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The Divisor summatory function is a function that is a sum over the divisor function.

$$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^u \lfloor\frac{x}{k}\rfloor - u^2, \;\;\text{with}\; u = \lfloor \sqrt{x}\rfloor$$

http://en.wikipedia.org/wiki/Divisor_summatory_function#Dirichlet.27s_divisor_problem

I am looking for a formula or an efficient algorithm (complexity less than $O(x)$) to calculate the sum of the dividers of the squares.

$$E(x)=\sum_{n\le x} d(n^2)$$

e.g. $$E(3)=d(1)+d(4)+d(9)=1+3+3=7$$

Eric Naslund
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wnvl
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  • What's "met"? Is that supposed to say "with"? Also, note that if you write out a word like that in $\TeX$, it gets interpreted as juxtaposed variable names and therefore italicized. To get proper formatting for text inside $\TeX$, use \text{...}. Also note that you can get displayed equations by enclosing them in double dollar signs. Displayed equations look nicer and are easier to read; single dollar signs are intended only for inline equations. – joriki Apr 18 '12 at 23:52
  • Your requirement $O(n)$ makes no sense, since $n$ is a dummy summation variable. Do you mean $O(x)$? – joriki Apr 18 '12 at 23:53
  • "met" is dutch for "with" I edited the text – wnvl Apr 19 '12 at 00:00
  • I mean indeed $O(x)$. Thanks for the $TEX$ hints. – wnvl Apr 19 '12 at 00:01
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    Two more $\TeX$ hints: You can get "$\TeX$" using \TeX, and you can see the $\TeX$ commands for anything you see on this site by selecting "Show Math As ... TeX Commands" in the context menu (right-click on the formula). – joriki Apr 19 '12 at 00:09
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    This is OEIS sequence A061503. The entry doesn't give an efficient algorithm. – joriki Apr 19 '12 at 00:32
  • An efficient algoritm should exist. A calculation for $10^{12}$ should be possible in a few minutes. The problem is related to http://math.stackexchange.com/questions/133105/counting-couples-having-least-common-multiple-less-than-a-number. – wnvl Apr 19 '12 at 00:45
  • This is answered at http://mathoverflow.net/questions/93916/sum-of-sum-k1ndk2 – Gerry Myerson Apr 19 '12 at 03:30
  • @Gerry: I only see asymptotics there; the second part of the question, "Are there good methods for calculating this sum quickly?", doesn't seem to have been answered. I took a look at the paper by Broughan mentioned there, but didn't find any practical algorithm there, either. – joriki Apr 19 '12 at 07:03
  • @joriki, quite right. So, where is Eric Naslund? – Gerry Myerson Apr 19 '12 at 12:57

1 Answers1

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The number of divisors of a square is the divisor function convolved with the square of the Möbius function (see $g(n)$ here)

$$e(n)=d(n^2)=(d\star\mu^2)(n)$$

by combining these three identities from the linked table

$$e=1\star1\star Q_2,\space d=1\star1,\space Q_2=\mu^2$$

and since

$$\mu^2(n)=\sum_{d^2|n}\mu(d)$$

therefore

$$e(n)=\sum_{a \left| n \right.} d \left( \frac{n}{a} \right) \sum_{b^2 \left| a \right.} \mu \left( b \right)$$

which can be simplified and rewritten as

$$e(n)=\sum_{b^2 \left| n \right.} \mu \left( b \right) d_3 \left( \frac{n}{b^2} \right)$$

where $d_3(n)$ is the number of ways that a given number can be written as a product of three integers. This identity can be verified by noting that $e(n)$ is multiplicative and checking at prime powers which yields $e(p^a)={2a+1}$ and can be compared with $d(p^a)={a+1}$. In particular note that $d_3(p^a)={\binom{a+2}{2}}$ (see $d_k$ here).

Then the summation of the number of divisors of the square numbers can be computed as:

$$E(x)=\sum_{n\le x} d(n^2) =\sum_{n \leq x} \sum_{b^2 \left| n \right.} \mu \left( b \right) d_3 \left( \frac{n}{b^2} \right)$$

which can be reorganized as:

$$E(x)=\sum_{b \leq \sqrt{x}} \mu \left( b \right) \sum_{n \leq x / b^2} d_3 \left( n \right)$$ $$E(x)=\sum_{a \leq \sqrt{x}} \mu \left( a \right) D_3 \left( \frac{x}{a^2} \right)$$

where $D_3$ is the summatory function for $d_3$. Since $D_3(x)$ can be computed in $O(x^{2/3})$ time complexity using the three dimensional analogue of the hyperbola method, $E(x)$ can also therefore be computed in

$$O(\int_{a=1}^\sqrt{x}{(\frac{x}{a^2})}^{2/3}da)=O(x^{2/3})$$

which is better than $O(x)$ as desired.

By taking an $O(x^{1/3})$ algorithm to compute $D(x)$ and using it for $D_3(x)$, this bound can be reduced to $O(x^{5/9})$. You can find such an algorithm and one formula for $D_3(x)$ in my article here which uses the notation $T(n)$ and $T_3(n)$.

Rick Sladkey
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