Let $q,a,b$ be integers, $q \geq 1$, $(q,a)=1$, and let $\chi$ be a Dirichlet character modulo $q$. Is there a formula for the sum $$ G_\chi(a,b,q) := \sum_{n(q)} e^{2\pi i \frac{(an^2 + bn)}{q}} \chi(n). $$ Any ideas on simplification or anything interesting to say of these?
2026-02-23 09:07:31.1771837651
Formula for twisted generalized quadratic Gauss sums
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