bibkey: coppolalaporta2015gallagher authors: Giovanni Coppola and Maurizio Laporta year: 2015 title: A generalization of Gallagher’s lemma for exponential sums doi: null url: https://arxiv.org/abs/1411.1739v1 claim: The weighted Gallagher lemma retains complex coefficients in one smoothed square. Its finite signed-measure application bounds an arithmetic head’s band allowance, without supplying that head’s sign or a cofinal Weil estimate. strata_touched: [] license: citation-only triage: anchor

A signed smoothing supplier for a finite arithmetic head

The inspected primary manuscript is arXiv:1411.1739v1, submitted 21 October 2014, 14 pages, SHA-256 9551081b48e9de583b49f160e9708701cc91691458ceef0339ab0f2fd94a1ffe. The arXiv metadata lists Šiauliai Mathematical Seminar 10 (18) (2015), 29–47. Locators below refer to the inspected preprint; the journal text is not claimed inspected.

The unnumbered Lemma in Section 1, printed p.2, equations $(\star \star )$ and $(\star \star \star )$, assumes an absolutely convergent exponential series with strictly increasing real frequencies and complex coefficients. For an integrable smoothing weight it bounds the series’ band mean square by the spatial mean square of the weighted coefficient sum, divided by the minimum squared Fourier modulus on that band. The proof is in Section 2. The source convention is $e(t)=e^{2\pi it}$.

Printed p.3 specializes to the Cesàro tent $C_{\delta }(t)=(1-|t|/\delta {)}_{+}$ and gives its Fourier transform $\delta {\mathrm{sinc}}^2(\pi \delta y)$, with $\mathrm{sinc}z=\sin z/z$ and value one at zero. The coefficients remain inside the same square. Replacing them by their absolute values is a different majorant and can lose cancellation.

The finite signed-head application uses the same Plancherel mechanism for a finite measure containing both prime-power atoms and a continuous main term. In the angular frequency convention its tent floor is ${\delta }^2{\mathrm{sinc}}^4(\delta T/2)$ when $0<\delta T<2\pi$. The finite-measure extension and the factor $2\pi$ are explicit applications, not the verbatim discrete source theorem.

The project’s directed arithmetic examples retain prime–prime, prime–continuum and continuum–continuum contributions. They estimate a finite-head coupling input; this source supplies neither a favorable low-block sign, a complete high–high estimate, nor the common cofinal parameter budget needed for RH or Robin.