Some zeros of the Titchmarsh counterexample

Abstract

Abstract. Zeros on and off the critical line are found for Titchmarsh's function As).Let s = a + it. E. C. Titchmarsh [1, pp. 240-244] With the help of programs for computing L and L' (Spira [3]), an exploratory computation of f(s) in the critical strip for 0 < t < 200 revealed the following zeros off the critical line:

Note on the Zeros of a Dirichlet Function

“…This can be seen by noticing that there are six curves turning around both zeros. The other points indicated in [4] are too far from the zeros appearing in this picture in order to be confused with one of them. However, even this point cannot be taken as one of them since then the other should be the symmetric with it with respect to the critical line, which is not the case.…”

“…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].…”

“…Applying it to a false zero provides a contradiction of the functional equation. Although this test is a rather experimental complement, it gave us a clue about non zeros of the points signaled in [4].…”

On Davenport and Heilbronn-Type of Functions

“…They even thought they had identified two such zeros, yet they acknowledged that "the calculations are very cumbrous, and can hardly be considered conclusive". However, after almost 60 years Spira's [2] calculations produced some more such zeros, which have found an undisputed place in the literature (see [3], [4], [5]), despite of a certain apparent ambiguity regarding the symmetric points with respect to the critical line. In [6] and [7] some more off critical line zeros of that function have been indicated and other functions obtained by a similar construction have been shown as possessing off critical line zeros.…”

Discretized Keiper/Li Approach to the Riemann Hypothesis

“…[11][30, § 10.25][29][3][6, § 5] We may then seek a testing ground for the RH-false branch of our asymptotic alternative (generalized, as (65)-(66)).Specifically, for φ = 1 2 (1 +√ 5) (the golden ratio), let τ ± def = −φ± 1 + φ 2 (τ + ≈ 0.28407904384, τ − = −1/τ + ≈ −3.52014702134); (75) ν ± (k) def = {1, +τ ± , −τ ± , −1, 0, . .…”