How many integers below $n$ can be expressed as a sum of $k$ $m$th powers?

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

• 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. – Christian Gaetz Jul 25 at 20:22