File:Divisor-summatory-big.svg

English: ;Divisor summatory function

This graph illustrates the divisor summatory function with the leading asymptotic terms subtracted. That is, it is a graph of

${\displaystyle D(x)-x\log x-x(2\gamma -1)}$

where ${\displaystyle D(x)}$ is the divisor summatory function

${\displaystyle D(x)=\sum _{n\leq x}d(n)}$

and ${\displaystyle d(n)}$ is the divisor function. Here, ${\displaystyle \gamma =0.577\ldots }$ is the Euler-Mascheroni constant. The graph extends up to ${\displaystyle x=10^{7}}$. The green lines that do not quite bound the picture are a graph of ${\displaystyle \pm 2x^{7/22}}$ and give a general scale for the rate of growth for this function. Curiously, note that the function is not centered on the zero axis, but seems to be closer to the upper curve. Although the graph appears to be noisy, notice that it is less noisy than the corresponding graph for smaller n, and is more noisy than a similar graph for larger n.