QuarticPairing: the pairing of the twists in Conjecture 1.2 of David, Dunn, Hamieh and Lin
Conjecture 1.2 of C. David, A. Dunn, A. Hamieh and H. Lin (arXiv:2306.11875) asserts that the sums S_beta(X; l) of normalized quartic Gauss sums g(c) over primary primes c of Z[i] with N(c) <= X and c = beta (mod 4), twisted by (conj(c)/|c|)^l and weighted by the von Mangoldt function, are b_beta X^(3/4) + O(X^(1/2+eps)) with b_beta nonzero when l = 0 and O(X^(1/2+eps)) when l is nonzero. The development proves that if g(c)^2 = -kappa c/|c| for a fixed unit kappa on the support of the weight (the identity (1.2) of the same paper), then g(c) conj(c)/|c| = -kappa conj(g(c)), so S_beta(X; 1) =…
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