= Divisor summatory function
{title2=$D(x)=\sum_{n\le x}\tau(n)$}
{wiki}
The <divisor summatory function> adds the <divisor function> up to a cutoff $x\ge1$. It counts all <ordered pairs> of <positive integers> whose product is at most $x$. For $N=12$, the twelve individual <divisor function> values are
$$
(\tau(1),\ldots,\tau(12))=(1,2,2,3,2,4,2,4,3,4,2,6),
\qquad D(12)=35.
$$
Counting columns of <lattice points> instead gives $12+6+4+3+2+2+1+1+1+1+1+1=35$. The <Dirichlet hyperbola method> needs only the first three column counts:
$$
D(12)=2(12+6+4)-3^2=44-9=35.
$$
The <arithmetic work of the Dirichlet hyperbola method> is much smaller even though the answer is identical.
This is useful for elementary averages: individual <divisor function> values fluctuate, but their <summation> has the regular behavior given by the <divisor summatory asymptotic>. Dividing that estimate by a <positive integer> $N$ gives
$$
\frac1N\sum_{n=1}^N\tau(n)
=\log N+2\gamma-1+O(N^{-1/2})\qquad(N\longrightarrow\infty),
$$
where $\log$ is the <natural logarithm>, $\gamma$ is the <Euler--Mascheroni constant>, and $O$ uses <big O notation>. This describes an average, rather than an estimate for every individual <divisor function> value. <Multiplicative number theory: The pretentious approach by Andrew Granville and K. Soundararajan>[§1.1.4, p. 8, equation (1.1.13): the divisor summatory asymptotic]{ref}
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