Sums of almost equal squares of primes

We study the representations of large integers n as sums , where p1,…,ps are primes with |pi−(n/s)1/2|⩽nθ/2, for some fixed θ<1. When s=5 we use a sieve method to show that all sufficiently large integers can be represented in the above form for θ>8/9. This improves on earlier work by Liu, Lü and Zhan (2006), who established a similar result for θ>9/10. 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 Lü and Zhai (2009), and when s=3 they appear to be first of their kind.