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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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