Unlocking the Riemann Hypothesis: Novel Approaches from Logarithmic Integral Approximations

Research Pathways for Approaching the Riemann Hypothesis Based on Ingested Mathematical Frameworks

Mathematical Frameworks and Their Application to the Riemann Hypothesis

1. Asymptotic Expansion of ali(x):
   • Formulation: The paper presents an asymptotic expansion for ali(e^x):

     \frac{\operatorname{ali}\left(e^{x}\right)}{x e^{x}}=1+\sum_{n=1}^{N} \frac{P_{n-1}(\log x)}{x^{n}}+\theta \cdot 20 \cdot\left(\frac{N}{e \log N}\right)^{N} \cdot \frac{\log ^{N} x}{x^{N+1}}, \quad\left(x>z_{N}\right)

     where P_n(y) are polynomials. This framework suggests exploring the distribution of the roots of these polynomials P_n(y).
   • Theorem to Construct: Establish a theorem relating the roots of P_n(y) to the non-trivial zeros of the Riemann zeta function. Specifically, show that the roots of P_n(y) accumulate near log(|ρ|) where ρ are the non-trivial zeros of ζ(s).
   • Connection to Zeta Function: If the real parts of all non-trivial zeros are 1/2, then the roots of P_n(y) should exhibit a specific symmetry around log(sqrt(abs(ρ))). This symmetry would be a strong indicator supporting the RH.
2. Polynomial Representation:
   • Formulation: The paper defines polynomials P_n(y) as:

     P_{n}(y)=\frac{(-1)^{n+1}}{n!}\left(a_{n, 0} y^{n}-a_{n, 1} y^{n-1}+\cdots(-1)^{n} a_{n, n}\right)=\frac{(-1)^{n+1}}{n!} \sum_{k=0}^{n}(-1)^{k} a_{n, k} y^{n-k}, \quad(n \geq 1)

     The coefficients a_{n,k} are of interest. The paper contains a table with numerical values that could provide a clue.
   • Lemma to Construct: Prove a lemma that links the coefficients a_{n,k} to the derivatives of the Riemann zeta function at s=1. This could involve complex analysis techniques, contour integration, and residue calculus.
   • Connection to Zeta Function: The RH is intimately tied to the behavior of the zeta function and its derivatives. Establishing a direct link between a_{n,k} and ζ'(1), ζ''(1), etc., could provide a new avenue for exploration.
3. Improved Approximation:
   • Formulation: The paper also gives an alternative approximation of ali(e^x) using polynomials U_k(log x):

     \frac{e^{x} \log \operatorname{ali}\left(e^{x}\right)}{\operatorname{ali}\left(e^{x}\right)}=1+\sum_{k=1}^{N+1} \frac{U_{k}(\log x)}{x^{k}}+\boldsymbol{\mathcal { O }}\left(\frac{\log ^{N+2} x}{x^{N+2}}\right)

     This suggests investigating the difference between the approximations using P_n and U_k.
   • Theorem to Construct: Prove a theorem demonstrating that the difference between the roots of P_n(log x) and U_n(log x) converges to zero as n approaches infinity if and only if the Riemann Hypothesis is true.
   • Connection to Zeta Function: The convergence rate would encode information about the distribution of the zeros of the zeta function.

Novel Approaches Integrating Existing Research

1. Hybrid Approximation and Functional Equation: Combine the asymptotic expansion from the paper with the functional equation of the Riemann zeta function.
   • Mathematical Foundation:
     1. Start with the asymptotic expansion:

        \frac{\operatorname{ali}\left(e^{x}\right)}{x e^{x}}=1+\sum_{n=1}^{N} \frac{P_{n-1}(\log x)}{x^{n}}+\mathcal{O}\left(\frac{\log ^{N} x}{x^{N+1}}\right)

     2. Relate ali(x) to the Riemann zeta function via its integral representation. Although there isn't a direct closed-form relation, the integral representation of ζ(s) involves a term similar to ali(x).
     3. Apply the functional equation of the Riemann zeta function:

        \zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)

