On a conjecture of Shanks (Q5894936)

Let \[ \phi_n:=n-\frac32-\pi^{-1}\theta(\gamma_n), \] where \(\gamma_n\) denotes the ordinate of the \(n\)-th zero of the Riemann zeta-function on the critical line. Shanks gave some numerical data concerning the average of the sum \(\sum_{n=1}^K\phi_n\). He conjectured that \[ \lim_{K\to\infty}\frac{\sum_{n=1}^K\phi_n}{K}=0 \] and that the stronger estimate \[ \lim_{K\to\infty}\frac{\sum_{n=1}^K\phi_n} {K^{\frac12}}=0 \] may hold. The author answers both of Shanks's conjectures in the affirmative by proving that if \(\alpha>0\) then \[ \lim_{K\to\infty}\frac{\sum_{n=1}^K\phi_n}{K^\alpha}=0. \]