Unlocking the Riemann Hypothesis: Novel Research Pathways via Limit Theorems and Divisor Functions

Research Pathways for Addressing the Riemann Hypothesis

Mathematical Frameworks

Framework 1: Connection to Divisor Functions

• Mathematical Formulation: Considering the sum of positive divisors function $\sigma(n)$, we can explore its relation to the distribution of zeros of the Riemann zeta function. We can define a new function derived from $\sigma(n)$, such as $\Sigma(s) = \sum_{n=1}^\infty \frac{\sigma(n)}{n^s}$ for $\Re(s) > 1$.
• Theoretical Construction: Develop and analyze properties of $\Sigma(s)$, especially in relation to $\zeta(s)$.
• Connection to Zeta Function: Investigate how $\Sigma(s)$ could provide new insights into the analytic continuation and zero distribution of $\zeta(s)$.

Framework 2: Asymptotic Limits Involving n and k

• Mathematical Formulation: Analyze and refine the limit expressions given in arXiv:XXXX.XXXXX, such as $\lim _{n \rightarrow \infty} \frac{1}{k-1} \geq n$ and its implications on the growth rates of sequences related to the primes.
• Specific Theorems: Propose theorems that link these asymptotic behaviors to the density and spacing of non-trivial zeros of $\zeta(s)$.
• Connection to Zeta Function: Explore potential correlations between these limits and the hypothetical regularity or irregularity in the spacing of zeros on the critical line.

Framework 3: Utilization of Infinite Product and Root Expressions

• Mathematical Formulation: Explore the implications of expressions like $1=\lim _{n \longrightarrow \infty} \sqrt[n H H_{n}]{n H_{n}}$ within the context of $\zeta(s)$.
• Specific Theorems: Develop theorems regarding the convergence properties of these sequences and their relation to the Euler product formula of $\zeta(s)$.
• Connection to Zeta Function: Analyze how these infinite product formulations relate to the properties of $\zeta(s)$, especially around its zeros.

Novel Approaches Combining Existing Research

Approach 1: Divisor Function and Analytic Properties

• Mathematical Foundation: Define $\Sigma(s)$ as above and relate it to $\zeta(s)$ via functional equations or convolution types.
• Step-by-Step Methodology:
  1. Prove analytic continuation of $\Sigma(s)$.
  2. Explore its zero distribution potentially mirroring that of $\zeta(s)$.
  3. Use contour integration to connect these properties explicitly to the Riemann Hypothesis.
• Predictions and Limitations: This might reveal new properties of $\zeta(s)$ at its critical zeros. Limitations include potential divergences requiring regularization or renormalization techniques.

Approach 2: Asymptotic Analysis and Zero Spacing

• Mathematical Foundation: Formalize the relationship between the asymptotic limits involving $n$ and $k$ and the distribution of zeros.
• Step-by-Step Methodology:
  1. Establish rigorous asymptotic bounds for sequences related to prime numbers.
  2. Correlate these bounds with estimates on the gaps between zeros of $\zeta(s)$.
  3. Apply statistical mechanics tools to model these distributions.
• Predictions and Limitations: Could predict regular patterns or discrepancies in zero spacings, potentially falsifying the hypothesis. Challenges include justifying the choice of sequences and managing complex asymptotics.

Tangential Connections

Connection 1: Computational Experiments with Divisor Sums

• Mathematical Bridges: Use computational power to evaluate $\Sigma(s)$ for large values and compare with known results of $\zeta(s)$.
• Conjectures: If certain patterns or regularities are observed computationally in $\Sigma(s)$, conjecture formal properties that could be proven analytically.

Connection 2: Infinite Products and Statistical Distribution

• Mathematical Bridges: Relate the infinite product expressions to statistical distributions known in quantum chaos theory, which has been suggested to analogize features of $\zeta(s)$.
• Conjectures and Experiments: Formulate conjectures on the statistical nature of zeros and validate them through numerical simulations of these product forms.

Detailed Research Agenda

1. Conjectures to Prove:
   • Convergence and analytic properties of $\Sigma(s)$ and its relation to $\zeta(s)$.
   • Asymptotic bounds derived from proposed limits and their implications for zero distributions.
2. Mathematical Tools:
   • Complex analysis, especially techniques for handling series and products.
   • Asymptotic analysis and potential use of probabilistic methods.
3. Intermediate Results:
   • Proofs of convergence or specific properties for the new functions and sequences introduced.
   • Empirical data supporting asymptotic conjectures.
4. Sequence of Theorems:
   • Establish basic properties and extensions of $\Sigma(s)$.
   • Link asymptotic behaviors to spacing of zeros.
5. Examples:
   • Start with simplified cases using smaller computational models or special cases of the functions proposed.

This plan provides a structured pathway using both established and innovative mathematical frameworks to explore and potentially prove the Riemann Hypothesis. This research is based on information found in arXiv:hal-01520127.