Exploring Riemann Hypothesis Pathways Through Complex Series and Bounds

The paper arXiv:hal-04697638 introduces several mathematical structures that may offer new perspectives on the Riemann Hypothesis. While seemingly distinct, these frameworks involving complex series, trigonometric functions, and bounds on related functions could potentially be combined to shed light on the distribution of the non-trivial zeros of the Riemann zeta function.

Key Mathematical Frameworks

Several core mathematical ideas from the paper appear promising for Riemann Hypothesis research:

Trigonometric Series Representations

The paper examines series expansions involving sine and cosine terms with varying phases and coefficients. The general form suggests that certain complex functions, perhaps related to the zeta or Dirichlet eta function, could be represented in this manner. If such a representation can be rigorously established for the zeta function within the critical strip, understanding the zeros of this trigonometric series could directly inform us about the zeros of the zeta function.

Bounds on the Dirichlet Eta Function

A specific lower bound on the magnitude of the Dirichlet eta function, dependent on the real part of the complex variable, is discussed. Since the eta function is closely related to the zeta function, establishing precise and tight bounds on the eta function could translate into constraints on the behavior of the zeta function, particularly near the critical line. A sufficiently strong lower bound could potentially restrict the possible locations of non-trivial zeros.

Series with Rotation

Another structure involves a series where each term includes a complex exponential factor that introduces a rotation dependent on the logarithm of the index. This type of series is related to the analytic continuation of functions like the Dirichlet eta function. Analyzing the convergence properties and behavior of this series, especially as the real part of the complex variable changes, could provide insights into the function's zeros.

Novel Approaches Combining Frameworks

Combining these structures with established knowledge about the Riemann Hypothesis suggests new research directions:

Integrating Trigonometric Series with the Functional Equation

The Riemann zeta function satisfies a well-known functional equation relating its values at s and 1-s. If we can represent the zeta function (or a related function like the eta function) as a trigonometric series, we could substitute this representation into the functional equation. This would yield a complex relationship between the coefficients, phases, and parameters of the series at s and 1-s. Analyzing the constraints imposed by this relationship, especially when the series value is zero, might reveal necessary conditions that hold only on the critical line.

• Methodology:
  1. Rigorous derivation of a convergent trigonometric series representation for the zeta or eta function.
  2. Substitution of this series into the functional equation.
  3. Analysis of the resulting equations relating series parameters at s and 1-s.
  4. Investigation of how the zero condition of the series constrains these parameters, aiming to show this constraint forces the real part of s to be 1/2.
• Potential Outcome: This could provide a new characterization of the critical line based on the properties of the trigonometric series components.

Leveraging Bounds to Constrain Zero Locations

The lower bounds on the Dirichlet eta function provide regions where the function cannot be zero. The series with rotation provides another way to represent the function. By analyzing the behavior of the rotation series off the critical line, one could attempt to show that if a zero were to exist in such a location, the magnitude of the series (and thus the eta function) would contradict the established lower bound. This would require proving sufficiently tight bounds and a strong connection between the rotation series and the eta function's zeros.

• Methodology:
  1. Establish a precise relationship between the series with rotation and the Dirichlet eta function, including bounds on any error terms.
  2. Refine the lower bounds on the magnitude of the eta function, particularly near the critical strip.
  3. Assume, for contradiction, the existence of a zero off the critical line.
  4. Show that the behavior of the rotation series at this assumed zero location violates the established lower bound for the eta function's magnitude.
• Potential Outcome: A successful contradiction would directly imply that all non-trivial zeros must lie on the critical line.

Research Agenda

A research agenda based on these ideas could involve:

• Conjectures:
  • The Riemann zeta function possesses a convergent trigonometric series representation whose coefficients and phases satisfy specific relationships dictated by the functional equation.
  • The lower bounds on the Dirichlet eta function, combined with the properties of the series with rotation, preclude the existence of zeros off the critical line.
• Mathematical Tools: Advanced complex analysis, Fourier analysis, techniques for analyzing the convergence and properties of complex series, rigorous inequality proofs, potentially numerical computation to explore series behavior and bounds.
• Intermediate Results: Explicit formulas for trigonometric series coefficients, tighter analytical bounds for the eta function, proof of a formal link between the rotation series and the eta function, demonstration of how the functional equation constrains series parameters.
• Logical Sequence: Start by formally defining and proving the existence of the proposed series representations. Then, apply the functional equation and bounds to derive necessary conditions for zeros. Finally, work towards showing that these conditions are met only on the critical line.

This structured approach, drawing on the frameworks presented in arXiv:hal-04697638, offers specific avenues for investigation that could contribute to solving the Riemann Hypothesis.