Unlocking the Riemann Hypothesis Through Novel Number Theoretic Functions

Mathematical Frameworks from the Paper Applied to the Riemann Hypothesis

This analysis builds upon insights from the paper arXiv:hal-00745160, which introduces specific number theoretic functions and their properties related to factorization and divisor sums.

Framework 1: Factorization and Divisor Sums

• Mathematical Formulation: The paper presents a table detailing numbers with their factorizations, sums of divisors (denoted as sigma(r)), and a function G(r). The Riemann Hypothesis is intricately linked to the distribution of prime numbers, which are fundamental to factorization and divisor sums.
• Connection to Zeta Function: The Euler product formula for the Riemann Zeta function (zeta(s) = Product_{p prime} (1 / (1 - p^-s))) directly connects to factorization properties. Studying sigma(r) and its relationship with prime factors can offer new perspectives on the non-trivial zeros of the zeta function.
• Proposed Theorems:
  • Theorem 1: For a natural number n, there exists a function F(n), structurally similar to G(r), that correlates directly with the non-trivial zeros of the zeta function.
  • Proof Strategy: Employ analytic techniques to express F(n) in terms of the zeta function's zeros and validate this expression against known numerical and theoretical results.

Framework 2: Analytical Properties of G(r)

• Mathematical Formulation: The function G(r), as defined in the paper using divisor sums and natural logarithms, exhibits analytical properties that may be analogous to those of the zeta function at specific points.
• Connection to Zeta Function: The growth conditions and bounds established for G(r) could mirror the analytical behavior of the Riemann zeta function, zeta(s), within the critical strip, offering a novel avenue for investigation.
• Proposed Theorems:
  • Theorem 2: There exists a specific bound B such that G(r) < B implies certain predictable behaviors of zeta(s) on the critical line.
  • Proof Strategy: Develop and rigorously analyze integral bounds for G(r) and establish formal connections to existing bounds and properties within zeta function theory.

Framework 3: Relationships between G(p) and G(2p)

• Mathematical Formulation: The derived inequality, G(2p) / G(p) > 1, suggests a scaling property that could be applicable to the values of the zeta function at scaled arguments.
• Connection to Zeta Function: Similar scaling properties might exist for zeta(s) when examining arguments such as 2s and s, particularly within the critical strip.
• Proposed Theorems:
  • Theorem 3: If the Riemann zeta function, zeta(s), fulfills certain scaling properties analogous to those observed for G(p), then all its non-trivial zeros must lie on the critical line.
  • Proof Strategy: Utilize functional equations and techniques from contour integration to establish these scaling properties within the critical strip for zeta(s).

Novel Approaches Combining Existing Research

Approach 1: Analytic and Combinatorial Analysis of Divisor Sums

• Mathematical Foundation: Define a new function H(n, s) = Sum_d|n d^s, where s is a complex variable. Analyze the analytical and combinatorial properties of H(n, s) in relation to the Riemann zeta function, zeta(s).
• Methodology:
  1. Establish fundamental analytical properties of H(n, s), including its continuity, differentiability, and bounds.
  2. Formally relate these properties to the critical line analysis of zeta(s).
  3. Employ computational methods to approximate and validate theoretical predictions for H(n, s) and its connection to zeta(s).
• Predictions and Limitations:
  • Prediction: The behavior of H(n, s) as s approaches the critical line will closely mimic the critical behaviors of zeta(s), potentially revealing new symmetries or constraints on zero locations.
  • Limitation: A direct, universal correlation may not hold for all natural numbers n; research should initially focus on highly composite numbers or superabundant numbers for clearer and more direct results.

Approach 2: Scale-Invariant Properties of the Zeta Function

• Mathematical Foundation: Explore the hypothesis that scale-invariant properties (similar to the G(p) and G(2p) relationship) can be rigorously defined for zeta(s), perhaps in the form of zeta(ks)/zeta(s) for a real constant k.
• Methodology:
  1. Define and rigorously prove or disprove specific forms of scale invariance for zeta(s) on the critical line.
  2. Utilize these proven properties to constrain the possible locations of zeta function zeros.
  3. Correlate theoretical findings with extensive numerical computations of zeta zeros to validate the approach.
• Predictions and Limitations:
  • Prediction: Discovering and proving specific scale invariance could lead to new symmetries in the zeta function, significantly restricting the locations of its non-trivial zeros to the critical line.
  • Limitation: This approach may necessitate deep, novel insights into the fundamental principles of analytic number theory; extensive computational verification will be crucial for guiding theoretical developments.

Tangential Connections

Connection 1: Statistical Mechanics and Zeta Zeros

• Mathematical Bridge: Apply concepts from statistical mechanics, such as partition functions (which bear a structural resemblance to infinite products found in some zeta function formulations), to model the distribution of the non-trivial zeros of the Riemann zeta function.
• Conjectures:
  • Conjecture 1: The distribution of the non-trivial zeros of the Riemann zeta function can be accurately modeled by a specific statistical ensemble, which predicts their density fluctuations and spacing.
• Computational Experiments: Conduct numerical simulations of proposed statistical ensembles to predict the distribution of zeta zeros, comparing the results rigorously with the known locations of computed zeros.

Detailed Research Agenda

This agenda outlines a structured pathway to rigorously investigate the Riemann Hypothesis using the proposed frameworks and novel approaches.

Conjectures to Prove:

• Establish the precise rigorous conditions under which the function H(n, s) reliably mimics the behavior of the Riemann zeta function, zeta(s), particularly near the critical line.
• Prove the existence and properties of scale invariance for zeta(s) as suggested by the G(p) relationships.

Mathematical Tools Required:

• Advanced techniques from analytic number theory, including complex analysis, Fourier analysis, and functional equations.
• Methods from statistical mechanics and probability theory for modeling distributions.
• High-performance computational simulations for numerical validation and exploration.

Potential Intermediate Results:

• Rigorous verification of the predicted analytical properties of H(n, s) for specific classes of numbers (e.g., highly composite numbers, superabundant numbers).
• Identification of specific functional equations or identities for zeta(s) that demonstrate scale-invariant properties.
• Development of a statistical model that accurately predicts the average spacing or density of zeta zeros within certain bounds.

Logical Sequence of Theorems to be Established:

1. Initial theorems proving the continuity and establishing tight bounds for the function H(n, s).
2. Theorems establishing scaling properties for H(n, s) and exploring their connection to the critical line.
3. Theorems formally correlating the analytical behavior of H(n, s) and the proposed scale invariance of zeta(s) to the location of its non-trivial zeros.
4. A final theorem that, based on the preceding results, provides a definitive statement on the critical line hypothesis for the Riemann zeta function.

Explicit Example of How the Approach Would Work on Simplified Cases:

A detailed analysis could be performed for n = 5040, a highly composite number mentioned in the source paper (arXiv:hal-00745160). This would involve:

• Calculating H(5040, s) and G(5040) precisely.
• Investigating how the properties of G(5040) (e.g., G(5040) >= e^gamma) translate into conditions for H(5040, s).
• Numerically evaluating H(5040, s) along the critical line (s = 1/2 + it) and comparing its behavior to that of zeta(s) to identify specific correlations and convergence properties in a simplified, yet highly relevant, setting.