On Davenport and Heilbronn-Type of Functions

Abstract

A correction is brought to the opinion expressed in a previous note published in this journal that the off critical line points indicated as being non trivial zeros of 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.

On The Generalized Riemann Hypothesis II

“…The off critical line non trivial zeros found for the Davenport and Heilbronn function show that the second class is not empty. We have shown in [18] that infinitely many functions of that type can be obtained by a similar construction. It is then interesting to put into evidence some general properties of these functions.…”

“…However, it has been obtained as a linear combination of series which were Euler products. We have shown in [18] that this function is not so special after all, since infinitely many functions of this type can be built using appropriate Dirichlet L-series. The question remains if all of them violate RH.…”