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Given $N\in\Bbb N$ such that $\prod_{i=1}^mp_i=N$ with $p_i$ being similar sized primes such that $p_i\neq p_j$ if $i\neq j$ where $m\in[1,\log\log N]$, consider $$r_4(N,[a,b])=|\{\alpha^2+\beta^2+\gamma^2+\delta^2=N:\alpha,\beta,\gamma,\delta\in[a,b]\cap\Bbb Z\}|.$$

What is $r_4(N,[2^{\frac{\log N-c}2},2^{\frac{\log N}2}])$ where fixed $c$ satisfies $0<c<\log\log N$?

Is this asymptotically $2^{\frac{\log N-c}{2m}}$?

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  • $\begingroup$ No. For example, if $N$ is a power of two, $r_4(N)$ is bounded. $\endgroup$ – GH from MO Aug 12 '15 at 0:39
  • $\begingroup$ holds for generic $N$? $\endgroup$ – Turbo Aug 12 '15 at 0:43
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    $\begingroup$ What do you mean by generic $N$? As long as $N$ is divisible by a large power of $2$, $r_4(N)$ is small. If $N$ is odd (say), then $r_4(N)$ is large, namely it is $N^{1+o(1)}$, but it can be significantly smaller or larger than $N$. This indicates strongly that your conjectured asymptotic formula is false, because the restricted count almost surely has the same kind of fluctuation as the unrestricted version. Also, $m$ does not influence these counts too much, e.g. for $m=10$ and $m=100$ the behavior will be similar. $\endgroup$ – GH from MO Aug 12 '15 at 3:46

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