Average of Hardy's function at Gram points

Let $Z(t)=χ^{-1/2}(1/2+it)ζ(1/2+it)=e^{iθ(t)}ζ(1/2+it)$ be Hardy's function and $g(n)$ be the $n$-th Gram points defined by $θ(g(n))=πn$. Titchmarsh proved that $\sum_{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log^{3/4}N)$ and $\sum_{n \leq N} Z(g(2n+1)) =-2N+O(N^{3/4}\log^{3/4}N)$. We shall improve the error terms to $O(N^{1/4}\log^{3/4}N \log\log N)$.

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