【摘要】：數(shù)論中的指數(shù)和,Kloosterman和,Guass和,Ramanujan和等和式都有緊密的聯(lián)系.近年來(lái),很多學(xué)者深入的研究了這些問(wèn)題,并且獲得了很多優(yōu)秀的研究成果.本文運(yùn)用簡(jiǎn)化剩余系,三角和,Dirichlet特征的正交性,Ramanujan和等性質(zhì)以及學(xué)者研究所利用的方法,對(duì)己有的各種指數(shù)和進(jìn)行推廣.研究結(jié)果如下:1.對(duì)于正整數(shù)q(q≥ 3),m,n,kk,s,其中n滿足(n,q)= 1,研究了(?)的均值,并且得到了 一個(gè)精確的計(jì)算公式.2.設(shè)t為正整數(shù),k1,k2,…,kt及m1,m2,…,mt,n均為正整數(shù)且p為素?cái)?shù),設(shè)q2為整數(shù),且滿足(m2,q)=…=(mt,q)=(n,q)= 1.令L =k1 + k2+…+kt,多項(xiàng)式φ(x)=m1xk1+ m2xk2+…+mtxkt,研究了(?)的均值,并且分別給出了當(dāng)(k1,q)= 1及k1|k2,k1|k3,…,k1|kt,正整數(shù)n滿足(n,q)= 1,或(k1,qφ(q))=1,正整數(shù)n滿足(n,p)= 1時(shí)的恒等式.3.設(shè)p,m,k,l,α是正整數(shù)且p是素?cái)?shù),那么對(duì)正整數(shù)n且(n,p)= 1,研究了(?)并獲得了一個(gè)等式.
[Abstract]:In number theory, the sum of exponents, Kloosterman and, Guass and, Ramanujan and sum are closely related. In recent years, many scholars have studied these problems in depth, and obtained a lot of excellent research results. In this paper, by using the orthogonality, Ramanujan sum and other properties of simplified residue systems, trigonometric sums, Dirichlet features and the methods used by scholars, we generalize the existing exponential sums. The results are as follows: 1. For a positive integer q (q 鈮，

本文編號(hào)：2333748