Harmonic Fluctuations and the Critical Line: A New Perspective on Zeta Zeros

Executive Summary

The Riemann Hypothesis (RH) remains the most significant unsolved problem in pure mathematics, asserting that all non-trivial zeros of the Riemann zeta function, denoted as ζ(s), lie on the critical line where the real part of s is 1/2. The source paper arXiv:2105.02492, titled "On the zeros of the Riemann zeta-function" by Nikolai A. Rozanov, proposes a framework for verifying this hypothesis by analyzing the integral representation of the zeta function and the behavior of its argument along the boundaries of the critical strip.

The key insight involves the application of the argument principle to a specific rectangular contour in the complex plane. By establishing rigorous bounds on the function S(T) — which represents the fluctuation of the argument of the zeta function on the critical line — the research attempts to demonstrate that the existence of zeros off the critical line would lead to a mathematical contradiction regarding the known growth rates of the zeta function. This approach is promising because it bridges the gap between local zero distribution and global analytic properties.

Introduction

The distribution of prime numbers is inextricably linked to the zeros of the Riemann zeta function, ζ(s). Defined for Re(s) > 1 by the Dirichlet series ∑ n^-s, the function possesses an analytic continuation to the entire complex plane with a simple pole at s = 1. The functional equation, established by Riemann, relates ζ(s) to ζ(1-s), implying a symmetry around the critical line Re(s) = 1/2.

The central challenge in analytic number theory is to prove that no zeros exist in the region 1/2 < Re(s) < 1. While numerical evidence confirms the first 10¹³ zeros lie exactly on the line, a general proof has remained elusive. Recent efforts have shifted toward refined estimates of the zero-counting function N(T) and its error term S(T). The analysis in arXiv:2105.02492 contributes to this discourse by re-examining the integral of the logarithmic derivative of ζ(s).

Mathematical Background

To evaluate the claims in arXiv:2105.02492, we must define the primary mathematical objects. The Riemann Xi function, ξ(s), is defined to satisfy the functional equation ξ(s) = ξ(1-s) and is given by ξ(s) = (1/2) s (s-1) π^-s/2 Γ(s/2) ζ(s). The zeros of ξ(s) correspond exactly to the non-trivial zeros of ζ(s).

According to the argument principle, the number of zeros N(T) in the rectangle 0 < Re(s) < 1 and 0 < Im(s) < T is determined by the change in the argument of ξ(s). This can be decomposed into an asymptotic main term and a fluctuating error term S(T) = (1/π) arg ζ(1/2 + iT). The behavior of S(T) is critical; if RH is true, S(T) grows very slowly, specifically S(T) = O(log T / log log T).

Main Technical Analysis: Integral Constraints and Zero Distribution

The core of the analysis in arXiv:2105.02492 rests on the evaluation of the integral of log ζ(s) along a contour that avoids the pole at s=1. The paper utilizes the fact that the change in the argument of ζ(s) along the path from infinity to 1/2 + iT determines the value of S(T).

The Argument Principle Application

By integrating around a rectangle with vertices at 1/2 + iT₁ and σ + iT₂ (where σ > 1), the research establishes a relationship between the real part of the integral and the distribution of zeros. If there were a zero ρ = β + iγ with β > 1/2, it would contribute a singularity that shifts the argument by 2π.

Bounds on the Function S(T)

Rozanov argues that the integral of S(t) from 0 to T remains bounded in a manner that is inconsistent with the presence of "off-line" zeros. Specifically, the paper investigates the integral J(T) = ∫ log |ζ(1/2 + it)| dt. Under the Riemann Hypothesis, this integral is constrained by the growth of the Gamma function via the functional equation. The author demonstrates that if a zero existed at 1/2 + ε + iT, the resulting spike in the logarithmic derivative would force J(T) to exceed the theoretical maximum allowed by the Lindelöf bound.

Novel Research Pathways

The methodology in arXiv:2105.02492 suggests several concrete avenues for future investigation:

• Pathway 1: Extension to the Selberg Class. Apply the Rozanov integral transform to the Ramanujan L-function or Dirichlet L-functions to see if argument constraints are universal.
• Pathway 2: Refinement of Littlewood’s Lemma. Use the specific bounds on S(T) derived in the paper to refine the error term in Littlewood's Lemma, potentially showing the number of zeros with real part greater than 1/2 is zero.
• Pathway 3: Spectral Interpretation. Connect the integral of log ζ(s) to the trace formula of a hypothetical Hilbert-Polya operator to determine if off-line zeros correspond to non-unitary transformations.

Computational Implementation

The following Wolfram Language implementation demonstrates the behavior of the Z-function and the Xi function zeros as discussed in the source paper.

(* Section: Visualization of Riemann Z-function and Xi Function Zeros *)
(* Purpose: Demonstrate the location of zeros on the critical line *)

Module[{tMax, zPlot, xiPlot, zeros},
  tMax = 50;
  
  (* Define the Hardy Z-function: Z(t) = exp(i theta(t)) zeta(1/2 + i t) *)
  zPlot = Plot[RiemannZ[t], {t, 0, tMax}, 
    PlotStyle -> Blue, 
    PlotRange -> All, 
    Frame -> True, 
    FrameLabel -> {"t", "Z(t)"}, 
    PlotLabel -> "Hardy Z-function on the Critical Line"];

  (* Define the Riemann Xi function on the critical line *)
  xiPlot = Plot[Log[Abs[RiemannXi[1/2 + I t]]], {t, 0, tMax}, 
    PlotStyle -> Red, 
    Frame -> True, 
    FrameLabel -> {"t", "Log|Xi(1/2 + it)|"}, 
    PlotLabel -> "Riemann Xi Function Magnitude"];

  (* Find the first few zeros of the Zeta function on the critical line *)
  zeros = Table[Im[ZetaZero[n]], {n, 1, 5}];
  
  Print["First five non-trivial zeros (imaginary parts):"];
  Print[zeros];
  
  GraphicsColumn[{zPlot, xiPlot}]
]

Conclusions

The analysis of the Riemann zeta function through its argument and integral representations offers a compelling perspective on the Riemann Hypothesis. By focusing on the constraints that the functional equation places on the growth of S(T), the research in arXiv:2105.02492 suggests that the existence of zeros off the critical line is incompatible with the analytic requirements of the zeta function’s phase relationship. Future work should focus on the rigorous verification of these integral bounds using higher-order terms of the Stirling approximation.

References

arXiv:2105.02492: Rozanov, N. A. "On the zeros of the Riemann zeta-function." (2021).

Titchmarsh, E. C. "The Theory of the Riemann Zeta-Function." Oxford University Press, 2nd ed., 1986.