Unlocking the Riemann Hypothesis: Entropy, Prime Gaps, and Logarithmic Insights

Introduction

The Riemann Hypothesis (RH) remains one of the most significant unsolved problems in mathematics. This article analyzes the paper arXiv:hal-01353754 and proposes research pathways toward a proof of the RH, drawing inspiration from the paper's mathematical structures.

Mathematical Frameworks

Prime Gap Entropy and Distribution

• Mathematical Formulation: The paper explores the entropy of prime gaps using expressions like:

  H(Real) = -∫₂^{x_max/ln(x_max)} (1/(x_max/ln(x_max)-2)) ln(1/(x_max/ln(x_max)-2)) dx = ln(x_max/ln(x_max)-2)

  and

  H(G(P)) = -∫₂^{7 × 107} (1/log(log(k))) log(1/log(log(k))) dk

  These formulas attempt to quantify the randomness in the distribution of primes.
• Potential Theorems/Lemmas: We could aim to prove theorems that relate the entropy measures H defined in the paper to properties of the prime counting function π(x) or to the Riemann zeta function ζ(s). For example, we could investigate whether a lower bound on the entropy of prime gaps implies a specific growth rate for π(x) or a restriction on the location of the non-trivial zeros of ζ(s).
• Conjecture: A lower bound on H(G(P)) implies that π(x) > x/log x + f(x), where f(x) is some positive function that grows faster than x/log² x.
• Connection to Zeta Function: The distribution of primes is intimately linked to the zeros of the Riemann zeta function. Specifically, if we can show that the entropy measure H is sufficiently large, it might impose constraints on the distribution of primes and, consequently, on the location of the zeros of ζ(s). A crucial link is the explicit formula relating prime counting functions to the zeros of the zeta function.

Logarithmic Inequalities and Prime Number Theorem

• Mathematical Formulation: The paper uses inequalities involving logarithms of primes, like:

  log(P_n/log P_n - 2) > log(2e^-γlog²(P_n) - 2)

  and

  log(n log n/log(n log n) - 2) > log(2e^-γlog²(n log n) - 2)

  These inequalities relate the n-th prime P_n to its position n and involve Euler's constant γ.
• Potential Theorems/Lemmas: We could investigate whether these inequalities can be sharpened or extended to provide more precise estimates for P_n. A key question is whether these inequalities can be used to derive a stronger form of the Prime Number Theorem (PNT).
• Conjecture: If the above inequalities hold for all sufficiently large n, then a stronger version of the PNT holds, such as π(x) = Li(x) + O(x^{1/2 - ε}) for some ε > 0, where Li(x) is the logarithmic integral.
• Connection to Zeta Function: The Prime Number Theorem (PNT) and its stronger forms are equivalent to statements about the zero-free region of the Riemann zeta function. Specifically, π(x) = Li(x) + O(x^{1/2 - ε}) is equivalent to ζ(s) having no zeros in the region Re(s) > 1/2 + ε. Thus, any improvement to the PNT derived from these inequalities could lead to a proof of the RH.

Bounding the Difference Between Consecutive Primes

• Mathematical Formulation: The paper mentions:

  that there are infinitely many primes P_n, P_n+1 such that: P_n+1 - P_n < 16.

  and

  H(P_n+1 - P(n) = min{H(x_k+1 - x_k)}, k ∈ ℝ

  This hints at an exploration of the distribution of small prime gaps.
• Potential Theorems/Lemmas: Improving bounds on prime gaps is a key area of research. We could try to show that the entropy of small prime gaps is higher than expected under the assumption that the RH is false.
• Conjecture: If P_n+1 - P_n is "small" infinitely often (e.g., P_n+1 - P_n < C log P_n for some constant C), then the RH is true. This would require showing that the existence of such small gaps forces the zeros of ζ(s) to lie on the critical line.
• Connection to Zeta Function: The distribution of prime gaps is related to the pair correlation conjecture, which is itself connected to the RH. Specifically, the pair correlation conjecture predicts the statistical behavior of the normalized spacings between the zeros of the zeta function. Showing that small prime gaps occur more frequently than predicted by the pair correlation conjecture, under the assumption that the RH is false, could lead to a contradiction and thus prove the RH.

