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 $O(\frac{n}{\log \log{n} } )$.
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