# 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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## 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:

.