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DISCREPANCY BOUNDS FOR THE DISTRIBUTION OF RANKIN–SELBERG L-FUNCTIONS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  18 October 2024

XIAO PENG Open the ORCID record for XIAO PENG [Opens in a new window]
XIAO PENG*
Affiliation:
School of Computer Science and Engineering, Macau University of Science and Technology, Macau, PR China
*
e-mail: pengxiao.must@gmail.com
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Abstract

We investigate the discrepancy between the distributions of the random variable $\log L (\sigma , f \times f, X)$ and that of $\log L(\sigma +it, f \times f)$, that is,

$$ \begin{align*} D_{\sigma} (T) := \sup_{\mathcal{R}} |\mathbb{P}_T(\log L(\sigma+it, f \times f) \in \mathcal{R}) - \mathbb{P}(\log L(\sigma, f \times f, X) \in \mathcal{R})|, \end{align*} $$

where the supremum is taken over rectangles $\mathcal {R}$ with sides parallel to the coordinate axes. For fixed $T>3$ and $2/3 <\sigma _0 < \sigma < 1$, we prove that

$$ \begin{align*} D_{\sigma} (T) \ll \frac{1}{(\log T)^{\sigma}}. \end{align*} $$

Keywords

discrepancyRankin–Selberg L-functionsrandom variables

MSC classification

Primary: 11F12: Automorphic forms, one variable
Secondary: 11K38: Irregularities of distribution, discrepancy

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 1 , February 2025 , pp. 32 - 42
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author is supported by The Science and Technology Development Fund, Macau SAR (File no. 0084/2022/A).

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