A. J. Irving
We consider almost-primes of the form $f(p)$ where $f$ is an irreducible polynomial over $\mathbb Z$ and $p$ runs over primes. We improve a result of Richert for polynomials of degree at least $3$. In particular we show that, when the degree is large, there are infinitely many primes $p$ for which $f(p)$ has at most $°f+O(\log°f)$ prime factors.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We adopt A. J. Irving's sieve method to study the almost-prime values produced by products of irreducible polynomials evaluated at prime arguments. This generalizes the previous results of Irving and Kao, who separately examined the almost-prime values of a single irreducible polynomial evaluated at prime arguments.
We improve Irving's method of the double-sieve by using the DHR sieve. By extending the upper and lower sieve functions into their respective non-elementary ranges, we are able to make improvements on the previous records on the number of prime factors of irreducible polynomials at prime arguments. In particular, we prove that irreducible quadratics over $\mathbb{Z}$ satisfying necessary local conditions are $P_4$ infinitely often.
We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However the underlying technique is different and allows us to obtain nontrivial results starting from much shorter ranges.
For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by $P^+(n)$ the greatest prime factor of $n$, it is also proved that $P^+(f(p))>p^{0.847}$ infinitely often.
In a recent joint work with D.A. Goldston and C.Y. Yildirim we just missed by a hairbreadth a proof that bounded gaps between primes occur infinitely often. In the present work it is shown that adding to the primes a much thinner set, called almost prime powers, the union of the set of primes and almost prime powers contains already infinitely many bounded gaps. More precisely it is shown that if we add to the set of primes either almost prime squares having exactly two, nearly equal prime factors or if we add to the set of primes almost prime cubes having exactly three, nearly equal prime fac…
By establishing an improved level of distribution we study almost primes of the form $f(p,n)$ where $f$ is an irreducible binary form over $\mathbb Z$.
In this paper we consider the equation $[p^{c}] + [m^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{29}{28}$, then it has a solution in a prime $p$ and an almost prime $m$ with at most $\left[ \frac{52}{29 - 28 c}\right] + 1 $ prime factors.
Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ is an irreducible polynomial of degree $d\ge 2$ then, $\log \operatorname{lcm} \{f(n)\mid n<x\} \sim (d-1)x\log x.$ In this article, we investigate the analogue of prime arguments, namely, $\operatorname{lcm} \{f(p)\mid p<x\}$ where $p$ denotes a prime and obtain non-trivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of $f(p).$
We study Gaussian primes lying in narrow sectors, and show that almost all such sectors contain the expected number of primes, if the sectors are not too narrow.
In this paper, we proved a theorem that every large enough odd number can be represented as the sum of three almost equal Piatetski-Shapiro primes.
In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.
We consider the function $G(n)=\frac{σ(n)}{n\log\log n}$ (where $σ(n)=\sum_{d|n}d$) and set an imposed condition on its argument $n$, the fulfillment of which is sufficient for the existence of a prime $p$, at which $G(np)>G(n)$. This inequality is of interest in connection with the Robin's inequality. The paper also presents the results of numerical experiment conducted with superabundant numbers.
Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes $p=[n^{1/γ}]$ with $\frac{85}{86}<γ<1$. Moreover, we use this result to prove that, for $0.9989445<γ<1$, there exist infinitely many Piatetski-Shapiro primes such that $p+2=\mathcal{P}_3$, which improves the previous results of Lu, Wang and Cai, and Peneva.
A positive integer is called an $E_j$-number if it is the product of $j$ distinct primes. We prove that there are infinitely many triples of $E_2$-numbers within a gap size of $32$ and infinitely many triples of $E_3$-numbers within a gap size of $15$. Assuming the Elliot-Halberstam conjecture for primes and $E_2$-numbers, we can improve these gaps to $12$ and $5$, respectively. We can obtain even smaller gaps for almost primes, almost prime powers, or integers having the same exponent pattern in the their prime factorizations. In particular, if $d(x)$ denotes the number of divisors of $x$, we…
We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the least prime or almost-prime primitive root (mod p) that holds uniformly for all p, analogous to Linnik's Theorem on a uniform upper bound for the least prime in a single arithmetic progression (mod p).
We investigate the problem of r almost-primes represented by sets of quadratic forms and give upper bounds for r. Our results extend work of Diamond and Halberstam in which they investigated the corresponding problem for polynomials.
