Almost Uniform Distribution Modulo 1 and the Distribution of Primes

Let (an), n = 1, 2, ... be a sequence of real numbers which is related with number theoretic functions such as Pn, the n-th prime. We study the distribution of the fractional parts of (an) using the concept of "almost uniform distribution" defined in [9]. Then we can show a generalization of the results of [2] on the convex property of log Pn. The method may be extended as well to other oscillation problems of number theoretical interest.     

Different Approaches to the Distribution of Primes

In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zeta-function.     

On Atkin and Swinnerton-Dyer congruence relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms. The research of the second author was supported in part by an NSA grant #MDA904-03-1-0069 and an NSF grant #DMS-0457574. Part of the research was done when she was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan. She would like to thank the Center for its support and hospitality. The third author was supported in part by an NSF-AWM mentoring travel grant for women. She would further thank the Pennsylvania State University and the Institut des Hautes études Scientifiques for their hospitality.     

On Atkin and Swinnerton-Dyer congruence relations (3)

It is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of SL₂(Z). Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.In this paper, we study an infinite family of spaces of noncongruence cuspforms (which includes the cases in [W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148; A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358]) under a general setting. It is shown that for each space in this family there exists a fixed basis so that the Fourier coefficients of each basis element satisfy certain weaker three-term congruence recursions. For a new case in this family, we will exhibit that the attached l-adic Scholl representations are modular and the p-adic Hecke operators can be diagonalized semi-simultaneously.     

On Products of Shifted Primes

Any rational representable as a product of shifted primes p+1 or their reciprocals, has a representation with exactly 19 terms.     

表大偶数为两个Piatetski-Shapiro素数之和

In this paper, we study the binary Goldbach problem in the set of the Piatetski-Shapiro primes. We obtain that for all most all large even integer n, the equationn = p1 p2, pi∈Pγi, i = 1, 2has solutions, where 0 < γ1,γ2 ≤ 1 are fixed real numbers, such that 73(1 - γ2) < 9, 73(1 - γ1) 43(1 - γ2) < 9.     

Annihilator Primes and Foundation Primes

ABSTRACT

Let R be a Noetherian ring and M a finitely generated R -module. In this article, we introduce the set of prime ideals Fnd  M , the foundation primes of M . Using the fact that this set is nicely organized by foundation levels, we present an approach to the problem of understanding Annspec  M , the annihilator primes of M , via Fnd  M . We show: (1) Fnd  M is a finite set containing Annspec  M . Further, suppose that moreover every ideal of R has a centralizing sequence of generators; now, Annspec  M is equal to the set Ass  M of associated primes of M. Then: (2) For an arbitrary P  ∈ Fnd  M , P  ∈ Annspec  M if and only if there is no Q  ∈ Annspec  M such that P contains Q , and at the same time, the minimal foundation level on which appears P is greater than the minimal foundation level on which appears Q .     

关于双随机循环矩阵中的素元(英文)

非负矩阵中的素元分类问题在控制和系统论中有重要的应用.本文将研究由G.Picci等所提出的关于双随机循环矩阵中素元的一个问题和一个猜想,得到了一个判别具有位数5的n阶双随机循环矩阵不是素元的充要条件,给出了猜想成立的一些充分条件.     

On Primes and Powers of a Fixed Integer

According to a 1904 conjecture of Dickson [2] unless preventedby congruence conditions, any finite collection of linear formsin Z[x] with positive leading coefficients infinitely oftensimultaneously represent primes. For the forms x, x+2 this includesthe conjectured infinitude of prime-pairs.     

