  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 points 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 $\sum_{n \leq N} Z(g(2n+1))
=-2N+O(N^{3/4}\log^{3/4}N)$. We shall improve the error terms to
$O(N^{1/4}\log^{3/4}N \log\log N)$.
