Sums of roots of unity vanishing modulo a prime (Q1849548)

Let \(\zeta_0 =1, \zeta_1, \dots,\zeta_{k-1}\) be \(k\) pairwise different roots of unity and \(0 \neq a_i \in \mathbb Q\) such that \(\sum_{i=0}^{k-1} a_i \zeta_i = 0\) but no proper subsum of this sum vanishes. \textit{J. H. Conway} and \textit{A. J. Jones} [Acta Arith. 30, 229--240 (1976; Zbl 0349.10014)] proved that in this situation \(Q = \text{lcm} \{ \text{ord} (\zeta_i) \mid 0 \leq i \leq k-1 \}\) must be squarefree and \(\sum_{p \mid Q} (p-2) \leq k-2\). In the present paper it is shown that the same result holds true when instead of being \(0\) one supposes that the above sum, but no proper subsum, is congruent to \(0\) modulo some rational prime \(l\).