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[4] viXra:2611.0019 [pdf] submitted on 2026-11-06 21:24:53
Authors: Ali Rıza Şahin
Comments: 13 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper establishes a rigorous real-analytic and geometric framework governing the distribution of the non-trivial zeros of the Dirichlet eta function $eta(s)$ across the critical strip $mathcal{S} = {s in mathbb{C} : 0 < Re(s) < 1}$. Diverging from traditional statistical and global zero-density bounds, we introduce a localized, deterministic algebraic mechanism defined as textit{Trigonometric Annihilation}. By decomposing $eta(s)$ into its strictly positive, monotonic real-axis baseline profile $eta(sigma)$ and two conditionally convergent, alternating quadratic half-angle residual series, we demonstrate that the vanishing of the real part at any exact root mathematically forces the underlying sine and cosine summation values into absolute scalar parity. This dual equilibrium inherently triggers the universal Pythagorean identity ($sin^2 theta + cos^2 theta = 1$) across the conditionally convergent series, entirely neutralizing the highly oscillatory complex frequency variable $t_0$. The most profound consequence of this annihilation is the textit{monotonic transformation} of the dynamic differences. Deprived of all dynamic oscillatory elasticity, the residual series irreversibly freeze onto the static real baseline. Consequently, the highly malleable dynamic differences $Delta S_{mathrm{Re}}$ and $Delta S_{mathrm{Im}}$ are structurally forced to inherit the strict monotonicity of the static disparity function $A(sigma) = eta(sigma) - eta(1-sigma)$. Under functional reflection symmetry, any hypothetical off-line zero must simultaneously satisfy $A(sigma_0) = Delta S_{mathrm{Re}} = Delta S_{mathrm{Im}}$. However, because Trigonometric Annihilation forces the $Delta S$ parameters to become strictly monotonic, they absolutely lose the geometric elasticity required to bridge the expanding static gap $A(sigma) eq 0$. Furthermore, to definitively resolve the Davenport-Heilbronn obstruction, we directly link this geometric rigidity to the arithmetic distribution of primes via the Euler product and the von Mangoldt function. This absolute monotonic constraint dictates that the dynamically frozen series can only satisfy the functional reflection condition if they simultaneously vanish at the unique geometric origin ($A=0$). Thus, the existence of non-trivial zeros off the critical line is reduced from a probabilistic density inquiry to a deterministic structural contradiction, unconditionally confining all non-trivial zeros to the critical line $Re(s) = 1/2$.
Category: Number Theory
[3] viXra:2611.0016 [pdf] submitted on 2026-11-05 17:34:05
Authors: Ryan Hackbarth
Comments: 16 Pages.
We investigate the complex geometry and Eulerian Beta--Gamma arc-length parameterization of the normalized Dirichlet eta terms. Whereas individual terms on the critical line undergo a nearly sixfold modulus oscillation driven by the Euler prefactor, the two-term partial sum $S_2(s) = t_1(s) + t_2(s)$ traces an exact M"obius circle for every fixed $sigma = operatorname{Re}(s) eq 1$; this circle is centered at the origin if and only if $operatorname{Re}(s) = 1/2$, where it reduces to a scaled Blaschke factor of constant radius. Factoring across odd--even dyadic pairs $P_m(s) = t_m(s) + t_{2m}(s)$ eliminates prefactor amplitude oscillations entirely, collapsing precessing annular rosettes into concentric origin-centered circles on the critical line with strictly decreasing radii. Removing all odd--even pairs leaves the complementary multiples-of-$4$ terms. We prove that on the critical line, the anchor term $t_4(s)$ traces an exact Lemniscate of Bernoulli with foci at $0$ and $1$ (uniquely for $operatorname{Re}(s) = 1/2$), pure powers of $2$ trace closed periodic multi-loop rational curves, and non-powers of $2$ trace precessing lemniscate rosettes. Meanwhile, the dyadic differences $D_{4m}(s) = t_{4m}(s) - t_{8m}(s)$ collapse both families into exact origin-centered circles, partitioning the series termwise and decomposing $zeta(s)$ via analytic continuation. Finally, we establish a pointwise velocity--modulus duality linking the central polar angle $alpha(b)$ of $t_4(1/2 + ib)$ to its fundamental perimeter $L(t_4)$, thereby parameterizing every individual term $t_n(s)$ and every concentric dyadic circle on the critical line in terms of the regularized inverse incomplete Beta function. As a geometric consequence, critical-line zeros are reframed as ordinates $b$ where the resultant of the inner concentric circles lands on the antipode of the initial circle of constant radius.
Category: Number Theory
[2] viXra:2611.0010 [pdf] submitted on 2026-11-03 20:11:43
Authors: Kouji Takaki
Comments: 4 Pages.
We derived an inequality based on the properties of the function log(x), and from that, we obtained a further inequality that serves as a sufficient condition for the conjecture to hold. By evaluating the function involved in this inequality, we demonstrated that it holds for n≧5. Consequently, we have proven that the conjecture is true.
Category: Number Theory
[1] viXra:2611.0006 [pdf] submitted on 2026-11-02 21:52:28
Authors: Andrey Belov
Comments: 9 Pages.
The article deals with the question of the permissibility of using infinite numbers in the formation of counterexamples for Fermat's Great Theorem. Counter examples are given for Fermat's Great Theorem using infinite numbers obtained without performing arithmetic operations on these numbers.
Category: Number Theory
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