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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Contents
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:
.