We obtain upper bounds for mixed exponential sums of the type $S(\chi ,f,p^m ) = \sum\nolimits_{x = 1}^{p^n } {\chi (x)e} _{p^m } (ax^n + bx)$ where pm is a prime power with m? 2 and X is a multiplicative character (m...We obtain upper bounds for mixed exponential sums of the type $S(\chi ,f,p^m ) = \sum\nolimits_{x = 1}^{p^n } {\chi (x)e} _{p^m } (ax^n + bx)$ where pm is a prime power with m? 2 and X is a multiplicative character (mod pm). If X is primitive or p?(a, b) then we obtain |S(χ,f,p m)| ?2np 2/3 m . If X is of conductor p and p?( a, b) then we get the stronger bound |S(χ,f,p m)|?np m/2.展开更多
We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive A-roots in any interval I of length ｜I｜ 〉〉 q^1/2＋ε Similar resu...We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive A-roots in any interval I of length ｜I｜ 〉〉 q^1/2＋ε Similar results are obtained for the distribution of ordered pairs （x, x^-1） with x a primitive λ-root, and for the number of primitive A-roots satisfying inequalities such as ｜x - x^-1｜≤ B.展开更多
基金Tsinghua University and the NNSF of China for supporting his visit to China during the Fall of 2000This work was supported by the National Natural Science Foundation of China (Grant No. 19625102).
文摘We obtain upper bounds for mixed exponential sums of the type $S(\chi ,f,p^m ) = \sum\nolimits_{x = 1}^{p^n } {\chi (x)e} _{p^m } (ax^n + bx)$ where pm is a prime power with m? 2 and X is a multiplicative character (mod pm). If X is primitive or p?(a, b) then we obtain |S(χ,f,p m)| ?2np 2/3 m . If X is of conductor p and p?( a, b) then we get the stronger bound |S(χ,f,p m)|?np m/2.
基金Project supported by the National Natural Science Foundation of China (No.19625102)the 973 Project of the Ministry of Science and Technology of China.
文摘We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive A-roots in any interval I of length ｜I｜ 〉〉 q^1/2＋ε Similar results are obtained for the distribution of ordered pairs （x, x^-1） with x a primitive λ-root, and for the number of primitive A-roots satisfying inequalities such as ｜x - x^-1｜≤ B.
