Lehmer pairs and binomial series

The Hardy function $Z(t)=\zeta (1\mathrm{/}2+it)e^{i\theta (t)}$ takes real values for real $t$ and its real zeros are zeros $\zeta (s)$ on the critical line $1\mathrm{/}2+it$. After discovering the critical value of the local maximum in 1956, Lehmer formulated the assumption that the Hardy function could have a negative local maximum or a positive local minimum. In the paper the Generalized Hardy function is defined as the real part of the Hardy function on any line ${\alpha }_{\nu }+it$ parallel to the critical line $1\mathrm{/}2+it$ $$Z_{{\alpha }_{\nu }}(t)=Re\zeta ({\alpha }_{\nu }+it)e^{i\theta (t)}$$ and established an distinct relationship between the zeros of the $\cos \theta (t)$ function and the zeros of the Generalized Hardy function. $$\mathrm{\forall }\mathrm{\Delta }{T}_{\lambda }=(t_{\lambda },t_{\lambda +1}],t_{\lambda }=2\pi {\lambda }^2,\lambda =1,2,3\mathrm{.}\mathrm{.}\mathrm{.}$$ $$\mathrm{\exists }{A}_{\lambda }:\mathrm{\forall }{\widehat{\alpha }}_{\lambda }>A_{\lambda }$$ $$\mathrm{\mid }\cos \theta (t)-Z_{{\widehat{\alpha }}_{\lambda }}(t)\mathrm{\mid }<\epsilon (A_{\lambda }),t\in \mathrm{\Delta }{T}_{\lambda }$$ Then the binomial series is used to establish a relationship between the values of the Generalized Hardy function on any two lines ${\alpha }_{\nu }+it$ and ${\alpha }_{\nu +1}+it$ parallel to the critical line. Thus, by induction between values $\sigma =1\mathrm{/}2$ and $\sigma ={\widehat{\alpha }}_{\lambda }>A_{\lambda }$ $${\alpha }_1^{(\lambda )}<{\alpha }_2^{(\lambda )}<{\alpha }_3^{(\lambda )}<\mathrm{.}\mathrm{.}\mathrm{.}<{\alpha }_{\nu }^{(\lambda )}<\mathrm{.}\mathrm{.}\mathrm{.}<{\alpha }_{{\mu }_{\lambda }}^{(\lambda )}$$ an distinct relationship has been established between the zeros of the function $\cos \theta (t)$ and the zeros of the Hardy function.