In 1998 John Friedlander and Henryk Iwaniec famously proved the asymptotic formula
$$\displaystyle \mathop{\sum \sum}_{a^2 + b^4 \leq x}\Lambda(a^2 + b^4) = \frac{4x^{\frac{3}{4}}}{\pi} \int_0^1 (1 - t^4)^{\frac{1}{2}} dt \left\{1 + O \left(\frac{\log \log x}{\log x} \right) \right\}. $$
Put
$$\displaystyle \kappa = \int_0^1 (1-t^4)^{\frac{1}{2}} dt.$$
Then their theorem asserts that the number of primes of the form $p = a^2 + b^4$ of size up to $x$ is approximately
$$\displaystyle \frac{4 \kappa x^{\frac{3}{4}}}{\pi \log x}.$$
Of course, the quality of the error term in the Friedlander-Iwaniec theorem is far from ideal. Indeed the error comes from the sieve machinery used to prove the asymptotic formula, and not from any deep structures involving primes. Compare this to the following formulation of the Riemann hypothesis:
$$\displaystyle \psi(x) = \sum_{n \leq x} \Lambda(n) = x + O_\varepsilon \left(x^{\frac{1}{2} + \varepsilon} \right),$$
for example. My question is, what is the expected order of magnitude of the difference
$$\displaystyle \mathop{\sum \sum}_{a^2 + b^4 \leq x} \Lambda(a^2 + b^4) - \frac{4 \kappa x^{\frac{3}{4}}}{\pi}?$$
Here the issue is not only the oscillation of prime (powers) but also the geometry of the shape of the region defined by $u^2 + v^4 \leq x$. For instance, the Gauss circle problem may be of relevance in this case.