More than five-twelfths of the zeros of $ζ$ are on the critical line

TL;DR

This work develops an unconditional framework to study the zeros of the Riemann zeta-function via mollification of a perturbed zeta operator $V(s)=oldsymbol{ ext ζ}(s)+ extstyleigoplus_{k e0}oldsymbol{ ext{λ}}_koldsymbol{ ext ζ}^{(k)}(s)/(oldsymbol{ ext log}T)^k$. Harnessing the autocorrelation of ratios of ζ-functions, the authors compute the twisted second moment with a general Dirichlet-polynomial mollifier of degree $d$, deriving a detailed main-term structure that is then analyzed via Dirichlet convolutions and Bell-polynomial combinatorics. They extend Feng’s mollifier framework to a full general $d$ and provide an explicit, computationally tractable recipe to extract main terms from high-dimensional contour integrals, including a thorough treatment of square-free versus non-square-free contributions and their impact on the mean value. This approach yields unconditional lower bounds on the proportion of nontrivial zeros on the critical line, surpassing five-twelfths (with their interim bound around $0.417293$) and offering a flexible, scalable method for longer mollifiers and potential extensions to other L-functions. Overall, the paper advances both the analytic technique and the numerical optimization of mollified moments, bridging rigorous control with predictions from ratios conjectures and random-matrix heuristics.

Abstract

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(μ\star Λ_1^{\star k_1} \star Λ_2^{\star k_2} \star \cdots \star Λ_d^{\star k_d})$ is computed unconditionally by means of the autocorrelation of ratios of $ζ$ techniques from Conrey, Farmer, Keating, Rubinstein and Snaith (2005), Conrey, Farmer and Zirnbauer (2008) as well as Conrey and Snaith (2007). This in turn allows us to describe the combinatorial process behind the mollification of \[ ζ(s) + λ_1 \frac{ζ'(s)}{\log T} + λ_2 \frac{ζ''(s)}{\log^2 T} + \cdots + λ_d \frac{ζ^{(d)}(s)}{\log^d T}, \] where $ζ^{(k)}$ stands for the $k$th derivative of the Riemann zeta-function and $\{λ_k\}_{k=1}^d$ are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (2017), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

Figures (11)

• Figure 2.1: $B_{3,3}(x_1,x_2,x_3)$ (extreme left), $B_{3,2}(x_1,x_2,x_3)$ (3 middle diagrams) and $B_{3,1}(x_1,x_2,x_3)$ (extreme right).
• Figure 2.2: $B_{4,k}(x_1,x_2,x_3,x_4)$ for $k=1,2,3,4$.
• Figure 3.1: Pictorial representation of \ref{['bellp1']}.
• Figure 3.2: Pictorial representation of \ref{['bellp2']}.
• Figure 3.3: Pictorial representation of \ref{['bellp3']}.

Theorems & Definitions (13)

• Conjecture 1.1: Conrey, Farmer and Zirnbaeur, 2006
• Theorem 1.1: Conrey and Snaith, 2007
• Conjecture 1.2: CFKRS, 2005
• Theorem 1.2
• Remark 1.1
• Theorem 4.1
• proof : Proof of Theorem \ref{['meanvalueintegral']}
• Theorem 7.1
• Remark 7.1
• Lemma 7.1