Sums of almost equal squares of primes

We study the representations of large integers $n$ as sums $p_1^2 + … + p_s^2$,where$p_1,…, p_s$areprimeswith$| p_i - (n/s)^{1/2} | \le n^{\theta/2}$,forsomefixed$\theta < 1$\mathrm{.}When$s = 5$ we use a sieve method to show that all sufficiently large integers $n\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}24)$ can be represented in the above form for $\theta >8\mathrm{/}9$. This improves on earlier work by Liu, L"{u} and Zhan, who established a similar result for $\theta >9\mathrm{/}10$. We also obtain estimates for the number of integers $n$ satisfying the necessary local conditions but lacking representations of the above form with $s=3,4$. When $s=4$ our estimates improve and generalize recent results by L"{u} and Zhai, and when $s=3$ they appear to be first of their kind.