Abstract: During the last decade, there have been a great deal of researches on Cauchy-like functional equation on positive integer. As we know, the arithmetic function is widely applied in the field of artificial intelligence, cryptography and engineering. This paper mainly discusses the characterization of a class of arithmetic functions. Firstly, we give a characterization under some condition, and then propose a conjecture and show that the result holds for several simple cases. Finally, we summarize the process of argument and give some valuable conclusions which have been found during the proof process.

Key words: arithmetic function, functional equation, characterization