Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N^2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q^(1/4+...Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N^2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q^(1/4+ε/2)N^(2/1)(log(6N))^(1/2)+N/(1/2)(loglog(6N)),where n is the multiplicative inverse of n such that nn ≡ 1(mod q),e(x)= exp(2πix),and τ(·)is the divisor function.展开更多
If n is a positive integer,let f (n) denote the number of positive integer solutions (n 1,n 2,n 3) of the Diophantine equation 4/n=1/n_1 + 1/n_2 + 1/n_3.For the prime number p,f (p) can be split into f 1 (p) + f 2 (p)...If n is a positive integer,let f (n) denote the number of positive integer solutions (n 1,n 2,n 3) of the Diophantine equation 4/n=1/n_1 + 1/n_2 + 1/n_3.For the prime number p,f (p) can be split into f 1 (p) + f 2 (p),where f i (p) (i=1,2) counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p<x f i (p),i=1,2,where p denotes the prime number.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11201117 and 11126150),supported by National Natural Science Foundation of China(Grant Nos.11371344 and 11321101)National Key Basic Research Program of China(Grant No.2013CB834202)
文摘Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N^2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q^(1/4+ε/2)N^(2/1)(log(6N))^(1/2)+N/(1/2)(loglog(6N)),where n is the multiplicative inverse of n such that nn ≡ 1(mod q),e(x)= exp(2πix),and τ(·)is the divisor function.
基金supported by National Natural Science Foundation of China (Grant No.11071235)
文摘If n is a positive integer,let f (n) denote the number of positive integer solutions (n 1,n 2,n 3) of the Diophantine equation 4/n=1/n_1 + 1/n_2 + 1/n_3.For the prime number p,f (p) can be split into f 1 (p) + f 2 (p),where f i (p) (i=1,2) counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p<x f i (p),i=1,2,where p denotes the prime number.
