Search

In the present work we use maximum entropy methods to derive several theorems in probabilistic number theory, including a version of the Hardy-Ramanujan Theorem. We also provide a theoretical argument explaining the experimental observations of Yang-Hui He about the learnability of primes, and posit that the Erd\H{o}s-Kac law would very unlikely be discovered by current machine learning techniques. Numerical experiments that we perform corroborate our theoretical findings., Comment: 10 pages; parts of arXiv:2308.10817 reworked and amended; author's draft; accepted in PLOS ONE

Suppose that $y>0$, $0\leq\alpha<2\pi$ and $0K$ and $P^-$ the set of primes $p$ such that $\cos(y\ln p+\alpha)<-K$ . In this paper we prove $\sum_{p\in P^+}\frac{1}{p}=\infty$ and $\sum_{p\in P^-}\frac{1}{p}=\infty$., Comment: 5 pages. v2 is a generalization of v1

We have shown previously that at each stage of Eratosthenes sieve there is a corresponding cycle of gaps $\mathcal{G}(p_0^\#)$. We can view these cycles of gaps as a discrete dynamic system, and from this system we can obtain exact models for the populations and relative populations of gaps $g < 2p_1$ if we can get the initial conditions from $\mathcal{G}(p_0^\#)$. In this addendum we have shown that we can produce the model for $g=2p_1$ from these initial conditions. This model requires one special iteration to track the count from $\mathcal{G}(p_0^\#)$ to $\mathcal{G}(p_1^\#)$, after which we can use the general model for these populations. As a specific example we exhibit the model for the gap $g=82$ using $\mathcal{G}(37^\#)$ for initial conditions. We show further that in order to produce the models for $g=2p_1+2$ and beyond from initial conditions in $\mathcal{G}(p_0^\#)$, we would have to track subpopulations of the driving terms until the general model applies, that is until $g < 2p_{k+1}$. This work serves as an addendum to the existing references "Patterns among the Primes" and "Combinatorics of the gaps between primes". We do not duplicate that background here, beyond summarizing a few needed results., Comment: 16 pages, 1 figure

Sums of the singular series constants that appear in the Hardy--Littlewood $k$-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo $r$ or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo $r$ of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats., Comment: 19 pages

The paper considers the asymptotic of the ratio of the number of primes not exceeding the primorial and the number of residues in the reduced system of residues for the given primorial. We study the relationship between asymptotic lower bounds for the values of the Jacobstal function and the maximum distance between successive primes. One algorithm for computing the Jacobstal function is given. The substantiation of the conjectures about the upper estimate of the maximum distance between successive prime numbers is given., Comment: 11 pages

We give a brief account of some of the most spectacular results established by James Maynard for which he has been awarded the Fields Medal., Comment: Fields Medal Laudatio

The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in this subset is constant and not equal to zero., Comment: 20 pages

In this article we show that the Erd\H{o}s-Kac theorem, which informally states that the number of prime divisors of very large integers converges to a normal distribution, has an elegant proof via Algorithmic Information Theory., Comment: There is an error in this paper that is addressed in arXiv:2308.10817

Building upon the work of Chebyshev, Shannon and Kontoyiannis, it may be demonstrated that Chebyshev's asymptotic result: \begin{equation} \ln N \sim \sum_{p \leq N} \frac{1}{p} \cdot \ln p \end{equation} has a natural information-theoretic interpretation as the information-content of a typical integer $Z \sim U([1,N])$, sampled under the maximum entropy distribution, is asymptotically equivalent to the information content of all primes $p \leq N$ weighted by their typical frequency. This result is established by showing that the correct information-theoretic interpretation of the entropy $\ln p$ may be deduced from the author's recent observation that the prime numbers have a uniform source distribution. Moreover, using Chaitin's theorem that almost all integers are algorithmically random we may extend this asymptotic result to the setting of algorithmic information theory and derive an upper-bound on the algorithmic information of a typical integer., Comment: The correct proof of Chebyshev's theorem is in: arXiv:2308.10817

Within the setting of rare event modelling, the method of level sets allows us to define an equivalence relation over rare events with distinct rates of entropy production. This method allows us to clarify the relation between the empirical density of primes and their source distribution which then allows us to address Cram\'er's conjecture, an open problem in probabilistic number theory. As a natural consequence, this analysis places strong epistemic limits on the application of machine learning to analyse the distribution of primes., Comment: The information-theoretic analysis used in this paper isn't strong enough to prove Cram\'er's Conjecture

We show that the assumption of a weak form of the Hardy-Littlewood conjecture on the Goldbach problem suffices to disprove the possible existence of exceptional zeros of Dirichlet L-functions. This strengthens a result of the authors named in the title., Comment: 8 pages

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vall\'ee Poussin's form: $$ \pi(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ 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 $\pi(x)$ by $\mathcal O(\sqrt x/\log x)$.

