Lehmer pairs and binomial series

TL;DR

The work develops a framework to relate zeros of the Hardy function on the critical line to zeros of $\cos\theta(t)$ by introducing a Generalized Hardy function $Z_{α\nu}$ on lines $α\nu+it$ parallel to the critical line. Using analytic continuation of $ζ(s)$, a binomial-series expansion, and a sequence of zero-transformations across increasing $α\nu$, the authors prove that zeros of $Z_{α\nu}$ converge to the zeros of $\cos\theta(t)$ and that all such zeros are real. They further establish monotone, real zero-sequences $\delta_{α^{(λ)}_ν,k,n}$ and show a structured mapping between the two zero-sets, linking the distribution of $\cos\theta(t)$ zeros to the zeros of the Hardy function and, by extension, to the zeros of the Riemann zeta function. The results provide a rigorous framework challenging Lehmer’s conjecture about negative local extrema of the Hardy function and offer a deeper, count-based connection between $\cos\theta(t)$ zeros and $ζ$-zeros.

Abstract

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.

Figures (7)

• Figure 1: Lehmer pair
• Figure 2: Hardy function $\zeta(1/2+it)e^{i\theta(t)}$
• Figure 3: Lehmer pair list
• Figure 4: Polygon
• Figure 5: Inlined polygons

Theorems & Definitions (2)

• Theorem 1
• Definition 1