A dynamic approach for the zeros of the Riemann zeta function - collision and repulsion

Abstract: For $N \in \mathbb{N}$ consider the $N$-th section of the approximate functional equation $$ \zeta_N(s)= \sum_{n =1 }^N B_n(s),$$ where $$ B_n(s)= \frac{1}{2} \left [ n^{-s} + \chi(s) \cdot n^{s-1} \right ].$$ Our aim in this work is to introduce a new approach for the Riemann hypothesis by studying the way pairs of consecutive zeros of $\zeta_N(s)$ change with respect to $N$. For the initial stage, it is known that the non-trivial zeros of $\zeta_1(s)$ all lie on the critical line $Re(s)=\frac{1}{2}$. In the region $2N \leq Im(s) \leq 2 \pi (N+1)$ the function $\zeta_N(s)$ serves as an approximation of $\zeta(s)$ itself, and it was conjectured by Spira that in this region $\zeta_N(s)$ also admits zeros only on the critical line. We show that the appearance of zeros of a section off the critical line can be realized as the result of two consecutive zeros meeting and pushing each other off the critical line as $N$ changes, a process to which we refer to as a collision of zeros. Based on a study of the properties of $\zeta_N(s)$, we suggest a way of re-arranging the order of summation of the elements $B_n(s)$ in $\zeta_{N}(s)$ with $N=\left [ \frac{Im(s)}{2} \right ]$ that is expected to avoid collisions altogether, we refer to such a re-arrangement as a repelling re-arrangement. In particular, establishing that the suggested repelling re-arrangement indeed avoids collisions for any pair of zeros would imply RH.

• The paper introduces a dynamic approach using ζₙ(s) approximations to examine zero collisions and repulsion, supporting the Riemann Hypothesis.
• It demonstrates that intentional series rearrangements can prevent collisions, keeping zeros confined to the critical line.
• Numerical analyses and visual plots show that repelling rearrangements yield zero distributions that align with both classical and novel analytic predictions.

A Dynamic Approach to the Zeros of the Riemann Zeta Function

The paper authored by Yochay Jerby proposes a novel approach to understanding the Riemann hypothesis (RH) through a study of the dynamic behavior of the zeros of approximations of the Riemann zeta function, denoted as $\zeta(s)$. The Riemann hypothesis conjectures that all non-trivial zeros of this function lie on the critical line $\sigma = \frac{1}{2}$ in the complex plane.

Key Concepts and Methodology

Jerby's approach focuses on $\zeta_N(s)$, the $N$-th section of the approximate functional equation for the Riemann zeta function, given by: $\zeta_N(s) = \sum_{n=1}^{N} B_n(s)$ where each term $B_n(s)$ includes a component related to both $n^{-s}$ and the product of a phase factor with $n^{s-1}$. This approximation, depending on a parameter $N$, evolves as $N$ increases, allowing the author to explore the sequential changes in zero distributions of $\zeta_N(s)$.

The exploration is divided into the "collision" and "repulsion" phenomena associated with these zeros. As $N$ changes, some zeros might undergo collisions—a process where pairs of consecutive zeros meet and diverge from the critical line. However, Jerby proposes that through intentional rearrangement of the series order in $\zeta_N(s)$, these collisions can be repelled, suggesting a method dubbed "repelling re-arrangement". If such rearrangements consistently prevent collisions, all zeros should theoretically remain on the critical line, thus supporting RH.

Numerical Analysis and Results

Jerby conducts numerical explorations to illustrate the dynamic behavior of zeros and the impact of rearrangement strategies. An example is provided where the zeros of $\zeta_N(s)$ for certain $N$-th sections initially lie on the critical line, adhere to collision events with ensuing divergence, and eventually return to the critical line through either managed series rearrangement or returning convergence.

An important analysis presented involves the numerical comparison to zeros obtained through classical methods and novel approximation techniques (e.g., through Euler's transformation of series), emphasizing that both classical and transformed methods share essential collision traits, but the transformed sections provide a smoother approximation profile. The paper demonstrates with visual plots how particular re-arrangements, opposed to standard consecutive summation, influence these interactions favorably.

Theoretical Implications

Theoretically, Jerby's work implies RH could be verified by virtue of order manipulation within approaches based on analytic continuations and functional equations. The conjectured success of the repelling re-arrangement foresees an operation framework where RH emerges naturally as a consequence of dynamic systemic constraints applied on zero behavior. This dynamic approach provides an insightful heuristic for RH, differing from traditional static conjectural arguments such as the Montgomery pair correlation perspective.

Future Prospects in Research

The introduction of this dynamic analysis opens up pathways to more rigorous explorations into the zero-distribution investigations of L-functions and other related analytic structures in number theory. Future work could explore formalizing the conditions under which repelling re-arrangements guarantee collision avoidance or extend these concepts to other zeta-like functions, including the challenging Davenport-Heilbronn $L$-functions.

In conclusion, this paper contributes significantly to the discourse on RH by suggesting fresh methodologies grounded in dynamic analysis, numerical validation, and conceptual reframing, potentially paving the way towards a stronger theoretical understanding of the zeta zeros' elusive nature.