Some Series Related to Extended Riemann Hypothesis for Dedekind Zeta Functions

2025/12/20 by Muhammad Zaigham Zaheer, Zaheer, Muhammad Atif
Mathematics · #11M06 #11R42 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Meromorphic and Entire Functions

paper · doi:10.48550/arxiv.2512.19761

openalex publication_date 2025/12/20 · openalex created_date 2025/12/25 · openalex updated_date 2026/07/28

Abstract

We obtain closed form of some infinite series involving derivatives of an analogue of the Riemann xi function for Dedekind zeta function and nontrivial zeros of Dedekind zeta function assuming the Extended Riemann Hypothesis. Conversely, we prove that if this closed form holds, then all of the zeros of Dedekind zeta function beyond a certain height lie on the critical line. This yields a large number of equivalent statements of Riemann Hypothesis.

Citations

• Some series equivalent to the extended Riemann hypothesis for Dedekind zeta functions

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