Vipul: Created page with '{{summatory function|divisor count function}} ==Definition== Let $x$ be a positive real number. The '''divisor count summatory function''' of $x$ is defin...'

2009-05-02T22:30:35Z

Created page with '{{summatory function|divisor count function}} ==Definition== Let <math>x</math> be a positive real number. The '''divisor count summatory function''' of <math>x</math> is defin...'

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{{summatory function|divisor count function}}

==Definition==

Let <math>x</math> be a positive real number. The '''divisor count summatory function''' of <math>x</math> is defined as the [[summatory function]] for the [[defining ingredient::divisor count function]]:

<math>x \mapsto \sum_{k \le x} \sigma_0(k)</math>,

where the sum is over positive integers <math>k</math>, and <math>\sigma_0(k)</math> is the number of positive integers dividing <math>k</math>.

This function can alternatively be defined as:

<math>\sum_{d \le x} \left[\frac{x}{d} \right]</math>.

Here, the square braces represent the greatest integer function. In other words, while the earlier expression counts for each element less than or equal to <math>x</math> the number of its divisors, the new expression counts for each possible divisor the number of multiples it has. Clearly, both expressions are equivalent.

==Behavior==

===Growth===

The divisor count function for a natural number <math>n</math> grows as:

<math>n \log n + (2\gamma - 1)n + O(n^\theta)</math>,

where <math>\gamma</math> is the [[Euler-Mascheroni constant]], and <math>\theta</math> is a (not completely determined) constant greater than <math>1/4</math>. The current best upper bound known for <math>\theta</math> is <math>131/416</math>.

Divisor count summatory function - Revision history

Vipul: Created page with '{{summatory function|divisor count function}} ==Definition== Let x be a positive real number. The '''divisor count summatory function''' of x is defin...'

Vipul: Created page with '{{summatory function|divisor count function}} ==Definition== Let $x$ be a positive real number. The '''divisor count summatory function''' of $x$ is defin...'