Mean-square values of the Riemann zeta function on arithmetic progressions

Hirotaka Kobayashi
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Abstract

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions \(\frac{1}{2} + i(a n + b)\). It reveals noticeable relation between the discrete moments and the continuous moment of the Riemann zeta function. Especially, when a is a positive integer, main terms of the formula are equal to those for the continuous mean value. The proof requires the rational approximation of \(e^{\pi k/a}\) for positive integers k.

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算术级数上黎曼zeta函数的均方值
我们得到了黎曼zeta函数在算术级数 \(\frac{1}{2} 上的第二离散矩的渐近公式。+ i(a n + b))。它揭示了黎曼zeta函数离散矩与连续矩之间的明显关系。特别是当 a 为正整数时,公式的主要项等于连续均值的主要项。证明需要对正整数 k 进行 \(e^{\pi k/a}\) 的有理逼近。
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