What is the density of integers which can be written as the sum of two positive integer powers?

Question

The power-free natural (asymptotic) density of $x^k$ is $1/\zeta(k)$ plus error terms.

I suppose this means that the number of integers less than $N$ that a have $k$ power in their factorization is:

$N(1-1/\zeta(k))$ plus error terms.

So, what is the asymptotic density of $x^k+y^l$?

Answer 1

Not a complete answer but a trivial lower bound. Too long to be a comment.

Let $f(x)$ be the number of positive integers $\le x$ which can be expressed as the sum of two powers. Then,

$f(x) > $ number of positive integers $\le x$ which can be expressed as the sum of two squares

or $f(x) > $ number of primes $\le x$ which can be expressed as the sum of two squares

or $f(x) > $ number of primes $\le x$ which are of the form $4k+1$.

Hence by Dirichlet theorem of primes in arithmetic progression,

$$ f(x) > \frac{x}{2\log x}. $$