of infinitely many such $p$ is another topical result in prime number theory) and prove that its distribution has a similar property.
We study the distribution of primes from a topological viewpoint. Certain conjecture is introduced, and we show that it is equivalent to the Riemann Hypothesis.
This is an expanded account of three lectures on the distribution of prime numbers given at the Montreal NATO school on equidistribution.
The shifted prime-divisor function over shifted primes
本文研究了移位素数除数函数 $ω_a^*$ 在移位素数上的矩，证明了第一矩的渐近公式及第二矩的上下界，揭示了其在移位素数与自然数上平均行为的相似性，并提出了关于第二矩的新猜想。
We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like fixed probability random events. As the sequence of primes grows, the probability decreases as the reciprocal of the count of primes to that point. The manner of the decrease is consistent with the Hardy--Littlewood Conjecture, the Prime Number Theorem, and the Twin Prime Conjecture. Furthermore, our probabilistic model, is simply parameterized. We discuss a sim…
On the distribution of $αp^2+β$ modulo one for primes $p$ such that $p+2$ has no more two prime divisors
本文针对 α 无理时 αp²+β 模 1 的分布问题，考察 p+2 为殆素数的素数子序列，证明该子序列保持与经典素数序列相似的分布性质。
We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the ``high jumper'' i.e., the separation with greatest likelihood (in terms of primes) is always expected to be zero.
The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of finite order. In this paper, we give an inequality between the order of the zeta function as a meromorphic function and the growth of the multiplicity in the prime distribution.
For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $φ(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $π(x)/φ(q)^2$. We begin to rigorously c…
Distributions of left prime truncations
研究了整数和多项式左素截断的分布特性，分析了其在所有ℓ位数或多项式中的比例、方差及最大比例。
Density-based structural frameworks for prime numbers, prime gaps, and Euler products
本文提出基于密度的结构框架，用于分析素数、素数间隙和欧拉乘积，揭示了Hardy-Littlewood、Cramer和PNT预测之间的张力，并通过截断欧拉乘积分析ζ函数的临界线与零点分布。
For an elliptic curve E/Q without complex multiplication we study the distribution of Atkin and Elkies primes l, on average, over all good reductions of E modulo primes p. We show that, under the Generalised Riemann Hypothesis, for almost all primes p there are enough small Elkies primes l to ensure that the Schoof-Elkies-Atkin point-counting algorithm runs in (log p)^(4+o(1)) expected time.
Regularized products of Gauss and Eisenstein integers and primes
受Muñoz García和Pérez-Marco工作的启发，采用欧拉式启发式计算，得到了高斯整数和艾森斯坦整数及其素数的正则化无穷乘积的显式表达式。
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