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 Mat. Zametki, 2007, Volume 82, Issue 1, Pages 84–98 (Mi mz3756)

Lower Bounds for Positive and Negative Parts of Measures and the Arrangement of Singularities of Their Laplace Transforms

A. Yu. Popov, A. P. Solodov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: For a real measure with variation $V(x)$ satisfying the estimate $V(x)\le c_0\exp(Cx)$ and with the Laplace transform holomorphic in the disk $\{|s-C|\le C\}$ and having at least one pole of order $m$, we obtain lower bounds for the positive and negative parts of the measure $V_\pm(x)>cx^m$, $x>x_0$. We establish lower bounds for $V_\pm(x)$ on “short” intervals. Applications to number theory of the results obtained are considered.

Keywords: real measure, positive and negative parts of a measure, Laplace transform, analytic function, pole of a meromorphic function, Möbius function, Riemann zeta function

DOI: https://doi.org/10.4213/mzm3756

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English version:
Mathematical Notes, 2007, 82:1, 75–87

Bibliographic databases:

UDC: 517.442

Citation: A. Yu. Popov, A. P. Solodov, “Lower Bounds for Positive and Negative Parts of Measures and the Arrangement of Singularities of Their Laplace Transforms”, Mat. Zametki, 82:1 (2007), 84–98; Math. Notes, 82:1 (2007), 75–87

Citation in format AMSBIB
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