There are infinitely many primes of the form $x^2+y^2+1$, as proved by Bredihin. Motohashi improved the result by showing that there were $\gg x/\log^2 x$ such primes up to $x$. But we expect $\Theta(x/\log^{3/2}x)$ primes up to $x$; has this result been proved? (Failing that, is anything else known about the density since the 70s? I know Sun & Pan proved the Green-Tao theorem for these primes.)
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$\begingroup$ I would expect further improvements to cite Motohashi's or Bredihin's work. I couldn't find the reference for Bredihin's work but there are only 5 papers referring to Motohashi's. $\endgroup$– Arnaud MortierCommented Mar 3, 2018 at 23:20
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$\begingroup$ @ArnaudMortier B. M. Bredihin, Binary additive problems of indeterminate type II. Analogue of the problem of Hardy and Littlewood (in Russian). Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 27 (1963), pp. 577-612. $\endgroup$– CharlesCommented Mar 3, 2018 at 23:21
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Iwaniec (Acta Arith. 1972) showed that the number of such primes has the order $x/(\log x)^{3/2}$. His result applies more generally to translates of binary quadratic forms. One can also show that short intervals contain the right density of such primes: for example, see the work of Matomaki (Acta Arith. 2007).
