Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analysis of the Riemann Zeta Function via Recursive Taylor Expansions

Yunwei Bai

Abstract

We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line $\text{Re}(s) = 0.5$. By defining a recursive path of Taylor expansions originating from the domain of absolute convergence, we translate the zeta function towards the critical region, which is an easy-to-understand form of the analytical continuation. We then assume the existence of off-critical-line (off-line) zeros, which exist in pairs symmetric by the critical line. If the pairs are zero in value, their real and imaginary components differences should be both zero. However, we derive a contradiction against the assumption via basic logical deduction, proving the non-existence of the off-line zeros.

Analysis of the Riemann Zeta Function via Recursive Taylor Expansions

Abstract

We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line . By defining a recursive path of Taylor expansions originating from the domain of absolute convergence, we translate the zeta function towards the critical region, which is an easy-to-understand form of the analytical continuation. We then assume the existence of off-critical-line (off-line) zeros, which exist in pairs symmetric by the critical line. If the pairs are zero in value, their real and imaginary components differences should be both zero. However, we derive a contradiction against the assumption via basic logical deduction, proving the non-existence of the off-line zeros.
Paper Structure (40 sections, 59 equations, 22 figures)

This paper contains 40 sections, 59 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Preliminary: The Riemann's Hypothesis
  3. Notation and Ordering Conventions
  4. Indices.
  5. Componentwise binomial.
  6. The Chained Disk Formulation
  7. Starting Point
  8. Path Steps
  9. Base Derivatives at $a_0 = 2+2i$
  10. Recursive Zeta Function Representation
  11. Derivation of RealDiff and ImagDiff
  12. Definitions of Exact Differences
  13. Updated RealDiff Formula
  14. Derivation of ImagDiff
  15. RealDiff and ImagDiff Simplification
  16. ...and 25 more sections

Figures (22)

  • Figure 1: We start from the location (2, 2) in the complex plane and taylor expand the zeta function to obtain a disk of radius 0.5 that does not touch the pole. We then always shift the base disk upward by 0.5 in distance, before finally shifting the disk leftward by three 0.5-distance steps and then a step less than or equal to 0.5, and the final vertical shift can be an arbitrary value less than or equal to 0.5 as well. All the disks are chained via overlapping points; the top tip of the previous disk becomes the center of the new disk. By starting at (2, 2) and constraining the disk radius to 0.5, we avoid the pole at (1, 0). Through the recursive formulation, we derive an alternative form of analytical continuation that covers the entire upper part above Im(s)$= 2$ of the critical strip. Note that all graphs are non-exact in details and only serve the illustration purpose.
  • Figure 2: Two symmetric points p1 and p2 shifted off the critical line by the same horizontal distance (i.e., d1 = d2). Their real/imaginary differences are evaluated.
  • Figure 3: Type B absolute graph, a positive, decreasing curve.
  • Figure 4: Type C absolute graph, a positive curve that initially increases until a peak turning point TP-a, and then decreases, in which there is an inflection point TP-b.
  • Figure 5: Type D absolute graph. Same as Type C except the existence of the inflection point TP-a.
  • ...and 17 more figures

Theorems & Definitions (1)

  • Definition 4.1: Coefficients at stage g