The Riemann Xi function, denoted as \(\Xi(s)\), is a special function closely related to the Riemann zeta function \(\zeta(s)\). It is defined to facilitate the analysis of the zeros of the zeta function, especially in the context of the Riemann Hypothesis.
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Removing the completion's poles produces an entire function with and . Its zeros are the nontrivial zeros of the Riemann zeta function. Its finite order allows Hadamard factorization, while the xi modulus criterion for the Riemann hypothesis characterizes their horizontal location.