Research Pathways Inspired by Prime Gap Analysis
This article details potential research directions inspired by a recent paper (arXiv 2007.15282v1) focusing on prime gaps and their distribution. We explore how these findings might relate to the Riemann Hypothesis (RH).
Framework 1: Prime Gap Distribution
The paper presents data on prime gaps, suggesting patterns in their distribution. This is relevant to RH because the distribution of prime numbers is intrinsically linked to the zeros of the Riemann zeta function, ζ(s).
- Formulation: Analyze the distribution of prime gaps g(p) using the provided data and explore its relationship with the distribution of zeros of ζ(s) on the critical line, Re(s) = 0.5.
- Potential Theorem: Develop a formal relationship between the density of prime gaps and the clustering of zeros. Specifically, investigate if a certain density of small prime gaps corresponds to a higher concentration of zeros on the critical line.
- Connection: Establish a quantitative link between the paper's prime gap bounds and the explicit formula connecting the prime-counting function π(x) to the zeros of ζ(s).
Framework 2: Logarithmic Bounds
The paper utilizes logarithmic bounds on prime-counting functions. These bounds are similar to those found in the Prime Number Theorem, which is closely related to RH.
- Formulation: Investigate how the paper's logarithmic bounds, such as x/(log x - 1.1), can be refined or extended to provide tighter constraints on the distribution of primes.
- Potential Theorem: Show that improved bounds on the prime-counting function directly impact the error terms in the explicit formula for π(x), leading to more precise estimations of zero locations.
- Connection: Explore whether these refined bounds can be used to establish new zero-free regions for ζ(s) or to improve existing bounds on the number of zeros in a given region.
Novel Approach 1: Combining Prime Gap Analysis with Existing RH Research
This approach combines the paper's prime gap analysis with known properties of ζ(s).
- Mathematical Foundation: Utilize the established connection between the distribution of primes and the zeros of ζ(s) via the explicit formula.
- Methodology: Develop a statistical model that correlates the distribution of prime gaps with the spacing between consecutive zeros of ζ(s) on the critical line.
- Predictions: Expect to find correlations between the density of small prime gaps and the clustering of zeros.
- Limitations: The probabilistic nature of prime gaps and the complexities of the explicit formula will require advanced statistical and computational techniques.
Tangential Connection 1: Computational Experiments
The paper's numerical data provides a starting point for computational experiments.
- Formulation: Develop algorithms to test the proposed correlations between prime gap distributions and zero locations using the provided data as a basis for initial testing.
- Conjecture: The observed patterns in prime gaps will correspond to predictable patterns in the distribution of zeros of ζ(s).
- Computational Experiments: Extend the analysis to much larger datasets to test the robustness of the observed correlations.