Coefficients of symmetric square \(L\)-functions
Let \(N \geq 1\) be a square-free integer and \(k \geq 2\) be an even integer. Let \(H_k^*(N)\) denote the set of all (holomorphic) primitive cusp forms of weight \(k\) and of level \(N\). The Fourier expansion of \(f \in H_k^*(N)\) at the cusp \(\infty\) can be written as \[ f(z) = \sum_{n=1}^{\infty} \lambda_f(n) n^{\frac {k-1}{2}} e^{2\pi in z} \;\;(\Im z > 0) \] where the \(n\)-th normalised Fourier coefficient \(\lambda_f(n)\) of \(f\) is the eigenvalue of the Hecke operator \(T_n\). It is well known that \(\lambda_f(n)\) is real multiplicative function. Writing \(F= \text{sym}^2 f\) for any \(f \in H_k^*(N)\), then the Dirichlet series (i.e Symmetric-square \(L\)-function) \[ L(s,F) := \sum_{n=1}^{\infty} \frac {\lambda_F(n)}{n^s} \;\;\text{for} \;\;\Re s > 1 \] has an Euler product and it satisfies a functional equation of the Rieman zeta type. The authors of the paper under review prove that : For any \(\varepsilon > 0\), there is a positive density of primes \(p\) such that \[ \lambda_F(p) \leq -\frac {5}{7} + \varepsilon \] and there is a positive density of primes \(p\) such that \[ \lambda_F(p) \geq 2 - \varepsilon. \] The also show that there exists two positive constants \(C_{\pm}(f)\) such that \[ \lambda_F(n) = \Omega_{\pm} \left ( \exp \left \{ C_{\pm}(f) \frac {\log n}{\log \log n} \right \} \right ) \] as \(n \rightarrow \infty\). It should be noted that this result is best possible except for the value of the constants \(C_{\pm}(f)\). They also establish lower bounds for the number of same signs in long intervals as well as in short intervals. The method of proof is basically the same as of [\textit{Y.-K. Lau} and \textit{J. Wu}, Math. Z. 263, No. 4, 959--970 (2009; Zbl 1242.11030), Theorem 2] but the situation is a bit delicate and certain difficulties are overcome with the tools developed in [\textit{J. L. Hafner}, Lect. Notes Math. 899, 148--165 (1981; Zbl 0476.10039); \textit{Y.-K. Lau} and \textit{K.-M. Tsang}, J. Reine Angew. Math. 544, 25--38 (2002; Zbl 1015.11048)] in the spirit of the works of \textit{K. Chandrasekharan} and \textit{R. Narasimhan} [Math. Ann. 152, 30--64 (1963; Zbl 0116.27001)].
scientific article; zbMATH DE number 5857689
scientific article