$, let $F_{i}, G_{i}: [0,+\infty) \to {\mathbb{R}}$ be fixed smooth compactly supported functions. Assume one of the following hypotheses:  -   (Elliott-Halberstam) There exists a fixed $0 < \vartheta < 0$ such that $\operatorname*{EH}[\vartheta]$ holds, and such that $$