The Riemann Hypothesis for Dirichlet $L$ Functions

This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is shown that the Riemann hypothesis holds for the Dirichlet beta function $L_{-4}(s)$ if and only if it holds for $ζ(s)$- a particular case of the Generalized Riemann Hypothesis.

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