Abstract

Let $\phi (s,\chi )$ be a function defined as the product of $\xi (s,\chi )$ and $\xi (s,\overline{\chi })$, where $\chi$ is a primitive character. We determine the positivity and negativity properties of the real part of the logarithmic derivative of $\phi (s,\chi )$ in the zero-free region. This result establishes an equivalence relation for the generalized Riemann hypothesis. Furthermore, we prove that the modulus of $\phi (s,\chi )$ is strictly increasing (respectively, decreasing) in the zero-free right (respectively, left) half-plane along every circle centered at the intersection of the real axis and the vertical boundary line of the zero-free right (respectively, left) half-plane. Using this result, we propose a new criterion for the generalized Riemann hypothesis.

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