In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal ...In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal and A is a non-zero ideal.In particular,we discuss the Kluyver formula and Hèolder formula forη(a,A)andη(B,A).We also prove the reciprocity formula enjoyed byη(B,A)and the orthogonality relations forη(a,A)in the last two parts.展开更多
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
基金Supported by the National Research and Development Program of China(Grant No.2018YFB1107402)。
文摘In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal and A is a non-zero ideal.In particular,we discuss the Kluyver formula and Hèolder formula forη(a,A)andη(B,A).We also prove the reciprocity formula enjoyed byη(B,A)and the orthogonality relations forη(a,A)in the last two parts.
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.
