On The Complex Zeros of The Riemann Zeta Function (1912.04058v3)

Abstract: Plotting $({\zeta},U(x,y),V(x,y),x,y,z)$ and numerical calculations show that ${\zeta}(s)=0$ finitely many times in the Interval $I$ with $k$ roots in the interval all of which are on the critical line which proves the Riemann Hypothesis in $I$. It is shown by using groups of intervals how to extend the proof of the Riemann Hypothesis in $I$ to intervals at infinity which extends the proof of the Riemann Hypothesis from the finite domain in the critical strip to the entire critical strip.

• The paper presents a new symmetric approach to analyzing the Riemann Zeta Function's zeros to advance understanding of the Riemann Hypothesis.
• It employs analytic continuation and Fourier analysis to construct a 'primed' function that mirrors the classical RZF properties.
• The research bridges advanced number theory with practical applications in physics and signal processing, suggesting innovative computational techniques.

Analyzing the Riemann Zeta Function's Complex Zeros

The paper by Devin Hardy offers an analytical pursuit of the Riemann Hypothesis (RH) through an exploration of the complex zeros of the Riemann Zeta Function (RZF). Hardy embarks on a discourse that seeks to redefine our understanding of the RZF's zeros by leveraging the relationship of these zeros to symmetry and analytic continuation within the complex plane.

Foundations and Significance of the Riemann Zeta Function

The Riemann Zeta Function, denoted by $\zeta(s)$, is central in analytic number theory, particularly in relation to the distribution of prime numbers. Hardy reiterates this classical definition: it is represented as a sum, $\sum_{n=1}^{\infty} n^{-s}$, and encompasses an Euler product over primes, converging for $\Re(s) > 1$. The RH conjectures that all non-trivial zeros of the RZF lie on the critical line $\Re(s) = 1/2$ in the complex plane.

Analytical Approach and Methodology

Hardy's approach is to explore a general proof of the RH, emphasizing symmetry inherent to the function. Hardy posits that by defining a function equivalent to the RZF with the same zeroes, the hypothesis can be verified through numerical alignment. This involves establishing a relationship between the RZF and Fourier Transform, potentially generalizing to encompass broader signal processing frameworks.

The paper explores establishing a symmetry-breaking technique associated with analytically continued functions. Hardy develops a symmetrical "primed" RZF, ostensibly similar in properties to the original, aiming to show an equivalence or relationship that underscores zero locations through symmetry. His analysis extends to a form of mathematical visualization—constructing complex dimensional plots that help depict function behavior over complex planes.

Engagement with the Functional Equation

Hardy significantly focuses on the functional equation intrinsic to the RZF, $\xi(s) = \xi(1-s)$, and its symmetric properties. The paper asserts that understanding this symmetry can result in a proof of the RH by demonstrating the critical line as a locus of maximum symmetry.

The analysis extends the classical definition by considering multivalued implications of complex logarithms within the boundary conditions defined by the RH. Hardy attempts to redefine or reinterpret analytic continuation paths, frequently confronting the limitations imposed by traditional real analysis.

Discussion of Symmetry and General Implications

In engaging with symmetry, Hardy suggests that the locations of the RZF's zeros are inherently tied to these symmetrical properties—an insight lured out by a systematic analytic expansion. This symmetry is crucial, as it translates into numerical processes potentially extending beyond number theory into physics, engineering, and even quantum mechanics.

The paper posits—and this is a novel claim—that understanding the RZF symmetry could yield insights akin to those obtained in certain spectral analyses in physics, touching upon quantum phenomena. Such introspection suggests that the methodology might be applied to develop numerical algorithms or signal distribution models predicated on this deep mathematical symmetry.

Conclusion and Outlook

Hardy's treatment of the RZF zeros embodies a deep investigation of mathematical symmetry, grounded in sophisticated theoretical analysis and envisaging practical applications. While asserting a systemic approach to proving the RH, the research also opens pathways to broader theoretical and practical applications. This potential bridges advanced mathematics and applied physics, and may inspire innovative computational techniques rooted in the older problem of prime distribution.

Future research directions could explore computational implementations of the symmetry principles Hardy outlines—particularly in signal processing domains. Conversely, the theoretical fronts may explore abstract spaces in mathematical physics or theoretical mechanics, building upon Hardy's foundational explorations.