Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.05347

Attempting to Prove the Riemann Hypothesis through the Reflection Formula

Farid Kenas
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Riemann Hypothesis, originally proposed by the eminent mathematician Bernard Riemann in 1859, remains one of the most profound challenges in number theory. It posits that all non-trivial zeros of the Riemann zeta function ζ(s) are concentrated precisely along the critical line where the real part equals 1/2. In this paper, our aim is to present an attempt to prove this conjecture. Our approach relies on the use of the reflection formula. By applying this tool with precision and insight, we can conclusively establish that ξ(s)^2 (Riemann's ξ-function) is valid only when Re(s)=1/2. As a direct consequence of this determination, we can assert that every zero of both ξ(s)^2 and ξ(s) has a real part equal to 1/2. This, in turn, leads us to the tentative conclusion that the real part of all non-trivial zeros of the zeta function is consistently 1/2.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Farid Kenas. 2024-08-25. Attempting to Prove the Riemann Hypothesis through the Reflection Formula. https://arxiv.org/abs/2403.05347

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. Liu proposed the conjecture \[ \sum_{\text{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \ge 2+\left(\frac{2r}{R}\right)^k,\qquad k>1, \] with the reverse inequality for $k<1$. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

math.GM↗
arXiv2026

A quadratic critical-value conjecture for the fifth Bessel moment

We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\mathbb{Q}(\sqrt{5})$ and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $\langle f,f\rangle_{60} = \frac{3(5-\sqrt{5})}{2π^4}|L(f,2)|^2$, the coefficient-conjugation relation $L(f^σ,2) = κL(f,2)$ with explicit $κ\in \mathbb{Q}(\sqrt{5},i)$, and the twisted symmetric-square evaluation $L(χ_{-4}\mathrm{Sym}^2 f,2) = \sqrt{15}\,π^2 \langle f,f\rangle_{60}$, together with $L(χ_{-4}\mathrm{Sym}^2 f,3) = π^4\langle f,f\rangle_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\mathrm{odd}} = \frac{3\sqrt{15}(5-\sqrt{5})}{2}|L(f,2)|^2$ is equivalent to the symmetric-square conjecture $D_{5,\mathrm{odd}} = π^2 L(χ_{-4}\mathrm{Sym}^2 f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\mathrm{even}} = π^2 D_{5,\mathrm{odd}}/(2\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.

math.GM↗