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Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are chosen uniformly from the set $S = \{-1, 1\}$. Does this sum equidistribute mod $p$ as $n$ goes to infinity? What would be the speed of equidistribution in terms of $n$? Is there any literature in this type of random sums? I would be surprised if not but I am unable to find anything related or similar to this.

One can also ask what is the probability of this sum being actually zero, but I also have no idea how to deal with it and thought that modulo a prime would be simpler.

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The condition $$\sum_{ 1 \leq i < j \leq n} a_ia_j \equiv 0 \pmod p$$ is equivalent to $$\left(\sum_{ 1 \leq i\leq n} a_i\right)^2 \equiv n \pmod p.$$ So a necessary condition is that $n$ is a quadratic residue modulo $p$ (including the zero residue). If $n$ is divisible by $p$, then the above condition says that the sum of the $a_i$'s is divisible by $p$. Otherwise, the condition says that the sum of the $a_i$'s is congruent to one of the two square-roots of $n$ modulo $p$. Now it is easy to see that the sum of the $a_i$'s is equidistributed modulo $p$ (think about what happens when an $a_i=1$ is switched to $a_i=-1$), hence in the first case the probability is $1/p+o(1)$, in the second case it is $2/p+o(1)$, as $n$ tends to infinity.

In fact the probabilities can be calculated explicitly as a linear combination of $n$-th powers of $p$ complex numbers (which only depend on $p$), since the sum of the $a_i$'s modulo $p$ is determined by $\#\{i:a_i=1\}$ modulo $p$, and vice versa. Compare with this post, where the role of $p$ is played by $4$. It follows, in particular, that the $o(1)$ terms above decay exponentially fast. For a more complete reference, see Theorems 8.7.2 & 8.7.3 in Wagner: A first course in enumerative combinatorics (AMS, 2020).

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  • $\begingroup$ Dear @GHfromMO, thanks a lot for the answer and for the reference. Your answer depends on the lucky coincidence that $\sum a_i^2 = n$, what about the case where $S \neq {-1,1}$?. Also,do you know of a trick that can help with the last sums which includes a_i's and b_i's?, observe that these sums are not symmetric and even in the case $S = {-1,1}$ I do not know how to approach them. $\endgroup$
    – shurtados
    Aug 18 '21 at 23:42
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    $\begingroup$ @shurtados: Thanks for your kind words. At MO, the policy is to ask one question in one post, and I just answered that. If you have further questions, please ask them in separate posts. Also, if you like my answer to your original (first) question, please accept it officially, so that it turns green. $\endgroup$
    – GH from MO
    Aug 19 '21 at 0:09
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    $\begingroup$ In that case, I'll accept the answer, edit the question and then post another question. Thanks again for the nice answer. $\endgroup$
    – shurtados
    Aug 19 '21 at 4:27

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