The aim of this survey is to introduce Waring's problem and some related topics including mean value theorems and Diophantine equations in prime variables.
Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(...Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(1),p_(2),p_(3),p_(4),p_(5)are prime numbers.This expands the previous results.展开更多
For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders ...For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders of this arithmetic function.展开更多
Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9,...Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9, or b=c=7, then R b,c (n)》n 5/6+1/b+1/c for all sufficiently large n.展开更多
It is established that all even positive integers up to N but at most O（N15/16＋ε） exceptions can be expressed in the form p1^2＋ p2^3＋ p3^4＋ p4^5,where p1,p2,p3 and p4 are prime numbers.
基金Supported by the NKRDPC(Grant No.2021YFA1000701)。
文摘The aim of this survey is to introduce Waring's problem and some related topics including mean value theorems and Diophantine equations in prime variables.
基金the National Natural Science Foundation of China(Grant No.11761048)the Natural Science Foundation of Jiangxi Province for Distinguished Young Scholars(Grant No.20212ACB211007)Natural Science Foundation of Jiangxi Province(Grant No.20224BAB201001).
文摘Let k≥1 be an integer.Assume that RH holds.In this paper we prove that a suitable asymptotic formula for the average number of representations of integers n=p^(k)_(1)+p^(3)_(2)+p^(3)_(3)+p^(3)_(4)+p^(3)_(5),where p_(1),p_(2),p_(3),p_(4),p_(5)are prime numbers.This expands the previous results.
文摘For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders of this arithmetic function.
文摘Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9, or b=c=7, then R b,c (n)》n 5/6+1/b+1/c for all sufficiently large n.
基金Supported by National Natural Science Foundation of China(Grant No.11326205)
文摘It is established that all even positive integers up to N but at most O（N15/16＋ε） exceptions can be expressed in the form p1^2＋ p2^3＋ p3^4＋ p4^5,where p1,p2,p3 and p4 are prime numbers.
