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Does there exist asymptotic formula for ways to write n as sum of four squares? Or can this be proved impossible? I can only find reference for sums of five squares.

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The function you are asking for is $r_4(n)$, the number of ways to write $n$ as a sum of four squares. The exact formula was discovered by Jacobi, and is given as

$$\displaystyle r_4(n) = 8\sum_{\substack{d | n \\ 4 \nmid d}} d.$$

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Note that $$L(x) = \sum_{n\leq x} r_4(n)$$ is the number of lattice points in the ball of radius $\sqrt{x}.$

It is known that $$L(x) = \frac{\pi^2}2 x^2 + O(x \log(x)).$$ (the error term can be improved slightly, but this is not important here). This means that that $r_4(n)$ (which is the number of lattice points on the sphere of radius $\sqrt{n})$ is bounded by $O( n \log n).$ An asymptotic formula does not exist, since for $n$ prime, Jacobi's formula gives $8(n+1),$ while Gronwall's theorem tells us that this is as big as $c n \log \log n$ infinitely often.

A nice reference (but far from the only one) is:

Ivi\'c, A.; Kr\"atzel, E.; K\"uhleitner, M.; Nowak, W.G., Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic, Schwarz, Wolfgang (ed.) et al., Elementare und analytische Zahlentheorie. Tagungsband. Stuttgart: Franz Steiner Verlag (ISBN 3-515-08757-5/hbk). Schriften der Wissenschaftlichen Gesellschaft an der Johann-Wolfgang-Goethe-Universit\"at Frankfurt am Main 20, 89-128 (2006). ZBL1177.11084.

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  • $\begingroup$ Your asymptotic implies $r_4(n)=L(n)-L(n-1)=O(n\log n)$ (assuming that $p$, whatever that is, is constant). Where does the $n^{3/2}$ come from? $\endgroup$ – Emil Jeřábek supports Monica Apr 12 '17 at 17:03
  • $\begingroup$ @EmilJeřábek Your point is well-taken :) $\endgroup$ – Igor Rivin Apr 12 '17 at 17:08
  • $\begingroup$ @EmilJeřábek fixed now. $\endgroup$ – Igor Rivin Apr 12 '17 at 17:11
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Stanley Yao Xiao gave a perfect answer, but let me remark that $r_4(n)$ also equals $n$ times the usual singular series (familiar from the circle method) when $4\nmid n$. The difference with five or more squares is that the singular series does not vary between two positive constants: instead, it varies between $1$ and a constant times $\log\log n$. Note also that $r_4(n)$ is very small when $n$ is divisible by a large power of $2$.

For more details, see Heath-Brown: A new form of the circle method, and its application to quadratic forms (J. reine angew. Math. 481 (1996), 149-206), especially Theorem 4 and Corollary 1.

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