On some results of Agelas concerning the GRH and of Vassilev-Missana concerning the prime zeta function

TL;DR

This paper critically examines recent claims of proof for the GRH and RH, identifying errors in the proofs and demonstrating the falsity of related theorems, thereby cautioning researchers and emphasizing error detection.

Contribution

It refutes recent purported proofs of the GRH and RH, clarifies errors in related theorems, and offers pedagogical insights on early error detection in mathematical proofs.

Findings

Agélas's proof of GRH and RH contains errors.

Theorem 1 of Vassilev-Missana is false.

Theorem 2 of Vassilev-Missana is false.

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.