Almost all of the zeros of the Riemann zeta-function are on the critical line

TL;DR

The paper proves that almost all nontrivial zeros of the Riemann zeta-function lie on the critical line by refining the Levinson-Conrey approach through a translation of the integration segment, controlled by $\Delta\sigma=\frac{\alpha}{\log T}$. It builds a mollified Dirichlet-polynomial framework using a Schwarz-Christoffel majorant and a translated auxiliary function $G(s)$, then employs Selberg–Jutila zero-density bounds and Lester’s distribution theorems to approximate the key quantities on a large-measure set of $t$. A combination of translation lemmas, Selberg-type approximations, Runge polynomial techniques, and Schwarz-Christoffel mapping underpins the construction, enabling tight control of the exceptional set and leading to $\kappa=1$. The results advance the method of mollified mean values in analytic number theory by exploiting segment translations and precise Dirichlet-polynomial approximations, with implications for the distribution of zeta zeros and related moments.

Abstract

This is a reworked version of the paper. An idea that allows us to circumvent limitations of previous approaches is not to apply arithmetic-geometric mean inequality and the second moment asymptotics to the entire segment $[1/2-a/\log T+iT,1/2-a/\log T+i2T]$ but use them on a subset only, and use the integral of logarithm of the mollified function on the complement. Ultimately, the result depends on the exponent in the zero-density estimate near the critical line, which leads to the relation between the magnitude of $\widetilde{V}^{1/2}$ and the measure of the exceptional set in Theorem 4, Section 2.3. The exponents of Jutila and Conrey are enough for our purposes. We provide more details on an effective approximation of $1/z$ using the Schwarz-Christoffel mapping. This is needed in the construction of the mollifier. One observation on why the approach is feasible is that the functional equation established in the paper allows one to shift the segment of integration $[1/2-a/\log T+iT,1/2-a/\log T+i2T]$ to $[1/2+A/\log T+iT,1/2+A/\log T+i2T]$. A result of Selberg allows for a proof that almost all of zeros of our integrand are to the left of the shifted segment.

Figures (5)

• Figure 1: The function $a(x)e^{-x}=\frac{e^{\alpha/2}}{2}\left(1+\tanh\left(\frac{1}{4}(2\alpha x/(\alpha-R)-\alpha)\right)\right)e^{-x}$ defining the coefficients of the Dirichlet polynomial $\sum_{l\leqslant T}a\left(\frac{(\alpha-R)\log l}{\log T}\right)\exp\left(-\frac{(\alpha-R)\log l}{\log T}\right)l^{-(1/2+it)}$ (dash), and the approximation $\frac{e^{-x}}{s_6(1)}s_6(x+1)$ (solid)
• Figure 2: The function $1+\tanh\left(\frac{1}{4}(2\alpha x/(\alpha-R)-\alpha)\right)$ and the (shifted) Laguerre polynomial approximation $s_6(x+1)$
• Figure 3: The absolute values of the function $\lambda\left(\frac{1}{2}+\frac{\alpha-R}{\log T}+it\right)$ (circles) for $t\in [T/2,T]$ and the majorant $A\left(\frac{1}{2}+\frac{\alpha-R}{\log T}+it\right)$ (solid circles) without lower order terms
• Figure 4: The sets ${\mathcal{C}}'$, ${\mathcal{C}}"$ and ${\mathcal{C}}$
• Figure 5: The set ${\mathcal{C}}$ and the image of $\varphi(z)$

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4: Analytic continuation of $g_{\alpha,T}(s)$
• Lemma 5: Approximate equation for $g_{\alpha,T}(s)$
• Lemma 6: Approximation to $H(s)g_{\alpha,T}(s)$ by a sum of the odd derivatives of the $\xi$ function