Average Orders Divisor Sums and Representation Counts — Examples

By The divisor functions arise by Dirichlet convolution, $\tau =\mathbf{1}\ast \mathbf{1}$. Therefore Dirichlet's hyperbola method for summatory convolutions applies with $F(y)=G(y)=\lfloor y\rfloor$.

The two arm sums are ${\sum }_{a\le 3}\lfloor \frac{12}{a}\rfloor =12+6+4=22,{\sum }_{b\le 4}\lfloor \frac{12}{b}\rfloor =12+6+4+3=25,$ and the overlap is $F(3)G(4)=3\cdot 4=12$.

Hence the hyperbola identity gives ${\sum }_{n\le 12}\tau (n)=22+25-12=35$. This matches the direct total $1+2+2+3+2+4+2+4+3+4+2+6=35$.

The exact divisor sums are obtained by direct counting: ${\sum }_{n\le 5}\tau (n)=10$, ${\sum }_{n\le 10}\tau (n)=27$, ${\sum }_{n\le 20}\tau (n)=66$, and ${\sum }_{n\le 50}\tau (n)=207$.

Substituting these four values into the displayed definition of $E(x)$ produces the residual column and then the scaled residual column $E(x)\mathrm{/}\sqrt{x}$.

The quotients in the last column stay on the scale of a bounded constant rather than growing like a positive power of $x$, which is exactly the $\sqrt{x}$ scale asserted by The summatory divisor-counting function is x log x plus (2 gamma - 1)x plus O(sqrt x).