This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erdős-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper arXiv:0803.2636, the authors showed that there are infinitely many positive integers $x$ such that both $x$ and $x+1$ have exponent pattern $\{2,1,1,1\}$, so $d(x)=d(x+1)=24$. Similar results were known for certain shifts $n$, i.e., $x$ and $x+n$ have the same fixed exponent pattern infinitely often. This was done for shifts $n$ which are either even or not divisible by the product of a pair of twin primes. The goal of this paper is to give simple proofs of results on exponent patterns for an arbitrary shift $n$.
We derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$π(x)$ --- the number of primes up to: $M_k(x)= Γ(k+1)x^k/π^{k-1}(x)+\mathcal{O}(x)$, We illustrate obtained results by computer data.
We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $π(x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\sim \frac{π^2(x)}{x-2π(x)}\log\Big(\frac{x}{2π(x)}\Big) \sim x \log \log(x)/\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\times 10^{18}$.
Let $α, β\geq 0$ and $α+ β< 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^β(p_{n+1}^α- p_n^α) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of Sándor and gives additional evidence towards a conjecture of Andrica. This follows directly from recent results on prime pairs from Maynard, Tao, Zhang.
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $π(x)$. As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than $x$, for the generalized Brun's constants and the first occurrence of a given gap $d$. Also the leading term $\log \log(x)$ in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.
Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.
Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many values of $n$. As it is shown, the key ingredient to obtain this behaviour is a consequence of an extension of the Kummer's characterization of convergent series of positive terms.
Jumping champions and gaps between consecutive primes
小于或等于$x$的连续素数之间最常见的差异称为跳跃冠军。偶尔也会有联系。因此，对于给定的$x$，可以有多个跳跃冠军。1999年A.奥德利兹科湾Rubinstein和M.沃尔夫提供了启发式和经验证据来支持这样一个猜想，即大于大于1的跳跃冠军是4，素数是2、6、30、210、2310.作为证明这一猜想的一步，他们引入了第二个较弱的猜想，即任何固定的素数$p$都可以整除所有足够大的跳跃冠军。本文推广了P.Erdens和E. G. Straus于1980年提出了第二个猜想，证明了第二个猜想是由G. H. Hardy和J.E.利特尔伍德
This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erdős-Woods conjecture for each k>2.
Recently Oliver and Soundararajan made conjectures based on computational enumerations about the frequency of occurrence of pairs of last digits for consecutive primes. By studying Eratosthenes sieve, we have identified discrete dynamic systems that exactly model the populations of gaps across stages of Eratosthenes sieve. Our models provide some insight into the observed biases in the occurrences of last digits in consecutive primes, and the models suggest that the biases will ultimately be reversed for large enough primes. The exact model for populations of gaps across stages of Eratosthenes…
We show that large gaps between smooth numbers are infrequent. The key new tool is a novel mean value bound for a special type of Dirichlet polynomial.
With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki-Radziwill theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of its long average. As part of this treatment, we use a variant of the Matomäki-Radziwill theorem for divisor-bounded multiplicative functions recently proven by the author. We consider two sets of applications of these methods. Our first application shows that for an additive function $g: \mathbb{N} \rightarrow \mathbb{C}$ any non-trivial savings in the size of …
A positive integer $n$ is called an abundant number if $σ(n)\ge 2n$, where $σ(n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H os proved that there are two positive constants $c_1$ and $c_2$ such that $c_1\log\log\log x\le E(x)\le c_2\log\log\log x$. In this paper, we resolve this old problem by proving that, $E(x)/\log \log\log x$ tends to a limit as $x\to +\infty$, and the limit value has an explicit form which is between $3$ and $4$.
Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$ for some small constant $c_m>0$. Furthermore, we also obtain a related result concerning the least primes in arithmetic progressions.
Let $n$ be a non-negative integer and put $p_{n}(x)=\prod_{i=0}^{n}(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$ y^m=p_{n}(x)+\sum_{i=1}^{k}p_{a_{i}}(x), $$ where $m\in\N_{\geq 2}$ and $a_{1}<a_{2}<\ldots <a_{k}<n$. This equation can be considered as a generalization of the Erdős-Selfridge Diophantine equation $y^m=p_{n}(x)$. We present some general finiteness results concerning the integer solutions of the above equation. In particular, if $n\geq 2$ with $a_{1}\geq 2$, then our equation has only finitely many …
Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.
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.
We study the value-distribution of Dirichlet polynomials on the critical line $\Re(s)=\tfrac{1}{2}$. As a consequence, we prove a corollary on small consecutive gaps between zeros of the Riemann zeta function. We also examine the distribution of zeros under the so-called alternative hypothesis and present a new approach to the problem of gaps between the zeros.
We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist).