We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.
An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an effective characterization of almost universal ternary quadratic polynomials with odd prime power conductor.
Zhang has shown there are infinitely many intervals of bounded length containing two primes. It appears that the current techniques cannot prove that there are infinitely many intervals of bounded length containing three primes, even if strong conjectures such as the Elliott-Halberstam conjecture are assumed. We show that there are infinitely many intervals of length at most $10^8$ which contain two primes and a number with at most 31 prime factors.
We show that as soon as $h\to \infty$ with $X \to \infty$, almost all intervals $(x-h\log X, x]$ with $x \in (X/2, X]$ contain a product of at most two primes. In the proof we use Richert's weighted sieve, with the arithmetic information eventually coming from results of Deshouillers and Iwaniec on averages of Kloosterman sums.
We improve Bombieri's asymptotic sieve to localise the variables. As a consequence, we prove, under a Elliott-Halberstam conjecture, that there exists an infinity of twins almost prime. Those are prime numbers $p$ such that for all $\varepsilon>0$, $p-2$ is either a prime number or can be written as $p_1 p_2$ where $p_1$ and $p_2$ are prime and $p_1<X^{\varepsilon}$, and we give the explicit asymptotic.
We prove that analogues of the Hardy-Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a product of exactly two primes which holds for almost all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$, under a restriction on the size of one of the prime factors of $n$ and $n+h$. Additionally, we consider correlations $n,n+h$ where $n$ is a prime and $n+h$ has exactly two prime factors, establishing an asymptotic formula which holds for …
We prove unconditionally that for each $\ell \geq 1$, the difference $ϕ(p-\ell) - ϕ(p+\ell)$ is positive for $50\%$ of odd primes $p$ and negative for $50\%$.
In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, the interval $[p_i, p_i + H]$ contains at least $m$ primes, for some $H = H(m)$. Second, every sufficiently large integer $N \equiv 3 \pmod{6}$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, $p_i + 2$ has at most two prime factors.
Goldston, Pintz and Yıldırım have shown that if the primes have `level of distribution' $θ$ for some $θ>1/2$ then there exists a constant $C(θ)$, such that there are infinitely many integers $n$ for which the interval $[n,n+C(θ)]$ contains two primes. We show under the same assumption that for any integer $k\ge 1$ there exists constants $D(θ,k)$ and $r(θ,k)$, such that there are infinitely many integers $n$ for which the interval $[n,n+D(θ,k)]$ contains two primes and $k$ almost-primes, with all of the almost-primes having at most $r(θ,k)$ prime factors. If $θ$ can be taken as large as $1-ε$, …
We show that there are infinitely many primes $p$ such that not only does $p + 2$ have at most two prime factors, but $p + 6$ also has a bounded number of prime divisors. This refines the well known result of Chen.
Gaussian almost primes in almost all narrow sectors
该研究在数论领域证明了复圆盘内不同面积的窄扇形中高斯殆素数的分布规律，论证依托Hecke特征和的相关分析工具，推进了高斯素数分布的研究。
Right-angled triangles with almost prime hypotenuse
本文研究整数边直角三角形中斜边为5-殆素数时的偶数直角边计数，证明其不同取值集合具有正确数量级的上下界，强化了Chow–Pomerance素数斜边猜想的类比结果。
Some explicit results on the sum of a prime and an almost prime
本文受 Rényi 经典结果启发，显式证明了每个不小于 4 的偶数可表示为一个素数与一个至多含 395 个素因子的殆素数之和，并在广义黎曼假设下将素因子数改进到 31。
Almost all primes are not needed in Ternary Goldbach
该研究针对三元哥德巴赫猜想，通过可容许同余扩张系统构造素数集$\u2119$，证明即便剔除几乎所有素数，仅用$\u2119$中的素数仍可满足该猜想，深化了对素数加法性质的认识。
Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the role of expansion coefficients. Founded on a known convergence acceleration for the ordinary prime zeta functions, the sums and first derivatives are tabulated with high precision for indices k=2,...,6 and integer powers s=2,...,8.
Jagy and Kaplansky exhibited a table of 68 pairs of positive definite binary quadratic forms that represent the same odd primes and conjectured that this list is complete outside of "trivial" pairs. In this article, we find all pairs of such forms that represent the same primes outside of a finite set.
