Arithmetical Functions, Code Theory and Distribution properties of the Residual Order

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From a code C, we can define the “sum of digits function” f_C(n), usually f_C(n) fluctuates quite irregularly. In this research, we proved “a one-to-one correspondence between arithmetical functions and the sum of digits function of code C’s”. This result enables us to control complex sum-of-digits functions by simple arithmetical functions. As an application, we can obtain the Delange-type average of sum-of-digits function of Gray code. We alsoconsider relations between sum-of-digits function and Automaton etc.