Average of Hardy’s function at Gram points

Let $Z(t)=\chi ^{-1/2}(1/2+it)\zeta (1/2+it)=e^{i\theta (t)}\zeta (1/2+it)$ be Hardy’s function and $g(n)$ be the $n$th Gram point defined by $\theta (g(n))=\pi n$. Titchmarsh proved that $$ \sum _{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log ^{3/4}N) $$ and $$ \