Unraveling the Riemann Hypothesis: Novel Pathways from Prime Number Sieves

Unraveling the Riemann Hypothesis

This article proposes novel research pathways towards proving the Riemann Hypothesis (RH) by leveraging insights from a recent paper (arXiv:1709.01146v1) focusing on prime number sieves. The paper's unique approach to prime number distribution offers potential connections to the distribution of the Riemann zeta function's zeros.

Framework 1: Prime Counting Function and the Sieve

The paper introduces a novel sieve method for analyzing prime number distribution. A key equation from the paper is:

lim sup (|π(n) - li(n)| / √(2 ∑_i=1ⁿ (1/(ln(i)(1 - 1/ln(i)))))) = 1

This equation relates the prime-counting function π(n) to the logarithmic integral li(n). The error term in this equation is crucial. We propose the following theorem:

Theorem 1: A refined analysis of the error term in the above equation, leveraging the sieve structure, can lead to tighter bounds on |π(x) - li(x)|, directly connecting to the Riemann Hypothesis. Specifically, if we can show that the error term is bounded by a function of order less than x^1/2+ε for any ε > 0, then the Riemann Hypothesis would follow.

Framework 2: Sieve Periodicity and Zeta Function Oscillations

The paper highlights the periodic nature of sequences generated by the sieve method. The period is given by Π(x_i), where x_i are sieve elements. This periodicity might reflect underlying oscillations in the Riemann zeta function.

Conjecture 1: The periodicity of the sieve method is directly related to the oscillations of the Riemann zeta function on the critical line. A formal connection between the sieve period and the spacing between consecutive zeta zeros could be established.

Framework 3: Statistical Approach to Error Term

The error term in the prime-counting function equation suggests a statistical approach. We propose a modified sieve operator:

S(x) = ∑ (a_i,j/x_i)

where a_i,j are the sieve elements. Analyzing the behavior of |S(x) - li(x)| / √(log log x) could reveal insights into the distribution of primes and its relation to zeta zeros.

Conjecture 2: The normalized error term |S(x) - li(x)| / √(log log x) converges to a specific distribution, providing a statistical perspective on the RH.

Research Agenda

• Phase 1: Rigorous analysis of the error term in the prime-counting equation, focusing on the connection to the sieve method.
• Phase 2: Establish a formal connection between sieve periodicity and zeta function oscillations. This may involve exploring Fourier analysis of the sieve sequences.
• Phase 3: Develop and test Conjectures 1 and 2 through extensive numerical computations. This would involve generating large sieve sequences and comparing their statistical properties to the distribution of zeta zeros.

Successfully navigating this research agenda could provide a novel pathway to proving the Riemann Hypothesis.