We formulate and prove the extension of the Rogers integral formula (Rogers [Acta Math. 94 (1955), 249–287]) to the adeles of number fields. We also prove the second moment formulas for a few important cases, enabling a number of classical and recent applications of the formula to extend immediately to any number field.

中文翻译：

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Adelic Rogers 积分公式

我们制定并证明了罗杰斯积分公式（Rogers [Acta Math. 94 (1955), 249–287]）到数域阿黛尔的推广。我们还证明了一些重要情况的二阶矩公式，使该公式的许多经典和最新应用能够立即扩展到任何数域。