Reading the rightmost zeros of the Davenport–Heilbronn function off its coefficients: a numerical companion to Bombieri–Ghosh

Bombieri and Ghosh computed the supremum of the real parts of the zeros of the Davenport–Heilbronn function and of its sieved versions: 1.120362 for the function itself, and 1.046712, 1.063679, 1.093482 and 1.106200 for q = 2, 3, 7, 13. We recover these numbers from the coefficients of −f′/f alone, computed exactly to n = 6·107, without computing any zero. A first registered test, comparing raw growth exponents with the suprema at tolerance 0.002, failed: the raw exponents overshoot by 0.009 to 0.023. The overshoot is described for all five functions at once by one empirical correction, exp(−K/((σ−1) log x)) with K ≈ 0.27–0.31; fitted on x ≤ 2·107 and tested blind on 2·107 < x ≤ 6·107, the corrected model held 4 of 4 registered criteria. The principle (the growth exponent is the rightmost singularity) is classical; we did not find the measurement or the correction law stated in the literature searched. K is empirical. Written jointly with Claude, an AI system made by Anthropic, which carried out much of the computation, the derivations and the drafting; the human author is responsible for the content. Not peer reviewed. The package contains the PDF, its LaTeX source, and the scripts, registered predictions and result files it cites.