Prime Number Bounds and the Riemann Hypothesis: A Novel Approach

Prime Number Bounds and the Riemann Hypothesis: A Novel Approach

The Riemann Hypothesis (RH) remains one of the most significant unsolved problems in mathematics. This article explores how the inequalities and prime number estimates presented in "hal-01560672" might contribute to a proof. This paper provides inequalities related to prime numbers and their distribution, which are intrinsically linked to the distribution of zeros of the Riemann zeta function.

Mathematical Frameworks

• Improved Prime Number Inequalities: The paper presents inequalities of the form:

  p(n) ≤ n(ln n + ln_2 n - C)

  where p(n) is the n-th prime number, ln is the natural logarithm, ln_2 n = ln(ln n), and C is a constant. Finding tighter bounds for C is a key focus.
• Bounding the Difference Between p(n) and its Approximation: The paper investigates the difference:

  d(n) = p(n) - n(ln n + ln_2 n - 1)

  aiming for sharper bounds for d(n).
• Logarithmic Interval Analysis: Based on inequalities such as:

  n / (n(ln n + ln_2 n - 1) + (129n/2500)) ≤ n / (p(n) - (n(ln n + ln_2 n - 1)))

  This structure suggests investigation of gaps between consecutive primes and distributional properties related to logarithmic intervals.

Novel Approaches

• Refined Explicit Formulas via Improved Prime Bounds: The Riemann-von Mangoldt explicit formula connects the prime counting function π(x) to the zeros of the Riemann zeta function ζ(s). This formula involves sums over the non-trivial zeros ρ = β + iγ of ζ(s):

  π(x) = li(x) - Σ li(x^ρ) + error terms

  Sharpening prime inequalities leads to more precise estimates for π(x), which, when substituted into the explicit formula, could reveal contradictions if the RH is false.
• Using d(n) to Bound the Zeta Function's Growth: Defining d(n) = p(n) - n(ln n + ln_2 n - 1), the Riemann Hypothesis has equivalent formulations related to the growth rate of the zeta function. Establish a connection between d(n) and the Chebyshev functions ψ(x) and θ(x). Use known relationships between the zeta function and the Chebyshev functions to express ζ(s) in terms of ψ(x) or θ(x). Using the bounds on d(n), derive an upper bound for |ζ(1/2 + it)|.

Tangential Connections

• Information Theory and Prime Distribution: The distribution of prime numbers can be viewed through the lens of information theory, where gaps between primes represent "information." Inequalities provide constraints on this information.
• Dynamical Systems and Prime Number Chaos: Dynamical systems exhibiting chaotic behavior can be statistically similar to prime number distributions. The inequalities can be interpreted as constraints on the "order" within this apparent chaos.

Research Agenda

A key conjecture is that an improved constant C* > 1 exists such that p(n) ≤ n(ln n + ln_2 n - C*) for all n ≥ N, and this tighter bound, when used in the Riemann-von Mangoldt explicit formula, leads to a contradiction if any non-trivial zero of the Riemann zeta function has a real part greater than 1/2.

This approach requires:

• Advanced analytic number theory techniques.
• Computational tools for high-precision calculations.

Potential intermediate results include:

• Successful implementation of refined π(x) expansions.
• Verification of new inequalities through computational models.

This structured approach ensures a comprehensive exploration of novel and existing methodologies to advance towards a proof of the Riemann Hypothesis. Citing the source paper as arXiv:hal-01560672.