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Let $S_4(n)$ be the number of ways of representing $n$ as a sum of 4 integer squares and suppose I can give you a proof of the result that $\sum_{i=1}^nS_4(i) \sim \frac{\pi^2}{2}n^2$.

Is this result enough to deduce that beyond some $N$, every integer can be represented as the sum of 4 squares (given that we are "weakly" suggesting that $\sum_{i=1}^nS_4(i) >> n$)?

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    $\begingroup$ It's not true that $S_4(n)\sim\frac{\pi^2}{2}n^2$. What is true is that $\sum_{k=1}^nS_4(k)\sim\frac{\pi^2}{2}n^2$. $\endgroup$ – Lord Shark the Unknown Dec 15 '18 at 16:16
  • $\begingroup$ Ah yes, I meant that! Let me just update this. $\endgroup$ – Isky Mathews Dec 15 '18 at 17:12
  • $\begingroup$ @LordSharktheUnknown: Can you answer the question, though? I know from experience on this site that you're quite a capable individual... $\endgroup$ – Isky Mathews Dec 15 '18 at 17:13
  • $\begingroup$ You need more than that asymptotic. $\endgroup$ – Lord Shark the Unknown Dec 15 '18 at 19:13

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