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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.

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    $\begingroup$ There should certainly be a power savings. I think my methods with Shusterman fall a bit short of proving power savings for the function field analogue but I'm not sure. I wouldn't be surprised if a variant of our approach could do it. If I had to guess one number for the true size of the power savings, I would say it's of square-root size, so $x^{ \frac{3}{8}+\epsilon}$. I would be shocked if it's not at least square-root size. Gauss-circle-like error terms should be irrelevant because they're smaller than this. $\endgroup$
    – Will Sawin
    Nov 23, 2022 at 0:39
  • $\begingroup$ The error term is very similar to that of Titchmarsh divisor problem. $\endgroup$
    – TravorLZH
    Nov 23, 2022 at 19:34

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