On some results of Agelas concerning the GRH and of Vassilev-Missana concerning the prime zeta function
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Publication:6363038
DOI10.7546/NNTDM.2021.27.2.49-50arXiv2103.09418MaRDI QIDQ6363038
Publication date: 16 March 2021
Abstract: A recent paper by Ag'elas [Generalized Riemann Hypothesis, 2019, hal-00747680v3] claims to prove the Generalized Riemann Hypothesis (GRH) and, as a special case, the Riemann Hypothesis (RH). We show that the proof given by Ag'elas contains an error. In particular, Lemma 2.3 of Ag'elas is false. This Lemma 2.3 is a generalisation of Theorem 1 of Vassilev-Missana [A note on prime zeta function and Riemann zeta function, Notes on Number Theory and Discrete Mathematics, 22, 4 (2016), 12-15]. We show by several independent methods that Theorem 1 of Vassilev-Missana is false. We also show that Theorem 2 of Vassilev-Missana is false. This note has two aims. The first aim is to alert other researchers to these errors so they do not rely on faulty results in their own work. The second aim is pedagogical - we hope to show how these errors could have been detected earlier, which may suggest how similar errors can be avoided, or at least detected at an early stage.
(zeta (s)) and (L(s, chi)) (11M06) Arithmetic functions; related numbers; inversion formulas (11A25) Nonreal zeros of (zeta (s)) and (L(s, chi)); Riemann and other hypotheses (11M26) Zeta and (L)-functions: analytic theory (11M99) Cyclotomy (11T22)
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