What would be the best lower bound for the number of squares modulo $p$ in an interval $[1,N]$ with $N<p$ that are prime?
Via the Burgess bound, I can find a lower bound for the number of squares modulo $p$ in $[1,N]$, but I would need a bound for the number of squares that are also prime. Since the size of $N$ matters, in my particular case I have
$$
N=\frac{\sqrt{p}}{2}.
$$
Thank you very much!
Counting squares modulo $p$ that are also prime in an interval
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