Let $α>1$ be irrational and of finite type, $β\in\mathbb{R}$. In this paper, it is proved that for $R\geqslant13$ and any fixed $c\in(1,c_R)$, there exist infinitely many primes in the intersection of Beatty sequence $\mathcal{B}_{α,β}$ and $\lfloor n^c\rfloor$, where $c_R$ is an explicit constant depending on $R$ herein, $n$ is a natural number with at most $R$ prime factors, counted with multiplicity.
On Piatetski-Shapiro primes from almost primes
该数论研究将Piatetski-Shapiro素数相关结果中殆素数的素因子上限从8优化至7，针对0.98353到1之间的$\u03b3$，证明存在无穷多符合条件的素数。
Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least almost-prime $\mathcal{P}_2$ which satisfies $\mathcal{P}_2\equiv a\pmod q$. In this paper, it is proved that for sufficiently large $q$, there holds \begin{equation*} \mathcal{P}_2(a,q)\ll q^{1.8345}. \end{equation*} This result constitutes an improvement upon that of Iwaniec, who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$.
On an additive problem involving fractional powers with one prime and an almost prime variables
本文研究分数次幂整数部分的二元加法表示，证明在指数范围内充分大整数可表为素数与素因子个数有界的殆素数之和，并改进已有结果。
Let $p_n$ denote the $n^{th}$ prime. Goldston, Pintz, and Yildirim recently proved that $ \liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0.$ We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that $\liminf_{n\to \infty} (q_{n+1}-q_n) \le 26.$ If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.
Sums of four generalized polygonal numbers of almost prime length
研究四类受限质数因子数的广义多边形数之和，证明大数可表示为此类和，参数限制最多988个质因子。
In his paper Almost-Primes Represented by Quadratic Polynomials, Iwaniec proved that the polynomial n^2 + 1 takes on values with at most two prime factors (counted with multiplicity) infinitely often. He states that "in order to avoid technical complications, we shall restrict our proof to the polynomial n^2 + 1.". In this exposition, we follow Iwaniec's proof and show that for any irreducible quadratic polynomial G(n) (satisfying some obviously necessary hypotheses), G(n) has at most two prime factors for infinitely many values of n.
Primes in almost all short intervals
本文改进了Jia的结果，证明在几乎所有长度为$n^{1/21.5+\varepsilon}$的短区间内必含素数，通过Watt均值界、筛法分解和精确积分估计实现。
Almost primes and primes that are sums of two squares plus 1
该研究用向量筛框架（含半线性筛、线性筛）得出满足p-1为两平方和且p+2素因子个数有界的素数p≤x的下界。
Almost primes in almost all short intervals II
该数论研究改进了短区间殆素数分布结果，将几乎所有含双素数乘积的短区间长度指数从3.51降至2.1，核心是借助Heath-Brown稀疏Dirichlet多项式均值定理证明新II型估计。
Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.
We make an observation which doubles the exponent of distribution in certain Affine Sieve problems, such as those considered by Liu-Sarnak, Kontorovich, and Kontorovich-Oh. As a consequence, we decrease the known bounds on the saturation numbers in these problems.
A Formula For the Number of $k$-almost Primes
本文将计算小于给定值$x$的半素数个数的公式推广至$k$-殆素数和无平方因子$k$-殆素数，并提供了文献中缺失的正式证明。
Infinitude of palindromic almost-prime numbers
本文证明了在任意基数下，存在无穷多个至多含六个较大素因子的回文数，通过等分布估计和矩上界实现。
Let $F=x^2+y^2-z^2$, $x_0 \in \mathbb{Z}^3$ primitive with $F(x_0)=0$, and $Γ\leq SO_F(\mathbb{Z})$ be a finitely generated thin subgroup. We consider the resulting thin orbits of Pythagorean triples $x_0 \cdot Γ$ - specifically which hypotenuses, areas, and products of all three coordinates arise. We produce infinitely many $R$-almost primes in these three cases whenever $Γ$ has exponent $δ_Γ>δ_0(R)$ for explicit $R$, $δ_0$.
