An asymptotic formula for primes of the form $u^2+v^2+1$

Abstract

We prove Motohashi's conjecture on primes of the form $u^2+v^2+1$, with Iwaniec's corrected constant. Let $b$ denote the characteristic function of the positive integers representable as a sum of two squares. Then, as $x\to \infty$, \[ \sum_{p\le x}b(p-1) =κ\Spr\frac{x}{(\log x)^{3/2}} +O\!\left(\frac{x}{(\log x)^{3/2}\sqrt{\log\log x}}\right), \] where $\kappa$ is the Landau--Ramanujan constant and $\Spr=\prod_{p\equiv3\,({\rm mod}\,4)}\bigl(1-1/(p(p-1))\bigr)$. The proof combines auxiliary prime factors with a mean square estimate for bilinear sums. We obtain the minor arc bound through harmonic switching and high moments, replacing arithmetic divisibility conditions by independent projective lines. We remove the auxiliary weights using Selberg's sieve.

arXiv comment44 pages