Generalization of the Ford-Zaharescu Theorem

We derive an asymptotic formula for the sum $$ H = \sum_{0<γ_k\leqslant T,\, 1\leqslant k\leqslant m}h(a_1γ_1+a_2γ_2+\cdots + a_mγ_m), $$ where $a_1, a_2, \ldots, a_m$ are integers whose sum equals zero, $γ_1, \ldots, γ_m$ independently run through the imaginary parts of the non-trivial zeros of the Riemann zeta function, each zero occuring in the sum the number of times of its multiplicity, and the function $h$ belongs to some special class.