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I'm trying to simplify this combinatorial looking sum: $$\sum_{ax+by=n}_{(a,x,b,y)\in \mathbb{N^4}}\max{\{a,b\}}$$ In terms of possibly some scaled divisor functions plus a Cauchy product/convolution of divisor functions.

I thought maybe I could break it up into the cases where $(a<b)\wedge (a>b) \wedge (a=b)$

Then separate the cases and use the representation:

$$\sum_{ax+by=n}_{(a,x,b,y)\in \mathbb{N^4}}a=\sum_{ax+by=n}_{(a,x,b,y)\in \mathbb{N^4}}b=\sum_{k=1}^{n-1}d(k)\sigma(n-k)$$

To evaluate the entire sum, but it's not really working out.

Can someone more experienced with this sort of thing help me simplify the first sum in terms of divisor function convolutions?

Or if it doesn't look like it can be simplified in such a closed form, give me some insight as to why it can't?

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Your sum is a special case of sums concidered by Halberstam, see

Halberstam, H. Four asymptotic formulae in the theory of numbers J. London Math. Soc., 1949, 24, 13-21

Halberstam, H. An asymptotic formula in the theory of numbers Trans. Amer. Math. Soc., 1957, 84, 338-351

Your sum is a complicated arithmetic function, that is why closed form is a problem. We can consider this sum as an everage value of some characterisic of reduced basis in a lattice $\Lambda\subset\mathbb{Z}^2$ with $\det\Lambda=n$. It is also characterize the work of Euclidean algorithm on numbers $a/n$ ($1\le a\le n$), see Euclidean algorithm. For example the sum $$\sum_{ax+by=n}_{(a,x,b,y)\in \mathbb{N^4}}1$$ is almost the same as a sum of all partial quotients in continued fraction expansions of numbers $a/n$, where $1\le a\le n\le N$.

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  • $\begingroup$ Do you have a subscription to the journal or do you know where I can get the pdf? I can't afford the $38 to view the article for one day. $\endgroup$
    – Ethan Splaver
    Commented Feb 22, 2014 at 8:05
  • $\begingroup$ Give me your e-mail [email protected] $\endgroup$
    – Alexey Ustinov
    Commented Feb 22, 2014 at 10:30
  • $\begingroup$ I can't seem to find the result, they seem to be proving asymptotics for convolutions of divisor functions. I was hoping there was a closed form for $\sum_{ax+by=n}_{(a,x,b,y)\in \mathbb{N^4}} \max(a,b)$ $\endgroup$
    – Ethan Splaver
    Commented Feb 23, 2014 at 0:23
  • $\begingroup$ It will be great if you'll find closed form for such sums. The binary additive divisor problem (see Motohashi, Y. The binary additive divisor problem Ann. Sci. École Norm. Sup. (4), 1994, 27, 529-572) looks more simple but we don't have closed form for it. $\endgroup$
    – Alexey Ustinov
    Commented Feb 23, 2014 at 3:47
  • $\begingroup$ Sorry, I forgot to tell you I actually did find a closed form in the last 2 hours using a combinatoral identity and some other techniques. It turns out that: $$\sum_{ax+by=n}_{(a,x,b,y)\in \mathbb{N^4}}\max{\{a,b\}}=nd(n)-\sigma_2(n)+2\sum_{k=1}^{n-1}d(k)\sigma(n-k)$$ $\endgroup$
    – Ethan Splaver
    Commented Feb 23, 2014 at 3:55

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