On Davenport and Heilbronn-Type of Functions

TL;DR

This paper corrects previous claims about zeros of Davenport and Heilbronn functions, demonstrating that certain off-critical line points are true zeros and that linear combinations of related L-functions do not disprove the Riemann Hypothesis.

Contribution

It clarifies the nature of zeros of Davenport and Heilbronn functions and refutes the idea that linear combinations of similar L-functions challenge the Riemann Hypothesis.

Findings

Off-critical line points are confirmed as true zeros.

Linear combinations of L-functions do not serve as counterexamples to RH.

Approximation errors in previous studies are addressed and corrected.

Abstract

A correction is brought to the opinion expressed in a previous note published in this journal that the off critical line points indicated by some authors as being non trivial zeros of the Davenport and Heilbronn function are affected of approximation errors and illustrations are presented which enforce the conclusion that they are true zeros. It is shown also that linear combinations of L-functions satisfying the same Riemann-type of functional equation do not offer counterexamples to RH, contrary to a largely accepted position.