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The Dirichlet hyperbola method turns a long summation over a Dirichlet convolution into two short ones by covering the lattice points below a hyperbola with two overlapping strips. Its simplest application is an exact count: for a positive integer $N$, every positive integer divisor $a$ of $n\le N$ corresponds to the ordered pair $(a,n/a)$. Thus the divisor summatory function counts precisely the ordered pairs of positive integers with $ab\le N$:Here $\tau$ is the divisor function, and the floor function gives the number of allowed values of $b$ for each $a$. The axes are excluded, while points on $ab=N$ are included.

Set $r=\lfloor \sqrt{N}\rfloor$, using the floor function and the positive square root. Every counted pair has $a\le r$ or $b\le r$. Otherwise $ab\ge (r+1{)}^2>N$. The first strip contains ${\sum }_{a=1}^r\lfloor N/a\rfloor$ points; swapping the two coordinates gives the same count for the second. Their overlap contains all $r^2$ pairs with $1\le a,b\le r$, since $r^2\le N$. The inclusion-exclusion principle therefore gives the exact identityThe subtraction is for the whole overlap square, rather than only the diagonal. The same Dirichlet hyperbola method identity holds for real $x\ge 1$, with $D(x)={\sum }_{n\le x}\tau (n)$ and $r=\lfloor \sqrt{x}\rfloor$. ^[ref]

The divisor summatory function adds the divisor function up to a cutoff $x\ge 1$. It counts all ordered pairs of positive integers whose product is at most $x$. For $N=12$, the twelve individual divisor function values areCounting 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: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 $N$ giveswhere $\log$ 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. ^[ref]