We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if $g$ mod $p$ generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present a conjecture for the asymptotic number $\pi_{d,g}(x)$ of primes $p \leq x$ such that both $p$ and $p+d$ are Artin primes for $g$. In particular, we identify a class of pairs $(d,g)$ for which $\pi_{d,g}(x) =0$. Our results suggest that the distribution of Artin prime pairs, amongst the ordinary prime pairs, is largely governed by a Poisson binomial distribution., Comment: 29 pages; Revised version. Appendix with relevant Mathematica code included. Accepted for publication in the Journal of Number Theory

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., Comment: 10 pages, 3 figures

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+\delta}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli., Comment: 79 Pages

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+\delta}$ with a 'convenient sized' factor. As a consequence, the expected asymptotic holds for all but $O(\delta Q)$ moduli $q\sim Q=x^{1/2+\delta}$ and we get results for moduli as large as $x^{11/21}$. Our proof extends previous techniques of Bombieri, Fouvry, Friedlander and Iwaniec by incorporating new ideas inspired by amplification methods. We combine these with techniques of Zhang and Polymath tailored to our application. In particular, we ultimately rely on exponential sum bounds coming from the spectral theory of automorphic forms (the Kuznetsov trace formula) or from algebraic geometry (Weil and Deligne style estimates)., Comment: 102 Pages; Corrected statement of Lemma 18.1 (main results unchanged)

The paper substantiates the conjecture of the asymptotic behavior of the largest distance between consecutive primes: $sup_{p_i \leq x}(p_{i+1}-p_i) \sim 2e^{-\gamma} \log^2(x)$, where $\gamma$ is the Euler constant. The Hardy-Littlewood conjecture about the number of prime tuples is investigated and the rationale for this conjecture is given, taking into account the fact that a large natural number is not divisible by primes. It also substantiates why the accuracy of this conjecture is not affected by another assumption about the probability of a natural number being prime, although such a probability does not exist. The paper also considers the distribution of prime tuples using a mathematical model based on the Hardy-Littlewood conjecture., Comment: 15 pages

The following is proven using arguments that do not revolve around the Riemann Hypothesis or Sieve Theory. If $p_n$ is the $n^{\rm th}$ prime and $g_n=p_{n+1}-p_n$, then $g_n=O({p_n}^{2/3})$., Comment: 6 pages

Let $\sigma+i\gamma$ be a zero of the Riemann zeta function to the right of the line $\frac{1}{2}+it$. We show that this zero causes large oscillations of the error term of the prime number theorem. Our result is close to optimal both in terms of the magnitude and in the localization of large values for the error term.

We discuss various recent advances on weak forms of the Twin Prime Conjecture., Comment: 29 Pages; old survey article from Takagi Lectures. arXiv admin note: text overlap with arXiv:1910.13450

We discuss recent advances on weak forms of the Prime $k$-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps between primes., Comment: 16 Pages; old survey paper for the ICM

We survey some different results on the digits of prime numbers, giving a simplified proof of weak forms of a result of Maynard and Mauduit-Rivat., Comment: 18 pages; Older survey article for European Congress of Mathematics

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer $k>1$ is the sum of two figurate primes. In this paper we give an equivalent proposition to the conjecture. By considering extreme value problems with constraints about the conjecture in the cases of odd and even integers and using the method of Lagrange multipliers, Cardano formula for cubic equations and the contradiction, we prove the conjecture., Comment: 14pages

We estimate the asymptotic density of the set $\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\bar{B} \subset \bar{A}$ such that for $p_1, p_2 \in \bar{B}$ no common prime divisor of $p_1(p_1 + 1)(p_1 - 1)$ and $p_2 (p_2 + 1)(p_2 - 1)$ is larger than $3$. Assuming a generalized Hardy--Littlewood conjecture, we prove that for both $\bar{A}$ and $\bar{B}$ the number of elements lesser than $x$ is asymptotically equal to a constant times $ x / (\log x)^3$., Comment: update references

The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the assumption to a more modest statement called the Parity conjecture. Let $p_n$ and $p_{n+1}$ be two consecutive odd primes, let $m$ be their midpoint fixed once for all. Conjecture: The largest multiple of $p_n$ not exceeding ${m_i}^2$ is odd for every integer $m_i$ in the interval $(p_n, m]$. Main result: We prove that the Parity conjecture implies Legendre's conjecture and Andrica's conjecture., Comment: In v3, we have reduced a difficult problem, that is the difference between two consecutive primes, to a much simpler problem where the focus is not in the gap or location of these primes, but simply in whether some particular numbers are even or odd

Computer experiments reveal that twin primes tend to center on nonsquarefree multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of nonsquarefree multiples of 6 to the number of squarefree multiples of 6 equal $\pi^2/3-1$, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427, a relative difference of ca $6.0\%$ measured against the expected value. A deviation from the expected value of this ratio, ca $1.9\%$, exists also for isolated primes. This shows that the distribution of primes is biased towards nonsquarefree numbers, a phenomenon most likely previously unknown. For twins, this leads to nonsquarefree numbers gaining an excess of $1.2\%$ of the total number of twins. In the case of isolated primes, this excess for nonsquarefree numbers amounts to $0.4\%$ of the total number of such primes. The above numbers are for the first $10^{10}$ primes, with the bias showing a tendency to grow, at least for isolated primes., Comment: 7 pages