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Solution distribution in a quadratic congruence

Fix a modulus $M$. For varying $m$, consider the (possibly empty) set $R_{m,M}$ of roots of $x^2\equiv m^2+2 \pmod{M}$. Are there good known bounds on $$\sum_{m\le M^{1/100}}\sum_{r\in R_{m,M}}\min(M^ …
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