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arXiv · 2503.05396

On primes represented by $aX^2+bY^3$

Jori Merikoski
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Abstract

Let $a,b>0$ be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form $ax^2+by^3$ with $x \leq X^{1/2}$ and $y \leq X^{1/3}$. The proof combines sieve methods with the theory of real quadratic fields/indefinite binary quadratic forms, the Weil bound for exponential sums, and spectral methods of GL(2) automorphic forms. We also discuss applications to elliptic curves.

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Jori Merikoski. 2025-03-07. On primes represented by $aX^2+bY^3$. https://arxiv.org/abs/2503.05396

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