     4. Express the asymptotic expansion in terms of ζ(s) and ζ(1-s). This is the most challenging step and likely requires approximating the Gamma function and sine function within the expansion.
   • Methodology:
     1. Express ali(x) as a Mellin transform. Mellin transforms are often used to analyze asymptotic expansions.
     2. Use the inverse Mellin transform to relate ali(x) to the Riemann zeta function. This requires careful contour integration.
     3. Apply the functional equation to the Riemann zeta function. This introduces ζ(1-s) into the equation.
     4. Analyze the resulting equation to find conditions under which the real parts of the zeros of ζ(s) must be 1/2. This may involve showing that if the real parts are not 1/2, the asymptotic expansion will diverge.
   • Prediction: This approach might reveal a relationship between the growth rate of the polynomials P_n(log x) and the location of the zeros of the zeta function. If the RH holds, the growth rate should be constrained.
   • Limitations: The primary limitation is the difficulty in directly relating ali(x) to the Riemann zeta function and handling the Gamma and sine functions within the asymptotic expansion. Overcoming this might involve using Stirling's approximation for the Gamma function and carefully analyzing the behavior of the sine function near the critical line.
2. Coefficient Analysis and Trace Formulae: Combine the coefficient analysis of the polynomials P_n(y) with existing trace formulae (e.g., the Guinand-Weil explicit formula).
   • Mathematical Foundation:
     1. Analyze the coefficients a_{n,k} of the polynomials P_n(y). Look for patterns or relationships between these coefficients.
     2. Establish a relationship (lemma) between these coefficients and the derivatives of the Riemann zeta function, as described earlier.
     3. Apply the Guinand-Weil explicit formula:

        \sum_{\rho} h(\rho) = h(0) + h(1) - \sum_{n=1}^\infty \frac{\Lambda(n)}{\sqrt{n}} \left( g(\log n) + g(-\log n) \right)

        where h(s) is a suitable test function, g(x) is its Fourier transform, ρ are the non-trivial zeros of the zeta function, and Λ(n) is the von Mangoldt function.
     4. Choose a test function h(s) that is related to the polynomials P_n(y). This could involve defining h(s) such that its Mellin transform is related to P_n(y).
   • Methodology:
     1. Compute the coefficients a_{n,k} for a large range of n and k. Look for patterns and relationships.
     2. Prove the lemma relating a_{n,k} to the derivatives of the zeta function.
     3. Choose a suitable test function h(s) for the Guinand-Weil formula. This is a crucial step.
     4. Evaluate the Guinand-Weil formula. This involves computing the sum over the zeros of the zeta function and the sum over the primes.
     5. Analyze the resulting equation to find conditions under which the real parts of the zeros must be 1/2. This may involve showing that if the RH is false, the Guinand-Weil formula will lead to a contradiction.
   • Prediction: This approach might reveal that if the RH is false, the coefficients a_{n,k} will exhibit specific growth behavior that contradicts the Guinand-Weil formula.
   • Limitations: The primary limitation is the difficulty in finding a suitable test function h(s) for the Guinand-Weil formula that is related to the polynomials P_n(y) and that allows for the evaluation of the formula. Overcoming this might involve using advanced techniques from harmonic analysis and number theory.

Tangential Connections and Conjectures

1. Quantum Chaos and Random Matrix Theory:
   • Mathematical Bridge: The distribution of zeros of the Riemann zeta function is conjectured to be statistically similar to the distribution of eigenvalues of large random matrices. The polynomials P_n(y) and U_k(log x) could be used to approximate the spectral density of these random matrices.
   • Conjecture: The roots of the polynomials P_n(y) and U_k(log x) converge to the same distribution as the eigenvalues of a large random matrix from the Gaussian Unitary Ensemble (GUE) as n approaches infinity.
   • Computational Experiments: Compute the roots of P_n(y) and U_k(log x) for large n. Compare their distribution to the known distribution of eigenvalues of GUE matrices.

Research Agenda

• Conjectures to Prove:
  • Conjecture 1 (Mellin Transform Representation): ali(e^x) can be represented as a Mellin transform of a function related to the Riemann zeta function. This requires finding a function F(s) such that:

    \operatorname{ali}\left(e^{x}\right) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} F(s) x^{-s} ds

    and F(s) is expressible in terms of ζ(s).
  • Conjecture 2 (Functional Equation Incorporation): Substituting the Mellin transform representation of ali(e^x) and applying the functional equation of the Riemann zeta function results in an equation where the location of the zeros of ζ(s) is explicitly linked to the asymptotic behavior of the polynomials P_n(log x). This requires proving that the functional equation transforms F(s) into a form that involves ζ(1-s).
  • Conjecture 3 (Contradiction): If the Riemann Hypothesis is false (i.e., there exists a zero with real part not equal to 1/2), then the resulting equation from Conjecture 2 leads to a divergence in the asymptotic expansion or a contradiction with other known properties of the Riemann zeta function. This is the core of the proof strategy.
• Mathematical Tools Needed: Complex analysis, Mellin transforms, Functional equation of the Riemann zeta function

This structured approach, combining rigorous analysis, computational experimentation, and theoretical insights, aims to advance our understanding of the Riemann Hypothesis significantly, referencing back to the mathematical frameworks presented in arXiv:1203.5413.