Sibo Huang

Exact Angular Critical-Point Counts for e^z ± 1

For f_a(z) = e^z + a, a = ±1, we count the distinct angular critical points of log |f_a| on every zero-free circle of radius r ≥ 12. Writing N_p(r) for the number of zeros in the disk, we prove that the critical-point count N_osc(r) satisfies 0 ≤ N_osc(r) − 2N_p(r) ≤ 4, with both...

Phase-Resolved Asymptotics for Proximity Functions of Finite Exponential Sums

For a finite exponential sum with constant coefficients, we determine the first-order asymptotic of the proximity function at every fixed value different from its constant term. The leading term is the angular measure of the strict decay region multiplied by the negative logarith...