## The distribution of Fourier coefficients of cusp forms over sparse sequences.(English)Zbl 1303.11055

Let $$f$$ be a normalized Hecke cuspidal eigenform of even weight $$k$$ on the full modular group $$\mathrm{SL}_2(\mathbb Z)$$ with Fourier coefficients $$a_f(n)$$ and normalized coefficients $$\lambda_f(n)= a_f(n)n^{-(k-1)/2}$$, so $$|\lambda_f(n)|$$ is bounded by the number of positive divisors of $$n$$. A. Ivić [in: IV International Conference “Modern Problems of Number Theory and its Applications”: Current Problems, Part II. Moscow: Mosk. Gos. Univ. im. M. V. Lomonosova, Mekh.-Mat. Fak., 92–97 (2002; Zbl 1230.11053)] initiated the study of the distribution of $$\lambda_f(n)$$ over sparse sequences by giving an estimate for the sum $$S_2(x)=\sum_{n\leq x}\lambda_f(n^2)$$. This was improved and generalized by other authors who estimated $$S_j(x)=\sum_{n\leq x}\lambda_f(n^j)$$ for $$j= 2,3,4$$. Much earlier, R. A. Rankin [Proc. Camb. Philos. Soc. 35, 357–372 (1939; Zbl 0021.39202)] and A. Selberg [Arch. Math. Naturvid. 43, No. 4, 1–4 (1940; Zbl 0023.22201; JFM 66.0377.01)] studied the average behaviour of $$\lambda_f(n)$$ and obtained an asymptotic formula for the sum $$\sum_{n\leq x}\lambda^2_f(n)$$. Combining both threads, the authors [Proc. Am. Math. Soc. 137, No. 8, 2557–2565 (2009; Zbl 1225.11059)] proved asymptotic formulas for $$\sum_{n\leq x}\lambda^2_f(n^j)$$, $$j= 2,3,4$$. In the paper under review they present their result with better error terms, namely, they show that $\sum_{n\leq x}\lambda^2_f(n^j)= c_jx+ O_f(n^{1-2/((j+1)^2- 1)})$ for $$j= 2,3,4$$, with suitable constants $$c_j$$ depending on $$f$$. An even sharper error term is given for $$j=2$$. For the proof, a classical lemma of Landau and analytic properties of symmetric power $$L$$-functions attached to $$f$$ are used.

### MSC:

 11F30 Fourier coefficients of automorphic forms 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations
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