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On the average behavior of the Fourier coefficients of \(j\)th symmetric power \(L\)-function over certain sequences of positive integers. (English) Zbl 07729543

Summary: We investigate the average behavior of the \(n\)th normalized Fourier coefficients of the \(j\)th (\(j\geq 2\) be any fixed integer) symmetric power \(L\)-function (i.e., \(L(s,\mathrm{sym}^{j}f)\)), attached to a primitive holomorphic cusp form \(f\) of weight \(k\) for the full modular group \(SL(2,\mathbb{Z})\) over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum \[S_j^*:=\sum_{\substack{{a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\leq x}\\{(a_1,a_2,a_3,a_4,a_5,a_6)\in\mathbb{Z}^6}}}\lambda ^{2}_{\mathrm{sym}^jf}(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2),\] where \(x\) is sufficiently large, and \[L(s,\mathrm{sym}^{j}f):=\sum_{n=1}^{\infty}\frac{\lambda_{\mathrm{sym}^{j}f}(n)}{n^s}.\] When \(j=2\), the error term which we obtain improves the earlier known result.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11F11 Holomorphic modular forms of integral weight
11F30 Fourier coefficients of automorphic forms
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