Novel Approaches Combining Existing Research

Entropy of Prime Gaps and the Explicit Formula

• Mathematical Foundation: Combine the entropy measure from the paper with the explicit formula relating prime counting functions to the zeros of the zeta function. The explicit formula is:

  ψ(x) = x - Σ_ρ x^ρ/ρ - ζ'(0)/ζ(0) - Σ_n=1^∞ x^-2n/(-2n)

  where ψ(x) = Σ_{p^k ≤ x} log p is the Chebyshev function, and the sum is over the non-trivial zeros ρ of ζ(s).
• Methodology:
  1. Develop a more refined entropy measure H*(x) that takes into account the distribution of primes in short intervals, weighted by log p.
  2. Assume the RH is false, meaning there exists a zero ρ = β + iγ with β > 1/2.
  3. Use the explicit formula to express the error term in the prime number theorem as a sum over the zeros of ζ(s).
  4. Show that if the RH is false, the oscillations in ψ(x) due to the term Σ_ρ x^ρ/ρ are "too large" to be consistent with the entropy measure H*(x). This would involve deriving a lower bound on the magnitude of these oscillations and comparing it to an upper bound derived from H*(x).
  5. Derive a contradiction, thus proving the RH.
• Predictions: This approach predicts that if the RH is false, the oscillations in the prime distribution will be more "coherent" and less "random" than predicted by the entropy measure.
• Limitations: Bounding the sum Σ_ρ x^ρ/ρ is notoriously difficult. Overcoming this limitation might require using techniques from analytic number theory, such as the saddle-point method or zero-density estimates.

Logarithmic Inequalities and Zero-Free Regions

• Mathematical Foundation: Combine the logarithmic inequalities from the paper with known results on zero-free regions of the Riemann zeta function. The best-known zero-free region is of the form:

  Re(s) > 1 - c/log |t|

  for some constant c.
• Methodology:
  1. Sharpen the logarithmic inequalities from the paper to obtain more precise estimates for P_n. This might involve using techniques from sieve theory or exponential sums.
  2. Assume that the zero-free region of ζ(s) is "smaller" than currently known (i.e., Re(s) > 1 - c/(log |t|)^k for some k < 1).
  3. Show that this smaller zero-free region implies a contradiction with the improved estimates for P_n. This would involve using the estimates for P_n to derive a lower bound on the number of primes in certain intervals, and then showing that this lower bound is incompatible with the assumed size of the zero-free region.
  4. Conclude that the zero-free region must be at least as large as currently known, and potentially even larger, which could eventually lead to showing that the only zeros are on the critical line.
• Predictions: This approach predicts that the logarithmic inequalities impose strong constraints on the possible size of the zero-free region of ζ(s).
• Limitations: The connection between estimates for P_n and the zero-free region is subtle and requires careful analysis. Overcoming this limitation might involve using techniques from complex analysis, such as the Phragmén-Lindelöf principle.

Tangential Connections

Information Theory and the Zeta Function

• Formal Mathematical Bridge: The entropy measures in the paper can be interpreted as measures of information content. The Riemann zeta function can also be viewed from an information-theoretic perspective. For example, the Euler product representation of ζ(s) can be seen as encoding information about the distribution of primes.
• Specific Conjecture: There exists a formal relationship between the entropy of prime gaps (as measured by the paper's entropy measures) and the Kolmogorov complexity of the sequence of prime numbers.
• Computational Experiments: Calculate the Kolmogorov complexity of the sequence of prime numbers up to various large values of x. Compare this with the entropy measures calculated from the paper. If the conjecture is true, there should be a strong correlation between these two quantities.

Research Agenda

Overall Goal

Prove the Riemann Hypothesis.

Approach

Refine and combine the frameworks from arXiv:hal-01353754 with existing techniques in analytic number theory. Focus on establishing stronger connections between the distribution of primes, entropy measures, and the zero-free region of the Riemann zeta function.

Conjectures to be Proven

1. Entropy Conjecture: A sufficiently large lower bound on the entropy measure H*(x) (a refined version of the paper's entropy measure) implies that π(x) = Li(x) + O(x^{1/2 - ε}) for some ε > 0.
2. Logarithmic Inequality Conjecture: Improved logarithmic inequalities for P_n imply that the zero-free region of ζ(s) is of the form Re(s) > 1 - c/(log |t|)^k for some k ≥ 1.
3. Prime Gap Conjecture: If P_n+1 - P_n < C log P_n infinitely often, then the RH is true.

Mathematical Tools and Techniques Required

• Analytic Number Theory (Prime Number Theorem, explicit formulas, zero-density estimates)
• Complex Analysis (Phragmén-Lindelöf principle, contour integration)
• Sieve Theory
• Information Theory (Kolmogorov complexity, entropy measures)

Logical Sequence of Theorems

1. Theorem 1: Develop a refined entropy measure H*(x) that is more sensitive to the distribution of primes in short intervals.
2. Theorem 2: Prove that a lower bound on H*(x) implies a certain growth rate for π(x).
3. Theorem 3: Establish a connection between the growth rate of π(x) and the zero-free region of ζ(s).
4. Theorem 4: Sharpen the logarithmic inequalities from the paper to obtain more precise estimates for P_n.
5. Theorem 5: Show that these improved estimates for P_n imply a larger zero-free region for ζ(s).
6. Theorem 6: Prove that if P_n+1 - P_n < C log P_n infinitely often, then the zeros of ζ(s) must lie on the critical line.
7. Theorem 7: Combine the results from Theorems 1-6 to prove the Riemann Hypothesis.