Unlocking Riemann's Secrets: New Pathways from Integral Symmetries and Functional Relations

Introduction

The Riemann Hypothesis stands as one of the most significant unsolved problems in mathematics, positing that all non-trivial zeros of the Riemann zeta function lie on the critical line with a real part of 1/2. A recent paper, arXiv:2007.07752, introduces several intriguing mathematical frameworks that could provide fresh perspectives and lead to a proof of this long-standing conjecture. This article synthesizes these insights into concrete research pathways.

Unveiling Zeros Through Functional Symmetry

A key insight from the paper lies in the symmetrical relationships observed between certain functions closely related to the Riemann zeta function, specifically q(s) and λ(s).

• Mathematical Foundation: The paper highlights relations such as:
  lims → se (1 - 2-s)ζ(s) / (2-s)ζ(s) = lims → se (1 - 2s̄-1)ζ(1 - s̄) / (2s̄-1)ζ(1 - s̄)
  and its simplification to:
  lims → se (1 - 2-s) / 2-s = lims → se (1 - 2s̄-1) / 2s̄-1
  where se denotes a zero of ζ(s). This simplifies to 2se - 1 = 21-s̄e - 1.
• Direct Implication for Zeros: The relation 2se = 21-s̄e directly implies that se = 1 - s̄e + 2πik / ln(2) for some integer k. For se to be a complex number with a real part, this equation forces 2σe = 1, meaning σe = 1/2, provided k=0.
• Research Challenge: The crucial next step is to rigorously demonstrate that this limiting process and the cancellation of ζ(s) terms are valid for all non-trivial zeros, including those of higher multiplicity, and that the k=0 case is the only relevant one for non-trivial zeros. This would provide a direct proof that all non-trivial zeros lie on the critical line.

Exploring Zeros via Novel Integral Representations

Another powerful framework presented in the paper, arXiv:2007.07752, involves an integral representation of the zeta function.

• The Integral Form: The paper introduces the representation:
  ζ(s) = (sin(πs / 2) / ((s-1)π1-s)) ∫0∞ x1-s (1/sin2(ix) + 1/x2) dx
• Function g(x) Analysis: The kernel g(x) = 1/sin2(ix) + 1/x2 requires detailed study. Key properties to investigate include its analytic behavior, asymptotic limits as x → 0 and x → ∞, and its relation to known special