UDC 511 We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the corresponding $\zeta$-functions. We use these results to prove prime number theorems and a Selberg formula for such semigroups.

中文翻译：

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关于加性算术半群数论的评注

UDC 511 我们处理加性算术半群，并给出相应 $\zeta$ 函数零点分布的新旧证明。我们使用这些结果来证明这些半群的素数定理和塞尔伯格公式。