Notes on Conjectures of Zhi-wei Sun

Conjecture 1 (1988-04-23). Let a0, . . . , an−1, b0, . . . , bn−1 ∈ N. Suppose that ∑n−1 r=0 are 2πir/n = ∑n−1 r=0 bre , and that the least prime divisor p = p(n) of n is greater than |{0 6 r < n : ar 6= 0}| and |{0 6 r < n : br 6= 0}|. Then ar = br for all r ∈ R(n) = {0, 1, . . . , n− 1}. Remark 1. M. Newman [Math. Ann. 1971] showed that if c0, . . . , cn−1 ∈ Q, ∑n−1 r=0 cre 2πir/n = 0 and |{0 6 r < n : cr 6= 0}| < p(n), then c0 = · · · = cn−1 = 0. Conjecture 2 (1988-04-23). For s = 1, . . . , k let ψs : Z → C be an arithmetical function with period ns ∈ Z. If ψ = ψ1 + · · ·+ ψk is not the zero function, then