For $m,k \geq 2$, let $C_{m,k}(n)$ denote the number of positive integers less than or equal to $n$ which can be expressed as a sum of $k$ $m$th powers.

I am interested in the asymptotic behavior of $C_{m,k}(n)$ for large $n$. What is currently known?

Here is what I could find.

  • $C_{2,2}(n) = \Theta(\frac{n}{\sqrt{\log{n} } } )$. This is due to a result by Landau and Ramanujan. By Lagrange's four squares theorem , $C_{2,k}(n) = \Theta(n)$ for $k \geq 4$.
  • Because even powers are in particular squares, by the previous bullet point, $C_{2m, 2}(n) = O(\frac{n}{\sqrt{\log{n} } } )$ for $m \geq 1$.
  • This question is related to Waring's problem. Let $G(m)$ be the least positive integer $k$ such that every sufficiently large integer can be expressed as a sum of $k$ $m$th powers. Then $C_{m,k}(n) = \Theta(n)$ for every $k \geq G(m)$.

Intuitively, I am expecting $C_{m,k}(n)$ to be very close to $\Theta(n)$, even for large $m$ and small $k$. Still, it would be interesting to know if there are any results about the factors of $\log$ which appear. For example, this might come down to the difference between $O(\frac{n}{\log{n}} )$ and $O(\frac{n}{\log \log{n} } )$.

Related questions are

  • 2
    $\begingroup$ For $m$ even you're certainly not going to get $\Theta(n)$ for small $k$. Since $m$-th powers are positive, there are at most $n^{1/m}$ possible summands, and so there are $O(n^{k/m})$ possible sums of $k$ $m$-th powers which are at most $n$. Thus your bound in the second bullet point can be improved a lot. $\endgroup$ – Christian Gaetz Jul 25 at 20:22

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