Denote by ω(n) and Ω(n) the number of distinct prime factors of n and the total number of prime factors of n,respectively.For any positive integer ι,we prove that ∑↑2≤n≤x1/ω（n)=ι↑∑↑κ=0(ι↑∑↑i=κ（-1)...Denote by ω(n) and Ω(n) the number of distinct prime factors of n and the total number of prime factors of n,respectively.For any positive integer ι,we prove that ∑↑2≤n≤x1/ω（n)=ι↑∑↑κ=0(ι↑∑↑i=κ（-1)^i-κCi^κF^(i-κ)(1)κ！x/（loglogx）^i+1+O(x/(loglogx)^ι+2) ∑↑2≤n≤xΩ（n)/ω（n)=x+ι↑∑↑κ=0ι↑∑↑i=κ∑↑p1/p^κ+2-p^κ+1（-1)^i-κCi^κF^(i-κ）（1）κ!)x/(loglogx)^i+1+O(x/(loglogx)ι+2） where F(z)=1/г(z)pⅡ（1+z/p-1)(1-1/p)^z,and the constant O despends on ι.This improves previous result of R.L.Duncan and Chao Huizhong.展开更多
Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|x...Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.展开更多
In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 ...In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 and any 0【ε【14,E k(x+H)-E k(x)≤H(log x) -Aholds for x 12-14k+ε≤H≤x, here the implies constant depends at most on A and ε.展开更多
It is shown that the Ramsey number r（K2,s＋1,K1,n）≤n＋√sn＋（s＋3）/2＋o（1）for large n, and r（K2,s＋1, K1,n） ∈{（q-1）^2/s ＋ 1,（q-1）^2/s＋2},where n =（q-1）^2/s -q＋2 and q is a prime power such that s｜...It is shown that the Ramsey number r（K2,s＋1,K1,n）≤n＋√sn＋（s＋3）/2＋o（1）for large n, and r（K2,s＋1, K1,n） ∈{（q-1）^2/s ＋ 1,（q-1）^2/s＋2},where n =（q-1）^2/s -q＋2 and q is a prime power such that s｜（q - 1）.展开更多
文摘Denote by ω(n) and Ω(n) the number of distinct prime factors of n and the total number of prime factors of n,respectively.For any positive integer ι,we prove that ∑↑2≤n≤x1/ω（n)=ι↑∑↑κ=0(ι↑∑↑i=κ（-1)^i-κCi^κF^(i-κ)(1)κ！x/（loglogx）^i+1+O(x/(loglogx)^ι+2) ∑↑2≤n≤xΩ（n)/ω（n)=x+ι↑∑↑κ=0ι↑∑↑i=κ∑↑p1/p^κ+2-p^κ+1（-1)^i-κCi^κF^(i-κ）（1）κ!)x/(loglogx)^i+1+O(x/(loglogx)ι+2） where F(z)=1/г(z)pⅡ（1+z/p-1)(1-1/p)^z,and the constant O despends on ι.This improves previous result of R.L.Duncan and Chao Huizhong.
文摘Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.
文摘In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 and any 0【ε【14,E k(x+H)-E k(x)≤H(log x) -Aholds for x 12-14k+ε≤H≤x, here the implies constant depends at most on A and ε.
文摘It is shown that the Ramsey number r（K2,s＋1,K1,n）≤n＋√sn＋（s＋3）/2＋o（1）for large n, and r（K2,s＋1, K1,n） ∈{（q-1）^2/s ＋ 1,（q-1）^2/s＋2},where n =（q-1）^2/s -q＋2 and q is a prime power such that s｜（q - 1）.
