{"id":"palomar/PALOMAR-2026-11-02-000015","url":"https://github.com/jkfarland-bot/QuarticPairing/tree/7ad815405be5e161e2c48dd9d4413ae5a214b9b8","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-02-000015-v1.json","repo":"https://github.com/jkfarland-bot/QuarticPairing","tags":["math.NT","msc:11L05","msc:11R18","msc:11N13","msc:68V20"],"score":0,"title":"QuarticPairing: the pairing of the twists in Conjecture 1.2 of David, Dunn, Hamieh and Lin","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"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) =…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-02","reactions":{},"formalized":true,"formalizes":[{"title":"The twists l = 0 and l = 1 in a conjecture on quartic Gauss sums, formalized in Lean","identifier":"","relationship":"formalizes"},{"title":"A note on Conjecture 1.2 of David, Dunn, Hamieh and Lin: the pairing of the twists, Suzuki's vanishing law, and a repair","identifier":"","relationship":"formalizes"}]}