Mathematical Frameworks for Zeta Function Analysis
The paper hal-01728058 provides a rigorous framework for analyzing the tail of the Riemann zeta function Dirichlet series through integral bounds on the Mertens function. By examining the relationship between the sum of the Mobius function and the reciprocal of the zeta function, researchers can define majorant sets to bound potential zero locations.
The Majorant-Infimum Duality
A central concept involves defining function classes, such as the set G, which contains functions that bound the absolute value of the integral of M(u) divided by u raised to the power of (sigma + it + 1). This framework transforms the search for zero-free regions into an optimization problem over functional spaces.
- The Supremum Function: Defining A(x) as the supremum over t of the product of the square root of (sigma squared + t squared) and the integral tail allows for a characterization of the critical strip behavior.
- Convergence Criteria: The relationship between simple and absolute convergence of the Dirichlet series for the Mobius function is critical for determining the supremum of the real parts of the zeros, denoted as beta-star.
Novel Research Pathways
Combining these integral bounds with spectral theory offers a way to connect the Mertens function growth directly to the zero-sum formulas. One approach involves analyzing the stability of the ratio between lower bounds derived from the density of square-free integers and upper bounds assumed under the Riemann Hypothesis. This connects the distribution of square-free integers directly to the tail of the zeta integral.
Proposed Research Agenda
To advance toward a formal proof, the following sequence is proposed based on the findings in hal-01728058:
- Phase 1: Foundation Verification. Validate the integral lower bounds and determine if they hold for all values of t or if phase oscillations in the Mertens function provide sufficient cancellation to maintain the hypothesis.
- Phase 2: Mertens Majorant Theorem. Establish a sequence of results linking the decay rate of the infimum of the set G to the existence of zeros in the critical strip, potentially refining the zero-free region.
- Phase 3: Resolution via Tauberian Theory. Utilize Tauberian theorems for oscillatory integrals to relate the decay of the integral tail to the singularities of 1/zeta(s), aiming to prove that the real part of all non-trivial zeros must be 1/2.