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arXiv 1008.4014math.CAmath.NT
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Minkowski问号函数的Fourier-Stieltjes系数

Fourier-Stieltjes coefficients of the Minkowski question mark function

Giedrius Alkauskas

备注：
15 pages, 4 figures. Typos fixed (e.g., in the formula (7)). To appear in: Analytic and Probabilistic Methods in Number Theory, Proceedings of the Fifth Intern. Conf. in Honour of J. Kubilius, Palanga, Lithuania, 4-10 September 2011
发表信息：
Anal. Probab. Methods Number Theory, Proceedings of the Fifth Intern. Conf. in Honour of J. Kubilius, Palanga, Lithuania, 4-10 September 2011 (2012), 19-33
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摘要：

本文研究了Minkowski问号函数的Fourier-Stieltjes系数。1943年，R.塞勒姆问这些系数是否在无穷远处消失。我们提出了一个猜想，它隐含着对Salem问题的肯定回答。此外，我们证明了这些Fourier-Stieltjes系数之间的无限线性恒等式。我们还提供了一种方法来数值计算相关zeta函数的整数的特殊值，具有高精度（超过30位）。

英文摘要：

In this paper we investigate the Fourier-Stieltjes coefficients of the Minkowski question mark function. In 1943, R. Salem asked whether these coefficients vanish at infinity. We propose the conjecture which implies the affirmative answer to Salem's problem. Further, we prove infinite linear identities among these Fourier-Stieltjes coefficients. We also provide a method to numerically calculate special values at integers of the associated zeta function with a high precision (more than 30 digits).

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