Zeros of the Davenport-Heilbronn counterexample

Abstract

Abstract. We compute zeros off the critical line of a Dirichlet series considered by H. Davenport and H. Heilbronn. This computation is accomplished by deforming a Dirichlet series with a set of known zeros into the DavenportHeilbronn series.

Note on the Zeros of a Dirichlet Function

“…Fig. 1 above suggests that for the function fs there are no off critical line non trivial zeros in the interval t  80, 180 considered in [4] and [1], despite of the obvious non alignment of the zeros. This last phenomenon is certainly due to the errors of approximation of tan .…”

“…Examples 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].…”

Symmetry of zeros of Lerch zeta-function for equal parameters

“…By Rouché's theorem, we have that for every sufficiently small open disc D with center at ρ in which the function L(λ 0 , λ 0 , s) has no other zeros except for ρ, there exists δ = δ(D) > 0 such that each function L(λ, λ, s), where λ ∈ (λ 0 − δ, λ 0 + δ), has exactly m zeros (counted with multiplicities) in the disc D (c.f. Theorem 1 in Balanzario and Sánchez-Ortiz [2] and Lemma 4.1 in Dubickas, Garunkštis, J. Steuding and R. Steuding [4]). If zero ρ is of multiplicity m = 1, then there exists a neighborhood of λ 0 and some function ρ = ρ(λ), which is continuous at λ 0 and, in addition, satisfies the relation L(λ, λ, ρ(λ)) = 0.…”

On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients

“…Finally, our method is computationally different from the other methods for searching the zeros with real part bigger than a given number (for example the secant method, used in [2], or the method of deforming Dirichlet series presented in [1]). The method itself is very fast and provides computations with arbitrary precision; however, it is not suitable for computations of zeros; it provides information about (eventual) existence of zeros in certain right half-planes.…”