Let ϕ(z)\documentclass[12pt]{minimal}
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\begin{document}$$\phi (z)$$\end{document} be a primitive Hecke–Maass cusp forms with Laplace eigenvalue 14+t2\documentclass[12pt]{minimal}
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\begin{document}$$\tfrac{1}{4}+t^2$$\end{document}. Denote by L(s,symmϕ)\documentclass[12pt]{minimal}
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\begin{document}$$L(s, \mathrm{sym}^m\phi )$$\end{document} the m-th symmetric power L-function associated to ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} and by λsymmϕ(n)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{\mathrm{sym}^m\phi }(n)$$\end{document} the n-th coefficient of the Dirichlet expansion of L(s,symmϕ)\documentclass[12pt]{minimal}
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\begin{document}$$L(s, \mathrm{sym}^m\phi )$$\end{document}. For any nonzero integer ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} we prove ∑n⩽xλϕ(n)λϕ(n+ℓ)≪ϕ,ℓx(logx)0.187(x⩾3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{n\leqslant x} \left| \lambda _{\phi }(n)\lambda _{\phi }(n+\ell )\right| \ll _{\phi , \ell } \frac{x}{(\log x)^{0.187}} \qquad (x\geqslant 3). \end{aligned}$$\end{document}This improves Holowinsky’s corresponding result, which requires 16\documentclass[12pt]{minimal}
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\begin{document}$$\tfrac{1}{6}$$\end{document} in place of 0.187. for all x⩾3\documentclass[12pt]{minimal}
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\begin{document}$$x\geqslant 3$$\end{document}. Further assuming that L(s,sym10ϕ)\documentclass[12pt]{minimal}
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\begin{document}$$L(s, \mathrm{sym}^{10}\phi )$$\end{document} and L(s,sym12ϕ)\documentclass[12pt]{minimal}
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\begin{document}$$L(s, \mathrm{sym}^{12}\phi )$$\end{document} are automorphic cuspidal, we obtain a conditional generalization to the symmetric square case: ∑n⩽xλsym2ϕ(n)λsym2ϕ(n+ℓ)≪ϕ,ℓx(logx)0.196(x⩾3).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{n\leqslant x} \left| \lambda _{\mathrm{sym}^2\phi }(n)\lambda _{\mathrm{sym}^2\phi }(n+\ell )\right| \ll _{\phi , \ell } \frac{x}{(\log x)^{0.196}} \qquad (x\geqslant 3). \end{aligned}$$\end{document}
