稀疏素数集中的Waring-Goldbach问题(Ⅱ)

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.     

An Application of Exponential Sum Estimates

Let p be an odd prime and let δ be a fixed real number with 0<δ<2. For an integer a with 0<a<p, denote by ā the unique integer between 0 and p satisfying a ā≡1 (mod p). Further, let {x} denote the fractional part of x. We derive an asymptotic formula for the number of pairs of integers (a, b with . This work is supported by N. S. F. of P. R. China (10271093)     

Prime in quadratic progressions on average

In this paper we study the distribution of primes in quadratic progressions on average. Our result improves a previous result of Baier and Zhao in some aspects.     

8-ranks of class groups of quadratic number fields and their densities

For \(F = \mathbb{Q}\left( {\sqrt {\varepsilon pq} } \right)\), ? ∈ {±1, ±2}, primes ?p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2 ⁿ , the quartic residue symbol \(\left( {\frac{p}{q}} \right)_4\) and binary quadratic forms such as q ^h(?2p)/4 = x ² + 2py ², where h(?2p) is the class number of \(\mathbb{Q}\left( {\sqrt { - 2p} } \right)\). The results are very useful for numerical computations.     

Selberg’s integral for Ramanujan automorphic representation

Let π _Δ be the automorphic representation of GL(2,ℚ_A) associated with Ramanujan modular form Δ and L(s, π _Δ) the global L-function attached to π _Δ. We study Selberg’s integral for the automorphic L-function L(s, π _Δ) under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.     

短区间的误差项的双线性形式

本文的目的在于改进 Iwaniec-Laborde 的关于短区间的误差项的双线性形式的估计.这类结果是用筛法和三角和方法研究短区间中殆素数 P_2的分布的关键.     

一个整除定理及其应用

众所周知,著名的费马小定理是:如果p是素数,那么对于任何整数a,都有P|(ap-a) 如果改动这个著名定理的条件,将p是素数放宽为p是奇数,会出现什么结论呢?这个结     

最高阶元素个数为 2p2 的有限群是可解群

本文讨论了最高阶元素个数 |M(G)|=2p2(p为素数) 的有限群, 证明了群G是可解群.     

与4/p = 1/(n1) + 1/(n2) + 1/(n3) 相关的素变量均值估计

如果n是正整数,我们用f（n）表示丢番图方程4/p=1/n_1＋1/n_2＋1/n_3的正整数解（n1,n2,n3）的个数.对于素数p,f（p）可以分解为f1（p）＋f2（p）,这里fi（p）（i=1,2）为分母n1,n2,n3中恰有i个能被p整除的解的个数.本文我们将研究关于均值∑p〈xfi（p）,i=1,2,的估计,其中p表示素数.     

梅森与梅森素数

数论问题中有许多关于素数的问题,在吸引人们去探索的同时又在磨砺着人类的智慧.许多素数问题的妙趣之处在于人们可以轻而易举地理解问题的表述,但是想要真正将问题解决,却需要坚强的意志、高超的技巧和艰苦的计算.如至今尚未完全解决的哥德巴赫猜想,历经几代数学家的苦苦求索直到1994年才得到求证的费尔马猜想(现在应该叫做费尔马大定理了),还有一个似乎不是那么著名的“梅森猜想”.提到“梅森猜想”,就要先从梅森其人谈起.梅森全名马林.梅森(Marin Mersenne,1588—1648),是法国圣弗朗西斯(St.Francis of Paola)所建的托钵僧团体中的修道…