It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}\_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one half lie in the interval $[1, (p-1)/2]$.

I remember reading this fact many years ago on a very popular book in number theory, where it was claimed that this is an easy consequence of a more sophisticated formula of analytic number theory.

Sadly I forgot both the formula and the book. So the purpose of the question is double:

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> 1) Has any simple way been found to derive the fact mentioned above?
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> 2) Does anybody know a reference for the analytic number theory route to the proof?
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