In short: the note shows that Montgomery’s large-value and moment conjectures for Dirichlet polynomials are false as stated for general coefficients. Short sums are enough to break them. It also shows that the bounded-coefficient version (1.6) is very strong: it would imply the Kakeya set conjecture in every dimension. The text you pasted has lost almost all of its displayed formulas in OCR. Below I fill in the most likely forms from context. Treat them as plausible reconstructions, not quotations.This looks like J. Bourgain, Remarks on Montgomery’s conjectures on Dirichlet sums (IHES preprint, early 1990s). I couldn’t confirm the exact reference online, because the search came back empty.1. The conjectures (likely forms)Throughout, and is 1-separated.
| No. | Statement | Status in the note |
|---|---|---|
| (1.2) | $\sum_{t\in\mathcal T} | D(t) |
| (1.3) | $\int_0^T | D(t) |
| (1.4) | $\int_0^T | D |
| (1.6) | $\sum_{t\in\mathcal T} | D(t) |
(1.3) holds for integer because is itself a Dirichlet polynomial of length , so (1.4) applies to it. Montgomery showed that bounds like these imply the density hypothesis .The other estimates mentioned:Lindelöf (1.8): .Halász–Montgomery (1.10): unconditionally. Under Lindelöf, becomes . Bourgain says this is essentially optimal for general coefficients.
