Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since $$9 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2$$ $$9 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2\phantom{1^2 +\;\,+ 1^2 + 1^2}$$ $$9 = 1^2 + 2^2 + 2^2\phantom{ + 1^2 + 1^2 + 1^2 + 2^2\;\,+ 1^2 + 1^2}$$ $$9 = 3^2 \phantom{+ 2^2 + 2^2 + 1^2 + 1^2 + 1^2 + 2^2\;\,+ 1^2 + 1^2}$$ and there are no other ways of writing $9$ as sum of squares.

It is known that $$\log p^{(s)}(n) \sim (s+1)\left(\frac1{2}\Gamma\!\left(1+\frac1{s}\right)\zeta\!\left(1+\frac1{s}\right)\right)^{s/(s+1)} n^{1/(s+1)},$$ as $n \to +\infty$ (See Hardy and Littlewood, Asymptotic formulæ in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVII, 1918, 75-115).

My question is: If $p_k^{(s)}(n)$ is the number of ways of writing the positive integer $n$ as a sum of exactly $k$ perfect $s$-powers, is there an asymptotic formula for $\log p_k^{(s)}(n)$ holding in a reasonable range of $n,k \to \infty$? I am particularly interested in the case of squares $s = 2$.

Thank you in advance for any suggestion.

  • Do you already know the asymptotics for fixed $k$ as $n\to\infty$, connected with "Waring's problem"? – Greg Martin May 15 at 16:42
  • So $9 = 3^2$ doesn't count, or should $p^{(2)}(9) = 4$? In the latter case the sequence is A001156. There are also sequences for sums of two nonzero squares A025426, three nonzero squares A025427, etc., which are each columns of the triangle A243148, although these do not mention asymptotic formulas. – Brian Hopkins May 15 at 19:02
  • @BrianHopkins Sorry, I forgot $9 = 3^2$, thanks! – Megan May 15 at 20:17
  • @GregMartin No, I don't. However, I am more interested in an asymptotic holding for $k \to +\infty$ (in some range of $n$ depending on $k$, or vice versa) – Megan May 15 at 20:18
  • It's going to be quite important to the answer to know the relative sizes of $n$ and $k$ ... what do you have in mind? – Greg Martin May 16 at 3:30

The folowing is a known result on Waring's problem: For fixed $k$ (sufficiently large), the number of ways to write $n$ as an ordered sum of $k$ squares is asymptotic to $$ \frac{\Gamma(3/2)^k}{\Gamma(k/2)} {\frak S}_2(n)n^{k/2-1}, $$ where ${\frak S}_2(n)$ (the "singular series") is a complicated but bounded function of $n$. Almost all of these representations use distinct squares, and so the number of unordered representations is asymptotic to this expression divided by $k!$.

For each such $k$, this asymptotic formula holds when $n$ is sufficiently large in terms of $k$. Therefore if we let $n$ go to infinity sufficiently quickly in terms of $k$, this asymptotic formula holds uniformly in $k$ and $n$. (Note that ${\frak S}_2(n)$ is not a smooth function—it depends on the factorization properties of $n$.)

  • "this asymptotic formula holds when $n$ is sufficiently large in terms of $k$" <- Then, I think the point is finding if somebody has worked out an explicit version of this statement, like "for $n \ge f(k)$", where $f$ is an explicit function. – Megan May 16 at 8:43
  • I totally agree that this is the right starting point for you! – Greg Martin May 16 at 17:10

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