Lehmer pairs and binomial series
The Hardy function $Z(t)=ζ(1/2+it)e^{iθ(t)}$ takes real values for real $t$ and its real zeros are zeros $ζ(s)$ 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 $α_ν+it$ parallel to the critical line $1/2+it$ $$Z_{α_ν}(t)=Re\ ζ(α_ν+it)e^{iθ(t)}$$ and established an distinct relationship between the zeros of the $\cosθ(t)$ function and the zeros of the Generalized Hardy function. $$\forall ΔT_λ=(t_λ, t_{λ+1}],\ t_λ=2πλ^2,\ λ=1,\ 2,\ 3\ ...$$ $$\exists A_λ:\forall \hatα_λ>A_λ$$ $$|\cosθ(t) -Z_{\hatα_λ}(t)|<ε(A_λ),\ t\in ΔT_λ$$ Then the binomial series is used to establish a relationship between the values of the Generalized Hardy function on any two lines $α_ν+it$ and $α_{ν+1}+it$ parallel to the critical line. Thus, by induction between values $σ=1/2$ and $σ=\hatα_λ>A_λ$ $$α^{(λ)}_1<α^{(λ)}_2<α^{(λ)}_3<...<α^{(λ)}_ν<...<α^{(λ)}_{μ_λ}$$ an distinct relationship has been established between the zeros of the function $\cosθ(t)$ and the zeros of the Hardy function.