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\newcommand*{\A}{\mathbb{A}}%.............................adele
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\newcommand*{\C}{\mathbb{C}}%.............................C
\newcommand*{\Ch}[1][\null]{\mathop{\mathbb{C}_{#1}^{\mathrm{h}}}\nolimits}%...............covariance.harmonique
\newcommand*{\Cheps}[1][\null]{\mathop{\mathbb{C}_{#1}^{\mathrm{h},\epsilon}}\nolimits}%...............covariance.harmonique.signee
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\newcommand*{\prp}{\mathsf{P}}%............................property.P
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\newcommand*{\sumh}{\mathop{\sideset{}{^\mathrm{h}}\sum}}%.somme.hyperbolique
\newcommand*{\sumsh}{\mathop{\sideset{}{^\sharp}\sum}}%.somme.sqrt{2}
\DeclareMathOperator{\sym}{Sym}%..........................Sym
\DeclareMathOperator{\TE}{TE}%............................TE
\DeclareMathOperator{\TP}{TP}%............................TP
\newcommand*{\un}{\mathop{\mathbb{1}}\nolimits}%...........................f.car
\newcommand*{\V}[1][\null]{\mathop{\mathbb{V}_{#1}}\nolimits}%.............................Variance
\newcommand*{\Vh}[1][\null]{\mathop{\mathbb{V}_{#1}^{\mathrm{h}}}\nolimits}%...............variance.harmonique
\newcommand*{\Vheps}[1][\null]{\mathop{\mathbb{V}_{#1}^{\mathrm{h},\epsilon}}\nolimits}%...............variance.harmonique.signee
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%...................................
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\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{convention}[theorem]{Convention} 
\newtheorem{conjecture}[theorem]{Conjecture} 
%................................
\newtheorem{propint}{Proposition}%...........................prop intro
\renewcommand{\thepropint}{\Alph{propint}}
\newtheorem{theoint}[propint]{Theorem}%.......................theo intro
\renewcommand{\thetheoint}{\Alph{thmint}}
\newtheorem{corint}[propint]{Corollary}%.............................corol intro
\renewcommand{\thecorint}{\Alph{corint}}
\newtheorem{lemint}[propint]{Lemma}%..................................lem intro
\renewcommand{\thelemint}{\Alph{lemint}}
%................................
\newtheorem*{hypothesis}{Hypothesis $\Nice(r,f)$}
\newtheorem*{hypCoMi}{Hypothesis $\sym^r(f)$}
\newtheorem*{ATRPS}{Hypothesis $\Hy_2(\theta)$}
\newtheorem*{hGRH}{Hypothesis $\GRH(r)$}
%...................................
\theoremstyle{erdfn}
%...................................
\newtheorem{definition}[theorem]{Definition}
\newtheorem*{notations}{Notations}
\newtheorem*{merci}{Acknowledgements}
%...................................
\theoremstyle{errem}
%...................................
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%................................

\date{Version of \today}
\subjclass{Primary 11F99, 20C08; Secondary 15A21.}
\keywords{???}
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