We prove that there are infinitely many solutions of $$ |λ_0+λ_1p+λ_2P_r|<p^{-τ}, $$ where $r=3,$ $τ=\frac1{118}$, and $λ_0$ is an arbitrary real number and $λ_1,λ_2\in\BR$ with $λ_2\neq0$ and $0>\frac{λ_1}{λ_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\floor{n^c}$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $τ=ρ(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$
We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are three main ingredients in the proof: a new transference principle like approach for sieves, a combination of the level of distribution estimates of Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce power savings, and a generalisation of the circle method arguments of …
On a multiplicative hybrid problem over almost-primes
本文研究殆素数上的乘法混合问题，证明在足够大的自然数N下，两个指定区间子集中的整数乘积可表示为殆素数平方加上一个较小误差项。
On almost primes in Piatetski-Shapiro sequences
本文研究 Piatetski-Shapiro 序列中的殆素数问题，证明在 γ 接近 1 时无穷多个相关取整值至多含 5 个素因子，改进了已有结果。
Let $\F$ be a family of $r$ integral forms of degree $k\geq 2$ and $\LL=(l_1,\ldots,l_m)$ be a family of pairwise linearly independent linear forms in $n$ variables $\x=(x_1,...,x_n)$. We study the number of solutions $\x\in[1,N]^n$ to the diophantine system $\F(\x)=\vv$ under the restriction that $l_i(\x)$ has a bounded number of prime factors for each $1\leq i\leq m$. We show that the system $\F$ have the expected number of such "almost prime" solutions under similar conditions as was established for existence of integer solutions by Birch.
We prove that the Kloosterman sum $S(1,1;c)$ can change sign infinitely often as $c$ runs over squarefree moduli with at most 10 prime factors, which improves the previous results of E. Fouvry and Ph. Michel, J. Sivak-Fischler and K. Matomäki, replacing 10 by 23, 18 and 15, respectively. The method combines the Selberg sieve, equidistribution of Kloosterman sums and spectral theory of automorphic forms.
Let $E_k$ be the set of positive integers having exactly $k$ prime factors. We show that almost all intervals $[x,x+\log^{1+\varepsilon} x]$ contain $E_3$ numbers, and almost all intervals $[x,x+\log^{3.51} x]$ contain $E_2$ numbers. By this we mean that there are only $o(X)$ integers $1\leq x\leq X$ for which the mentioned intervals do not contain such numbers. The result for $E_3$ numbers is optimal up to the $\varepsilon$ in the exponent. The theorem on $E_2$ numbers improves a result of Harman, which had the exponent $7+\varepsilon$ in place of $3.51$. We will also consider general $E_k$ n…
Almost primes of the form $[p^{1/γ}]$
本文研究素数幂次取整后的殆素数问题，证明当$0.989<γ<1$时有无穷多个素数$p$使$[p^{1/γ}]$至多含7个素因子，将已有结果的8个因子改进为7个。
The transference principle of Green and Tao enabled various authors to transfer Szemerédi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers, mostly sparse sets of primes. In this paper, we provide a transference principle which applies to general affine-linear configurations of finite complexity. We illustrate the broad applicability of our transference principle with the case of almost twin primes, by which we mean either Chen primes or "bounded gap primes", as well as with the case of primes of the form $x^2+y^2+1$. Thus, we show that in these sets of…
Extreme central $L$-values of almost prime quadratic twists of elliptic curves
该研究证明椭圆曲线殆素数二次扭曲的中心 $L$-函数极值，并在 BSD 猜想下推出其二次扭曲族中 Tate--Shafarevich 群的极值。
Primes and almost primes between cubes
研究立方数间素数的存在性，通过大规模计算证明n^3至(n+1)^3间存在素数，且利用筛法论证证明存在至多2个素因子的数，采用Richert的对数加权方法提升数值效果。
Let $k\ge 2$ and $Π(n)=\prod_{i=1}^k(a_in+b_i)$ for some integers $a_i, b_i$ ($1\le i\le k$). Suppose that $Π(n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that $Ω(Π(n))\le r_k$ holds for some integer $r_k$ which is asymptotic to $k\log{k}$. We use a new kind of weighted sieve to improve the possible values of $r_k$ when $k\ge 4$.
An Explicit Result for the Sum of Two Almost Primes
研究证明任意大于等于2的数可表示为两个正整数之和，其乘积的总素因子数不超过40，方法为显式下界Selberg筛法结合计算与优化。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。