Why are there usually an even number of representations as a sum of 11 squares

Question

Question: Why do so few $n\equiv 3 \bmod 8$ have an odd number of representations in the form $$n=x_0^2 + x_1^2 + \dots + x_{10}^2$$ with $x_i \geq 0$?

Note that $x_i\geq 0$ spoils the symmetry enough that this is not the usual "number of representations as a sum of squares" question. The question may be ill-posed, too: I do not know that there are few such $n$, but computations suggest that their counting function is $\sim cx/\log x$.

The question I'd like to answer is this: which $n$ have an odd number of representations of the form $$n=x_0^2+2x_1^2+4x_2^2+ \dots = \sum_{i=0}^\infty 2^i x_i^2,$$ where $x_i \geq 0$? Does the set of such $n$ have 0 density? The motivation for this question is that it is nontrivially the same as this question.

If $n$ is even and has an odd number of reps, then (nice exercise) $n$ has the form $2k^2$. If $n\equiv 1\pmod4$ and has an odd number of reps, then (challenging but elementary) $n$ has a special sort of factorization, and there are very few such $n$.

The question I'm asking here is for $n\equiv 3 \bmod 8$. Reducing modulo 8 reveals that $x_2$ must be even. If it isn't a multiple of 4, then we can pair off the two reps: $$(x_0,x_1,x_2,x_3,x_4,x_5,\dots) \leftrightarrow (x_0,x_1,2x_4,x_3,x_2/2,x_5,\dots).$$ If $x_2$ is a multiple of 4 but not 8, then we can make a similar pairing with $x_2$ and $x_5$, and so on. This only leaves the situation when $x_2$ and $x_4,x_5,\dots$ are all zero.

This leaves us at: Which $n\equiv 3\bmod 8$ have an odd number of representations in the form: $$n = x_0^2 + 2x_1^2 + 8 x_3^2 \; , \; \; \mbox{with} \; \; x_0, x_1, x_3 \geq 0 \;?$$ Some play with the parities of binomial coefficients reduces this to the question I led with.

Answer 1

Throughout $N>0,$ and $N \equiv 3 \pmod 8.$ Let $I$ be the number of ordered triples $(a,d,e) \;\mbox{with} \; a,d,e \geq 0,$ such that $$a^2+2 d^2+8 e^2=N.$$ I'll use a result of Gauss on sums of 3 squares to show that if there are 3 or more primes whose exponent in the prime factorization of $N$ is odd, then $I$ is even. As a consequence those $N$ for which $I$ is odd form a set of density 0; in fact the number of such $N < x$ for positive real $x$ is $$ O \left( \frac{x \; \log \log x}{\log x} \right). $$
Let $ R = R(N) $ be the number of triples $ (a,b,c) \; \mbox{with} \; a,b,c > 0 $ and
$$ a^2+b^2+c^2=N,$$ and let $r(N)$ be the number of such triples with the $ \gcd(a,b,c) = 1.$ Then $R$ is the sum of
the $r(N/k^2),$ the sum running over all $k>0$ for which $k^2 | N.$ Now in Disquisitiones, Gauss shows that if
$N>3, \mbox{then} \; r(N)/3$ is the number of classes (under proper equivalence) of positive primary binary forms of discriminant $- N.$ (Or if you prefer, the number of classes of invertible ideals in the quadratic order of discriminant $- N$). Now these classes form a group, and Gauss uses genus theory to show that the order of this group is divisible by $2^{M-1}$ where $M$ is the number of primes that divide $N.$ So if 3 or more primes have odd exponent in the prime factorization of $N,$ then all these primes divide $N/k^2,$ the corresponding group has order divisible by $2^{3-1}=4,$ so 4 divides each $r(N/k^2),$ and 4 divides $R.$ $$ $$ Now let $S=S(N)$ be the number of pairs $(a,d) \; \mbox{with} \; a,d > 0$ and
$$a^2+2 d^2=N,$$ and $s(N)$ be the number of such pairs with $ \gcd(a,d) = 1.$ Then $S$ is the sum of the $s(N/ k^2).$ Using the fact that $\mathbb{Z} \left[ \sqrt{-2} \right]$ is a UFD we can calculate $s(N/k^2);$ it is zero when some prime $p \equiv 5,7 \pmod 8$ divides $N/k^2.$ When this doesn't happen there are 3 or more
primes $q \equiv 1,3 \pmod 8$ dividing $N/k^2,$ so 4 divides each $s(N/ k^2)$ and 4 divides $S$ as well as $R.$ We conclude the proof by showing that $$2I=R+S.$$

