Journal of the Ramanujan Mathematical Society
Volume 39, Issue 4, December 2024 pp. 307–321.
On sign changes of the Fourier coefficients of symmetric power L-functions
Authors:
Huafeng Liu and Rui Liu
Author institution:School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, Shandong, P. R. China.
Summary:
Let λ{f} (n) and λ{sym} {j} {f} (n) be the normalised Fourier
coefficients of the Hecke L-function and the j-th symmetric power L-function
associated with a primitive holomorphic cusp form f of even integral weight k
≥ 2 for the full modular group S L2(Z), respectively. In this paper, we
study the sign changes of the sequences {λsyma f (n)}, {λsyma f
(n)λsymbg(n)} and {λf (n)λg(n)λh(n)} in short intervals,
and establish lower bounds for the number of these sign changes for n ≤
x, where a, b ∈ ℕ{+}, a, b ≥ 2, and f, g, h are three
different primitive holomorphic cusp forms for SL{2}(ℤ). In particular,
some results improve previous results.
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