一类数论函数的新进展

Further Research Progress on a Class of Arithmetic Functions

进一步研究了满足条件f(u^2+kv^2)=f^2(u)+kf2(v)的数论函数f(n),证明了其在k=6的情形下可分成3类,进而验证了关于f(n)的猜想在k=6时是正确的.进一步地,总结了研究过程中出现的一些有趣结果,指出了该数论函数在参数k的不同取值下,其证明过程中出现的一些联系与区别,旨在为猜想的完全证明提供一些可能的理论支撑.

The arithmetic function f（n） which satisfies f（u^2＋kv^2） = f^2（u） ＋k f^2（v） is further studied. It is proved that f（n） can be divided into three classes with k = 6,and thus the conjecture of f（n） in the case of k = 6 is verified. What＇s more,some interesting results are presented. Some relations and distinction in the process of the proof with different values of parameter k are pointed out,which aims at providing some possible theoretical support for the complete proof of the conjecture.