一个素变数丢番图不等式

本文证明了如果1相似文献   

On Some Nonlinear Diophantine Inequalities with Primes

Mathematical Notes -     

稀疏素数集中的Waring-Goldbach问题

本文研究了Ｗａｒｉｎｇ Ｇｏｌｄｂａｃｈ问题（ｋ＝２）与Ｐｉａｔｅｔｓｋｉ Ｓｈａｐｉｒｏ素数定理的混合问题，从而进一步深化了华罗庚教授的经典结果     

素变数线性三角和的估计

设实效满足（ａ，ｑ）＝１：｜θ｜≤１．Ｎ≥３是一个整数。记ｒ＝ｌｏｇＮ，ｅ（ａｎ）＝ｅ￣（２πｉａｎ），为Ｍａｎｇｏｌｄｔ函数。在本文中，我们证明了     

关于特殊形式素数$p$的$\alpha p^2$ 模$1$的分布

证明了如果$0<\theta < \frac {2}{375}$, 则对于无理数$\alpha$, 存在无限个素数$p$, 使得$p+2$不超过4个素因子, 并满足不等式$\|\alpha p^2+\beta\|相似文献   

On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues if m and n are restricted by arbitrary congruence conditions,providing that there is still no fixed prime divisor. Two immediate consequences for the solvability of diagonal cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).     

On the Upper Bound of the Number of Primes in Arithmetic Progression

Let a and q be relatively prime positive integers and \pi(x,q,a) stand for the number of primes p\leq x congruent to a and q. H. Iwanice proved that \pi(x;q,q)<\frac{(2+\varepsilon)x}{\phi(q)log D} for any \varepsilon>0, x>x_0 (\varepsilon) and q\leq x^9/20-\varepsilon , where D=xq^-3/8. The author applies an improved estimation of the error term in the linear sieve, proves that for any \varepsilon>0, x>x_0 (\varepsilon) and q\leq x^5/11-\varepsilon , (1) is true.     

On the Primes of Simple Imaginary Quadratic Fields and Real Euclidean Quadtic Fields

As is well known,an Euclidean fiels is a simple field.As regards the quadratic field Q(√D),where Q is the ring of rational integers and D is a square-free rational integer,we have know that an imaginary quadratic field Q(√D) is simple iff D∈｛-1,-2,-3,-7,-11,-19,-43,-67,-163｝,and a real quadratic field is Euclidean iff D∈｛2,3,5,6,,7,11,13,17,19,21,29,33,37,41,57,73｝.This paper will discuss the primes of Q(√D) when D belongs to the set QD=｛-1,-2,-3,-7,-11,-19,-43,-67,-163,2,3,5,6,,7,11,13,17,19,21,29,33,37,41,57,73｝.     

On the Greatest Prime Divisor of the Sum of Two Squares of Primes

One of the most famous theorems in number theory states thatthere are infinitely many positive prime numbers (namely p =2 and the primes p 1 mod4) that can be represented in the formx²₁+x²₂, where x₁ and x₂ are positive integers. In a recentpaper, Fouvry and Iwaniec [2] have shown that this statementremains valid even if one of the variables, say x₂, is restrictedto prime values only. In the sequel, the letter p, possiblywith an index, is reserved to denote a positive prime number.As p²₁=p²₂ = p is even for p₁, p₂ > 2, it is reasonable toconjecture that the equation p²₁=p²₂ = 2p has an infinity ofsolutions. However, a proof of this statement currently seemsfar beyond reach. As an intermediate step in this direction,one may quantify the problem by asking what can be said aboutlower bounds for the greatest prime divisor, say P(N), of thenumbers p²₁=p²₂, where p₁, p₂ N, as a function of the realparameter N 1. The well-known Chebychev–Hooley methodcombined with the Barban–Davenport–Halberstam theoremalmost immediately leads to the bound P(N) N^1–, if N N_o(); here, denotes some arbitrarily small fixed positivereal number. The first estimate going beyond the exponent 1has been achieved recently by Dartyge [1, Théorème1], who showed that P(N) N^10/9–. Note that Dartyge'sproof provides the more general result that for any irreduciblebinary form f of degree d 2 with integer coefficients the greatestprime divisor of the numbers |f(p₁, p₂)|, p₁, p₂ N, exceedsN^_d–, where _d = 2 – 8/(d = 7). We in particular wantto point out that Dartyge does not make use of the specificfeatures provided by the form x²₁+x²₂. By taking advantage ofsome special properties of this binary form, we are able toimprove upon the exponent ₂ = 10/9 considerably.