Note on the Zeros of a Dirichlet Function

Abstract

The existence of non trivial zeros off the critical line for a function obtained by analytic continuation of a particular Dirichlet series is studied. Our findings are in contradiction with some results of computations which were present in the field for a long time. In the first part of the note we illustrate how the approximation errors may have had as effect inexact conclusions and in the second part we prove rigorously our point of view. AMS subject classification: 30C35, 11M26Examples of functions obtained by analytic continuation of Dirichlet series, which have zeros off the line Re s  1/2 are important for the purpose of circumscribing the field where the Riemann Hypothesis might be true. Such a candidate is attributed by some mathematicians to Davenport and Heilbronn (1936) (see [1]) and by others to Titchmarsh [5] (see [4]). In [6], R. C. Vaughan provided an elementary clear presentation of the respective example using the robust theory from [3]. He starts with that Dirichlet character  modulo 5 for which 2  i. For this characterRepeating the same computation for the character  it can be easily found that   . With the notation used in Mathematica implementation of Dirichlet L-functions, L5, 2, s   n1  nn s and L5, 3, s   n1  nn s for Re s  1, it is known that L5, 2, s and L5, 3, s have analytic continuations to the whole complex plane and they are entire functions. Moreover, by corollary 10.9 of [3], they verify the functional equations:(3) L5, 2, s  2 s  s1 5 1/2s 1  scos s 2 L5, 3, 1  s (4) L5, 3, s  2 s  s1 5 1/2s 1  scos s 2 L5, 2, 1  s

On Davenport and Heilbronn-Type of Functions

“…We made known our findings in [8]. However later we discovered that inadvertently one of the Dirichlet characters we were supposed to use in the construction of the Davenport and Heilbronn function was wrong, hence our function did not satisfy a Riemann-type of functional equation and, normally this fact produced the lack of symmetry.…”

On The Generalized Riemann Hypothesis II

“…We were wrong assuming that the deviation from the critical line of Spira's and Balazario's points is due to errors of approximation (see [13], [14] ) and they are not true non trivial zeros. This assumption was supported by the fact that in [13] we took inadvertently a wrong Dirichlet character in building what we considered to be the Davenport and Heilbronn function.…”