Average of Hardy's function at Gram points

Abstract

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