Given a prime $p=3m+1$, $(p-1)/3$ of the residues mod $p$ are cubic residues. So heuristically, for any given integer $k>1$ not a perfect cube, we would expect that about 1/3 of the primes $\equiv1\pmod3$ up to $x$ would have $k$ as a cubic residue. Is this known, and what kind of error term has been proved?



Counting cubic residues mod p
This is true. Such primes are exactly the primes that split in the field $\mathbb Q(\mu\_3, \sqrt[3]{k})$, and they split into exactly $6$ prime ideals of norm $p$ since the extension is Galois of degree $6$, and thus their density among the primes is $1/6$ by the [Landau prime ideal theorem](https://en.wikipedia.org/wiki/Landau_prime_ideal_theorem) (i.e. their density among the primes congruent to $1$ mod $3$ is $1/3$.)


The error term is exp-root-log size.


