A Proof of the Riemann Hypothesis via the Divisibility of Entire Functions
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This paper presents a proof of the Riemann Hypothesis (RH), in which the divisibility of entire functions arising from the symmetric functional equation, the uniqueness of zero multiplicities, and the Euler product property (which guarantees the divisibility of entire functions by confining all the zeros of the completed Riemann zeta function \(\xi(s)\) to the critical strip) play pivotal roles. We first develop a new absolutely and uniformly convergent expression for \(\xi(s)\) by pairing the conjugate zeros \(\rho_i\) and \(\bar{\rho}_i\), counted with their multiplicities, in the Hadamard product, i.e. \[ \xi(s) = \xi(0) \prod_{\rho} \Bigl(1 - \frac{s}{\rho}\Bigr) = \xi(0) \prod_{i=1}^{\infty} \Bigl(1 - \frac{s}{\rho_i}\Bigr) \Bigl(1 - \frac{s}{\bar{\rho}_i}\Bigr) = \xi(0) \prod_{i=1}^{\infty} \Bigl( \frac{\beta_i^2}{\alpha_i^2 + \beta_i^2} + \frac{(s-\alpha_i)^2}{\alpha_i^2 + \beta_i^2} \Bigr)^{m_i} \] where \(\rho\) ranges over all zeros of \(\xi(s)\), \((\rho_i, \bar{\rho}_i)\) range over all conjugate zero pairs of \(\xi(s)\), \(\rho_i = \alpha_i + j\beta_i\) with \(0 < \alpha_i < 1\), \(0 < |\beta_1| \le |\beta_2| \le \cdots\), \(\sum_{i=1}^{\infty} \frac{1}{\beta_i^2} < \infty\), and \(m_i \ge 1\) is the multiplicity of the \(i\)-th quadruplet of zeros \((\rho_i, \bar{\rho}_i, 1-\rho_i, 1-\bar{\rho}_i)\). Then, according to the symmetric functional equation \(\xi(s) = \xi(1-s)\), we obtain \[ \prod_{i=1}^{\infty} \Bigl( 1 + \frac{(s-\alpha_i)^2}{\beta_i^2} \Bigr)^{m_i} = \prod_{i=1}^{\infty} \Bigl( 1 + \frac{(1-s-\alpha_i)^2}{\beta_i^2} \Bigr)^{m_i} \] which, owing to the divisibility of entire functions, uniqueness of \(m_i\), and the irreducibility of each real quadratic polynomial factor, is finally equivalent to \[ \alpha_i = \frac{1}{2}, \quad 0 < |\beta_1| < |\beta_2| < |\beta_3| < \cdots, \quad i = 1, 2, 3, \dots \] Thus, we conclude that the RH is true.