Divisor power sum function

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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).
View a complete list of arithmetic functions

Definition

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.

Particular cases

The case

The case gives the divisor count function, i.e., the function that counts the number of positive divisors of .

The case

The case gives the divisor sum function, i.e., the sum of all the positive divisors.

Dirichlet series

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:

.