On the distribution of zeros of a certain Dirichlet series
Abstract
[Abstract placeholder — two or three sentences. We establish a bound on the real parts of the non-trivial zeros of the series in question, show that the bound is sharp under a mild growth hypothesis, and give a counterexample when the hypothesis is dropped.]
1. Introduction
[Body placeholder.] Let s denote a complex variable and let the series be defined as usual on the half-plane of absolute convergence. One may assume, following the classical treatment, that the analytic continuation exists; we do not require it, and the argument below is elementary in the sense that it invokes no result later than 1932.
[Body placeholder.] Whence the strategy: bound the partial sums, transfer the bound across the critical strip, and observe that the transferred bound is incompatible with a zero off the expected line. Sections 2 and 3 carry out the two halves; Section 4 exhibits the counterexample.
2. The partial-sum bound
[Body placeholder.] Let N be a positive integer and define the truncation in the obvious way. The estimate we need is uniform in the imaginary part, and the constant is explicit — a point on which one referee insisted, correctly, and which improved the paper.
Referee reports
Published in full, signed, per Axiom III[Report placeholder.] The argument in Section 2 is correct as written. I asked the authors to make the constant explicit; they did. My AI assessment is one step: the exposition shows signs of machine polishing, which the authors declared, and the mathematics does not.
[Report placeholder.] I recommended rejection of the first version and I am recorded as having done so. The revised Section 4 answers my objection. I record my AI assessment as one step and note that it agrees with the authors’ declaration.