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- W4366245893 abstract "Based on the Hadamard product $xi(s)= xi(0)prod_{rho}(1-frac{s}{rho})$, a new absolute convergent expression of $xi(s)$ is obtained by paring $rho_i$ and $bar{rho}_i$, and putting all the $rho_i$ related multiple zeros together in one factor, i.e., $$xi(s)=xi(0)prod_{i=1}^{infty}Big{(}frac{beta_i^2}{alpha_i^2+beta_i^2}+frac{(s-alpha_i)^2}{alpha_i^2+beta_i^2}Big{)}^{d_{i}}$$ where $xi(0)=frac{1}{2}$, $rho_i=alpha_i+jbeta_i$ and $bar{rho}_i=alpha_i-jbeta_i$ are the complex conjugate zeros of $xi(s)$, $0<alpha_i<1$ and $beta_ineq 0$ are real numbers, $d_igeq 1$ are the real multiplicities of $rho_i$, $beta_i$ are in order of increasing $|beta_i|$, i.e., $|beta_1|leq|beta_2|leq|beta_3|leq cdots$. Then, by the functional equation $xi(s)=xi(1-s)$, we have $$xi(0)prod_{i=1}^{infty}Big{(}frac{beta_i^2}{alpha_i^2+beta_i^2}+frac{(s-alpha_i)^2}{alpha_i^2+beta_i^2}Big{)}^{d_{i}} =xi(0)prod_{i=1}^{infty}Big{(}frac{beta_i^2}{alpha_i^2+beta_i^2}+frac{(1-s-alpha_i)^2}{alpha_i^2+beta_i^2}Big{)}^{d_{i}}$$ i.e., $$prod_{i=1}^{infty}Big{(}1+frac{(s-alpha_i)^2}{beta_i^2}Big{)}^{d_{i}}=prod_{i=1}^{infty}Big{(}1+frac{(1-s-alpha_i)^2}{beta_i^2}Big{)}^{d_{i}}$$ which, by Lemma 3, is equivalent to $$begin{cases}&alpha_i=frac{1}{2}, i =1,2,3, cdots, infty & |beta_1|<|beta_2|<|beta_3|< cdots end{cases}$$ Thus, we conclude that the Riemann Hypothesis is true." @default.
- W4366245893 created "2023-04-20" @default.
- W4366245893 creator A5080014014 @default.
- W4366245893 date "2023-04-17" @default.
- W4366245893 modified "2023-09-30" @default.
- W4366245893 title "A Proof Of The Riemann Hypothesis Based On A New Expression Of The Completed Zeta Function" @default.
- W4366245893 doi "https://doi.org/10.20944/preprints202108.0146.v25" @default.
- W4366245893 hasPublicationYear "2023" @default.
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