A note on Fourier coefficients of Hecke eigenforms in short intervals

Sanoli Gun, Sunil Naik
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Abstract

In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in an interval of length \(\frac{x}{(\log x)^A}\) for any \(A>0\), modifying an earlier work of Balog and Ono. Furthermore, we need to strengthen a work of Rouse-Thorner to derive a lower bound for the largest prime factor of Fourier coefficients in an interval of length \(x^{1/2 + \epsilon }\) for any \(\epsilon >0\).

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关于短区间赫克特征形式傅里叶系数的说明
在这篇文章中,我们研究了短区间内非 CM 归一化 cuspidal Hecke 特征形式的傅里叶系数的大质因数。新内容之一涉及在任意 \(A>0\) 的长度区间内推导出切波特列夫密度定理的显式版本(\frac{x}{(\log x)^A}\ ),修改了巴洛格和小野的早期工作。此外,我们还需要加强 Rouse-Thorner 的一项工作,为任意 \(\epsilon >0\) 的长度为 \(x^{1/2 + \epsilon }\) 的区间中傅里叶系数的最大质因子推导出一个下限。
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