The Riemann Hypothesis remains one of the most significant unsolved problems in mathematics. This article analyzes frameworks and techniques presented in arXiv:hal-04578467, formulating research pathways that could lead to a proof.
arXiv:hal-04578467 presents a complex integral representation involving a function f_τ(x+yi):
f_τ(x+yi) = ∫₀^∞ g_τ(x,y,t)(√(1+t²))^(-x)[cos(y ln(√(1+t²))) - i sin(y ln(√(1+t²)))]dt
This connects to the Riemann Hypothesis through its resemblance to the Riemann zeta function's integral representation and the potential to relate the zeros of f_τ(x+yi) to those of the zeta function.
The paper also presents:
Δ̄(x,y) = h₀(x,y) + ω₀(x,y)∑_{k=0}^∞ (-1)^k δ̄_{2k}(x,y)/(2k)! y^(2k)
This structure suggests a new way to analyze the behavior of related functions near the critical line x=1/2 and a potential connection to the functional equation of the zeta function through symmetry properties.
The paper decomposes a rational function as:
(1-x-yi)/(1+x+yi) = (1-x²-y²)/(1+2x+x²+y²) - 2yi/(1+2x+x²+y²)
This decomposition could provide insights into the distribution of zeta zeros and symmetry properties around the critical line.
This approach combines the paper's integral transform with the zeta representation. A proposed theorem states: Let F(s) = f_τ(s)ζ(s). Then, F(s) = ∫₀^∞ K(s,t)g_τ(x,y,t)dt, where K(s,t) is a kernel function to be determined.
The steps involve establishing properties of K(s,t) needed for the Riemann Hypothesis, proving F(s) has zeros only on the critical line, and showing this implies the Riemann Hypothesis for ζ(s). A limitation is the need for a precise form of g_τ(x,y,t).
Using the Δ̄(x,y) expansion to study Z(s) = ζ(s)Δ̄(Re(s),Im(s)) could reveal new functional relationships near the critical line.
This approach focuses on the integral representation of f_τ(x+yi) and attempts to characterize the kernel g_τ(x, y, t). We propose decomposing g_τ(x, y, t) into a sum of simpler kernels:
g_τ(x, y, t) = g₀(x, y, t) + g₁(x, y, t)
where g₀(x, y, t) captures the essential properties of the zeta function on the critical line, and g₁(x, y, t) is a "correction term" that vanishes on the critical line.
The research agenda involves proving convergence properties of the series expansion for f_τ(s), establishing a relationship between g_τ(x,y,t) and known zeta function properties, and developing computational methods to verify proposed relationships.
The Riemann zeta function is conjectured to have statistical properties similar to the eigenvalues of large random matrices. The kernel g_τ(x, y, t) can be viewed as a correlation kernel, potentially related to the eigenvalue statistics of random matrices.
The Riemann zeta function is connected to the spectrum of quantum chaotic systems. The kernel g_τ(x, y, t) might be related to the classical orbits of a quantum chaotic system.