summatory function of arithmetic function
It may be proved that the summatory function of a multiplicative function is multiplicative.
Proof. The first equality follows from the fact that any positive divisor of is got from where is a divisor of . Further, let where . Then and . This defines a bijection between the prime classes modulo and such values of in for which . The number of the latters . Furthermore, the only with and is , and , by definition. Summing then over all possible values yields the second equality.
|Title||summatory function of arithmetic function|
|Date of creation||2013-03-22 19:31:53|
|Last modified on||2013-03-22 19:31:53|
|Last modified by||pahio (2872)|