黎曼猜想和哥德巴赫猜想有什么联系?
数学算法俱乐部
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·
2020-12-29 18:23
数学算法俱乐部
日期 : 2020年12月28日
正文共 :5347字
作者 : 郁林成森严格上讲黎曼猜想与哥德巴赫猜想并没有特别明显的联系(至少现在应该没有什么神奇的定理表明二者是等价的),不过在对哥德巴赫猜想的研究过程中黎曼猜想确实扮演了类似敲门砖的作用。先讲黎曼猜想(the Riemann Hypothesis):
一、黎曼函数
所谓的黎曼
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猜想一:在
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二、黎曼函数与素数分布
熟悉初等数论的人都知道欧拉(L.Euler)在1737年发表的一个著名公式
![ac7bc626acf1a3f964e0c01c24d3f68b.webp](https://filescdn.proginn.com/45f95f32d914e0d72b3b05fdc6781e76/ac7bc626acf1a3f964e0c01c24d3f68b.webp)
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三、素数定理
对素数规律的探求一直是数论领域的核心问题。对于
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四、广义黎曼假设(GRH)
即使研究黎曼猜想受阻,但依然拦不住数学家们想要高飞的心。所谓的广义黎曼猜想,就是黎曼猜想的2.0版本,不过其研究对象由黎曼
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五、研究进展
基本离证明还差得远呢……(好吧好吧,我承认是来凑字数的) 不过有好多有希望的想法,有复变函数论的(黎曼猜想多半是个复变函数问题),有解析数论的,有非对易几何的(代表人物法国大数学家孔涅,不过希望渺茫)还有量子力学的!!!(没错,确实有量子力学的,参见“希尔伯特—波利亚猜想”)但怎么有种有生之年看不到的感觉……
简单介绍一下孔涅的研究:(严格上说孔涅的证明思路是属于量子力学的,但他在研究过程中确实也用了非对易几何,具体效果如何恐怕不容乐观。)孔涅写出了一组方程,用其构造了一个量子力学体系,这个体系的本征值恰好对应着黎曼ζ函数在临界线上的非平凡零点,如果孔涅能证明出了对应本征值的零点外没有其他非平凡零点了,那也就相当于证明了黎曼猜想了,但就目前来看要做到这一点难比登天。
六、哥德巴赫猜想(Goldbach Problem)
在1742年给欧拉的一封信中,哥德巴赫提出了两个猜想,欧拉用稍微简练的语言改下后表述如下:(哥德巴赫猜想)每一偶数
![049276f35416c1412c77ee036f0af973.webp](https://filescdn.proginn.com/58b13411b5c79f27785a47ec2d950f56/049276f35416c1412c77ee036f0af973.webp)
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七、弱哥德巴赫猜想与GRH
数学家们首先向弱哥德巴赫猜想发起冲锋。第一次重大突破发生在20年代,哈代(Hardy)和李特尔伍德(Littlewood)在其“算术分拆”的系列文章中创立并发展了“圆法”即把方程
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八、哥德巴赫猜想与GRH
对哥德巴赫猜想的研究主要是围绕圆法进行的,以华罗庚为代表的中国解析数论学派在其中发挥着举足轻重的作用。筛法源于公元前250年的Eralosthenes筛法,Eralosthenes用该方法制作出了世上第一张素数表。1919年,布伦对传统筛法进行了大幅度改进,并首先将其应用于哥德巴赫猜想的研究,他证明了 每一个充分大的偶数都是两个素因子个数不超过9的整数之和,简记为“9+9” 我们可以类似定义
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九.黎曼猜想可能构成哥德巴赫猜想的证明吗?
我的感觉是不太可能,且容我缓缓道来…… 就目前在整个数学上的地位来讲(我从对数学的发展角度出发说一点不成熟的见解),哥德巴赫猜想肯定是无法与黎曼猜想匹敌。因为哥德巴赫猜想横竖就是个数论问题,再牛B也就是个数论问题,而且从目前来看它也并未对除堆垒数论以外的数论分支产生重大影响,在这一点上它连费马大定理(FLT)也比不过。而黎曼猜想则不同,其证明不但对数论领域有深刻影响,而且可以对复变函数论的发展起相当积极的推动作用(前面说过了,黎曼猜想多半是个复分析问题),也就是说黎曼猜想是数学界最重要的问题,而哥德巴赫猜想则更像是某个智力竞赛题。那么,到底黎曼猜想可能构成哥德巴赫猜想的某种证明吗?要回答这一问题,首先就要回顾一下哥德巴赫猜想的历史(翻前文):迄今为止,对哥德巴赫猜想的并未用到黎曼猜想,而是清一色用的是更厉害的广义黎曼猜想。原因很简单,黎曼猜想在这个问题上不够强!!另外,很有可能单从证明上讲黎曼猜想就要比哥德巴赫猜想难得多,更别提广义黎曼猜想。有可能若干年后,出了一位不世出的天才以不世出的方法证明了哥德巴赫猜想,但黎曼猜想仍然悬而未决。
最后说一句,哥德巴赫猜想跟孪生素数猜想有着极为深刻的联系,哥德巴赫猜想的相关结果一般而言是可以转换成孪生素数猜想的相关结果的,比如陈景润也曾证明过这样一个定理:存在无穷对素数
![bf7a5c550a30106bfb6774ec1f4f2edb.webp](https://filescdn.proginn.com/d727bd74a205ece5d1341278d7c5fb8c/bf7a5c550a30106bfb6774ec1f4f2edb.webp)
![ffa7145a73195effbaef9fa42bbb6966.webp](https://filescdn.proginn.com/af4fe9ff9219fadc17d128255398e1af/ffa7145a73195effbaef9fa42bbb6966.webp)
这跟他的1+2很像,也跟孪生素数猜想很接近。
— THE END —
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☞大学数学(高数线代)直观理解(一)☞读博期间强烈怀疑自己所在学科的价值是一种什么样的体验?☞罗杰·科恩伯格:基础科学——人类进步的希望☞太巧了!学霸夫妻携手进入武大读博,两人的硕士导师也是一对夫妻☞21岁MIT本科生推动数学重要问题新进展,曾获阿里数学竞赛奖
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