【作者】 张蕊；

【导师】 刘志新；

【作者基本信息】 天津大学 ， 基础数学， 2020， 硕士

【摘要】 Waring-Goldbach（华林-哥德巴赫）问题作为堆垒素数论的一个重要问题备受数论学家的广泛关注.它主要研究满足必要同余条件的正整数n表示为素数方幂之和的可能性,即方程n=p₁^k+···+p_s^k的可解性,其中k是事先给定的正整数,p_i是素数（i=1,2,···,s）.对于满足必要同余条件的充分大的正整数n,设H（k）为使上述方程有素数解的最小s值.为了考虑变量个数小于H（k）的相关问题,通常的做法是考虑它的例外集E（N）,E（N）是指在满足必要同余条件的不大于N的正整数中不能表示为上述方程的正整数个数.作为经典华林-哥德巴赫问题的推广,数论学界开始关注一般形式的华林-哥德巴赫问题,即方幂次数不一定相同的华林-哥德巴赫问题.本文的主要工作是改进了不等次幂华林-哥德巴赫问题n=p₁²+p₂²+p₃³+p₄³+p₅⁴+p₆⁴的例外集结果.证明了除至多O(N^17/192+ε)个例外,所有不超过N的偶数n均可以表示为两个素数的平方,两个素数的立方和两个素数的四次方之和（即定理1.1）.本文的结果改进了张敏和李金蒋^[1]给出的O(N^13/16+ε),赵晓东^[2]给出的O(N^17/42+ε).本文首先利用例外集之间关系的最新结果,将问题转化为研究方程n=p₁²+p₂³+p₃⁴+p₄⁴的例外集,然后利用Hardy-Littlewood圆法研究转化后问题的例外集.由于对于主区间的研究已经非常成熟和完备,本文利用最先进的技术和结论去处理余区间,改进了例外集的结果.同时,利用相似的方法,本文改进了刘玉辉^[3]关于方程n=p₁²+p₂²+p₃³+p₄⁴+p₅⁴+p₆⁴的例外集结果,将O(N^61/144+ε)改进为O(N^13/96+ε).此外,利用定理1.1中处理余区间的方法,本文还研究了两个其他的一般型华林-哥德巴赫问题,改进了Br¨udern^[4]和李太玉^[5]关于几个素数的平方与一个素数的k次方之和的例外集结果,得到了不等次幂华林-哥德巴赫相关问题n=a₁p₁+a₂p₂²+a₃p₃³+a₄p₄⁴+a₅p₅⁵的Baker常数,并且简单叙述了证明方法.更多还原

【Abstract】 The Waring-Goldbach problem as an important part of additive prime number theory,has attracted the attention of many number theory scholars.It mainly researches the possibility of positive integers n satisfying necessary congruence conditions can be represented as the sum of powers of primes,that is the solvability of the equation n=p₁^k+···+p_s^k,where k is a positive integer given in advance,and p_iare primes（i=1,2,···,s）.For every sufficiently large positive integer n satisfying the necessary congruence conditions,let H（k）be the smallest s that make the above equation have prime solutions.In order to consider the related problems that the number of variables less than H（k）,we usually consider its exceptional set E（N）,E（N）denotes the number of positive integers satisfying the necessary congruence conditions not greater than N,which cannot be represented as the above equation.As a generalization of the classical Waring-Goldbach problem,the field of number theory began to pay more attention to the general Waring-Goldbach problem,that is the Waring-Goldbach problem with powers not necessarily the same.The main work of this paper is to improve the result of the exceptional set of a general Waring-Goldbach problem n=p₁²+p₂²+p₃³+p₄³+p₅⁴+p₆⁴.It is proved that with at most O(N^17/192+ε)exceptions,all even positive integers n up to N can be represented as sums of two squares,two cubes and two biquadrates of primes（Theorem 1.1）.The result of this paper improves the result O(N^13/16+ε)given by Zhang Min and Li Jinjiang^[1],and the result O(N^17/42+ε)given by Zhao Xiaodong^[2].In this paper,we first use the latest results of the relations between exceptional sets to transform the problem into researching the exceptional set of equation n=p₁²+p₂³+p₃⁴+p₄⁴,then we use Hardy-Littlewood circle method to study the exceptional set of transformed problem.Because the method of dealing with the major arcs has been very mature and standard,we use the most advanced technology and results to deal with the minor arcs to improve the result.Meanwhile,by using the similar method,we improve the result of exceptional set of the equation n=p₁²+p₂²+p₃³+p₄⁴+p₅⁴+p₆⁴due to Liu Yuhui^[3]by replacing O(N^61/144+ε)with O(N^13/96+ε).Furthermore,we also study two other general Waring-Goldbach problems by us-ing the similar method of dealing with minor arcs in Theorem 1.1,improving the results of the exceptional sets of sums of squares of primes and a k-th power of prime due to Br¨udern^[4]and Li Taiyu^[5],getting the Baker constant of the unlike powers Waring-Goldbach related problem n=a₁p₁+a₂p₂²+a₃p₃³+a₄p₄⁴+a₅p₅⁵,and giving a brief description of the proof methods.更多还原