We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progression. We prove a result analogous to previous work of Fouvry-GangulyKowalski-Michel and Kowalski-Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution which comes from the asymptotic mixed distribution of Salié sums.

中文翻译：

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算术级数中半积分权的模形式的傅立叶系数

我们研究算术级数中傅立叶系数之和的概率行为。我们证明了类似于 Fouvry-GangulyKowalski-Michel 和 Kowalski-Ricotta 在半积分权重全纯尖点形式和素功率模量的背景下先前工作的结果。我们实际上表明，这些和在合适的范围内遵循混合高斯分布，该分布来自 Salié 和的渐近混合分布。