Journal of the Ramanujan Mathematical Society
Volume 38, Issue 4, December 2023 pp. 355–367.
Averages of double shifted convolution sum of half-integral weight cusp forms
Authors:
Manish Kumar Pandey and Lalit Vaishya
Author institution:
SRM University AP, Andhra Pradesh, India.
Summary:
In this article, we obtain an estimate of the following sum associated to the
Fourier coefficients of cusp forms of half-integral weight:
{1/H} ∑
{h ≤ H} V
(h/H) ∑{n ≤ N}
λ {f1} (n) λ {f2} (n+h) λ {f3} (n+2h)W (n/N)
where λ{fi} (n) are the normalized Fourier coefficients of half-integral
weight cusp forms (newforms) f{i} of weight k{i} + 1/2 for i = 1,2,3 on the
congruence subgroup of level 4M, k{i} ≥ 2 and V andW are smooth bump functions
supported on [1,2]. We obtain a non-trivial upper bound estimate using a
variant of circle method.
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