Lehmer pairs and binomial series

Abstract

The Hardy function $Z(t)=\zeta(1/2+it)e^{i\theta(t)}$ takes real values for real $t$ and its real zeros are zeros $\zeta(s)$ on the critical line $1/2+it$. 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 $1/2+it$ $$Z_{\alpha_\nu}(t)=Re\ \zeta(\alpha_\nu+it)e^{i\theta(t)}$$ and established an distinct relationship between the zeros of the $\cos\theta(t)$ function and the zeros of the Generalized Hardy function. $$\forall \Delta T_\lambda=(t_\lambda, t_{\lambda+1}],\ t_\lambda=2\pi\lambda^2,\ \lambda=1,\ 2,\ 3\ ...$$ $$\exists A_\lambda:\forall \hat\alpha_\lambda>A_\lambda$$ $$|\cos\theta(t) -Z_{\hat\alpha_\lambda}(t)|<\epsilon(A_\lambda),\ t\in \Delta T_\lambda$$ 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 $\sigma=1/2$ and $\sigma=\hat\alpha_\lambda>A_\lambda$ $$\alpha^{(\lambda)}_1<\alpha^{(\lambda)}_2<\alpha^{(\lambda)}_3<...<\alpha^{(\lambda)}_{\nu}<...<\alpha^{(\lambda)}_{\mu_\lambda}$$ an distinct relationship has been established between the zeros of the function $\cos\theta(t)$ and the zeros of the Hardy function.