Divisor power sum function
This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
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Let be a real number (typically an integer). The divisor power sum function (sometimes called the divisor function) is defined as the following arithmetic function from the natural numbers to the real numbers:
The sum is over all the positive divisors of .
Definition in terms of Dirichlet product
The divisor power sum function is defined as:
Here is the power function, and is the all ones function.
The case gives the divisor count function, i.e., the function that counts the number of positive divisors of .
The case gives the divisor sum function, i.e., the sum of all the positive divisors.
Further information: Formula for Dirichlet series of divisor power sum function
The Dirichlet series for is given by:
This is related to the Riemann zeta-function by the following identity, that holds both formally and for the corresponding meromorphic functions: