Correlation laws and identifiability of multiplicative functions on quadratic traces
We study which completely multiplicative sign functions can be identified from the statistics of $h(n^2+1)$, $h(n^2+2)$ and $h(n^2-2)$. In a class of finite products of quadratic characters of odd prime conductor, completed without zeros and modified at primes with summable inverse square roots, diagonal correlations of orders two and four determine the function. The second-order data recover the conductor and prime corrections, leaving exactly the ramified local signs undetermined. Fourth-order detectors recover those signs, including at the small primes; the inverse also extends to…
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