Arithmetic functions | Mathematical series

Bell series

In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function and a prime , define the formal power series , called the Bell series of modulo as: Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem: given multiplicative functions and , one has if and only if: for all primes . Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions and , let be their Dirichlet convolution. Then for every prime , one has: In particular, this makes it trivial to find the Bell series of a Dirichlet inverse. If is completely multiplicative, then formally: (Wikipedia).

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Related pages

Prime number | If and only if | Geometric series | Formal power series | Dirichlet convolution | Mathematics | Multiplicative function | Prime omega function | Liouville function | Möbius function | Constant function | Divisor function | Arithmetic function