Suppose $N \equiv 3 \pmod 8$ and $a^2+b^2+c^2=N,$ with $a,b,c>0.$ Of course $a,b, c$ are odd. If $b \equiv c \pmod 4,$ let $d=(b+c)/2$ and $e = | (b-c)/4 |.$ Otherwise let $d = | (b-c)/2 |$ and $e=(b+c)/4.$ Then $$a^2+2 d^2+8 e^2=a^2+b^2+c^2=N.$$ Furthermore $(a,b,c)$ and $(a,c,b)$ map to the same $(a,d,e).$ The fiber of the map $(a,b,c) \mapsto (a,d,e)$ has 1 element when $e=0$ and 2 elements otherwise. So $2I=R+S.$ $$ $$ If $N = p q$ where $p$ and $q$ are primes congruent to 5 and 7 $ \pmod 8$
respectively, with $ (q | p ) = -1$ it can be shown that $R \equiv 2 \pmod 4,$ so that $I$ is odd. This should
allow one to get a lower bound for the number of $ N < x $ with $I$ odd that's a constant multiple of the upper
bound mentioned above. But whether the number is asymptotic to a constant multiple of $x \; \log \log(x)/ \log (x)$ as Jagy's calculations suggest isn't clear.

Answer 2

This is not an answer to your stated question, but a relevant remark. Suppose that $a(x)$ is a unipotent formal power series in the formal power series ring $(\mathbb{Z}/p)[[x]]$. (Here unipotent just means constant term 1, so you could say topologically or adically unipotient.) Then obviously $a(x)^n$ is well-defined for any integer $n$. What is somewhat less obvious, but not hard and an interesting general principle, is that $a(x)^d$ is well-defined for any $p$-adic integer $d$. Namely, if $d$ has digits $\ldots d_2d_1d_0$, then $$a(x)^d := a(x)a(x^{d_1p})a(x^{d_2p^2})\cdots$$ It's not hard to check that this formula is (a) correct when $d$ is an integer using the Frobenius map, (b) convergent when $a(x)$ unipotent, and (c) continuous in $d$ and $a$. Therefore (d) it satisfies $a^{d+e} = a^d a^e$ and $(ab)^d = a^db^d$.

You use this formula twice in your question. You use it with $d=-1$ when you call your power series "nontrivially the same" as your other question. You use it again with $d=11$ with the phrase "some play with parities of binomial coefficients". Of course in both cases you're using the power series from your paper, $$a(x) = 1+x+x^4+x^9 + x^{16} + \cdots \in (\mathbb{Z}/2)[[x]].$$ I hadn't thought of this style of $p$-adic exponentiation and I think that it's cute, and it could be a unifying principle for some of what you are doing.

Answer 3

ORIGINAL:The counting function looks better with an extra factor of $ \log \log n $ in the numerator. That is, the cumulative count of numbers $ \equiv 3 \pmod 8 $ up to some $x$ that have an odd number of your representations resembles $$ \frac{ C \;x \; \log \log x}{\log x} $$

I do not know how to keep this in columns.

        n      odd    even    n / odd      / log n     * log log n
        3       1       0    3           2.73071768   0.256818066
     7995     405     595   19.7407407   2.1966932    4.82334827  
    15995     775    1225   20.6387097   2.13209118   4.83998588  
    23995    1147    1853   20.9197908   2.07422357   4.79375621 
    31995    1495    2505   21.4013378   2.06311065   4.826108     
    39995    1831    3169   21.8432551   2.06136319   4.86589867   
    47995    2166    3834   22.1583564   2.05572506   4.88766328   
    55995    2509    4491   22.3176564   2.041308     4.88237466   
    63995    2860    5140   22.3758741   2.02193578   4.86058798   
    71995    3177    5823   22.6613157   2.02616261   4.89220133   
    79995    3524    6476   22.7000568   2.01068387   4.87368161   
    87995    3848    7152   22.8677235   2.00857731   4.88546218   
    95995    4179    7821   22.9708064   2.00232771   4.88550694   
   103995    4499    8501   23.1151367   2.00094706   4.89605147   
   111995    4831    9169   23.1825709   1.99399218   4.89178523   
   119995    5142    9858   23.3362505   1.99536902   4.90696952   
   127995    5472   10528   23.3908991   1.98906491   4.90241326  
   135995    5782   11218   23.5204082   1.98981939   4.91450488   
   143995    6107   11893   23.5786802   1.98514948   4.9125476   
   151995    6432   12568   23.6310634   1.98054391   4.91014581