The Ramanujan Journal ( IF 0.7 ) Pub Date : 2020-07-21 , DOI: 10.1007/s11139-020-00268-9 Moni Kumari , Jyoti Sengupta
Let f and g be two Hecke–Maass cusp forms of weight zero for \(SL_2({\mathbb {Z}})\) with Laplacian eigenvalues \(\frac{1}{4}+u^2\) and \(\frac{1}{4}+v^2\), respectively. Then both have real Fourier coefficients say, \(\lambda _f(n)\) and \(\lambda _g(n)\), and we may normalize f and g so that \(\lambda _f(1)=1=\lambda _g(1)\). In this article, we first prove that the sequence \(\{\lambda _f(n)\lambda _g(n)\}_{n \in {\mathbb {N}}}\) has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of f and g.
中文翻译:
Hecke-Maass形式的傅立叶系数的第一个同时符号变化
令f和g为\(SL_2({\ mathbb {Z}})\)的权重为零的两个Hecke-Maass尖点形式,具有Laplacian特征值\(\ frac {1} {4} + u ^ 2 \)和\ (\ frac {1} {4} + v ^ 2 \)。然后两者都具有真实的傅立叶系数,即\(\ lambda _f(n)\)和\(\ lambda _g(n)\),我们可以对f和g进行归一化,以便\(\ lambda _f(1)= 1 = \ lambda _g(1)\)。在本文中,我们首先证明序列\(\ {\ lambda _f(n)\ lambda _g(n)\} _ {n \ in {\ mathbb {N}}} \\)有无数种符号变化。然后,根据f和g的拉普拉斯特征值,得出相同序列的第一个负系数的界。