This paper investigates the analytic structure of the family of Dirichlet Lfunctions on the critical line from a geometric perspective, proposing a zero-counting mechanism based on geometric structure. By introducing a unitary normalization factor W (χ) -1/2 , we construct a real-valued function Ξ Λ (s, χ) with conjugate symmetry. On this basis, we establish the zero-counting principle of "Anchoring at Two Ends, Counting in the Middle," and derive an integer-type counting formula independent of the asymptotic error term S(T, χ): In addition, this paper applies the proposed universal geometric counting framework to an extended context, demonstrating that it naturally encompasses the classical case of the Riemann ζ(s) function.

Let a,b,c be positive integers such that a²+b²=c², 2|b,gcd(a,b)=1. In 1956, Jesmanowicz conjectured that for any positive integer w, the only solution of (aw)^x+(bw)^y=(cw)^z in positive integers is (x, y, z) = (2, 2, 2). In this paper, based on Gaussian integer ring, we show that Je$\acute{s}$manowicz' conjecture is true for any positive integer

The research shows that Riemann proved that all of zeros of Riemann's zeta function are on $\sigma=1/2$ based on the functional equation \begin{align*} \pi^{-\frac{s}{2}}\Gamma \left( \frac{s}{2} \right) \zeta(s)&={\frac{1}{s(s-1)} + \int\limits_1^\infty \psi(x) \left( x^{\frac{s}{2} - 1} + x^{-\frac{1+s}{2}} \right) \,dx,}\quad\qquad{s}=\sigma+it, \end{align*} which is in Riemann's ``\"{U}ber die Anzahl der Primzahlen unter einer gegebenen Grosse". According to the geometric meaning of the functional equation and the argument principle, we obtain the number of zeros $N_0(T)$ of the Riemann zeta function on the critical segment $\sigma=1/2,0\leq{t}\leq{T}$ and the number of zeros $N(T)$ of the Riemann zeta function in the rectangular region $-1\leq\sigma\leq{2},0\leq{t}\leq{T}$, respectively. The result is \begin{align*} N(T)&=N_0(T)=\frac{\arg{\left[\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2} \right)\zeta(s)\right]}}{\pi}+1\\ &=\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+O(\log{T}),\qquad{s=1/2+iT}. \end{align*}