Fundamental properties of symmetric square \(L\)-functions. II.
The author continues his investigation of the symmetric square \(L\)-functions attached to a holomorphic cusp forms over the full modular group \(\text{SL}(2,\mathbb Z)\) which was began by himself in [J. Math. 46, No. 1, 23--43 (2002; Zbl 1116.11032)]. For an arbitrary primitive Dirichlet character \(\psi\), the symmetric square \(L\)-function attached to the holomorphic cusp form \[ f(z)=\sum_{n=1}^{\infty}a_n e^{2 \pi i n z}, \quad z=x+iy, \] of even integral weight \(k\) for \(\text{SL}(2,\mathbb Z)\) is defined by \[ D(s)=\sum_{n=1}^{\infty}a_{n^2}n^{-s}, \quad | a_{n^2}| \leq d(n^2)n^{k-1}, \] where \(a_n\) are the eigen-values of all Hecke operators and \(a_1=1\). The series is absolutely convergent in a certain half-plane, and \(D(s)\) is analytically continuable to an entire function. In the paper, author proves few theorems concerning the density of the zeros of \(D(s)\) on the right of the critical line. For example, it is proved that, for \(\sigma \geq k-\frac{1}{2}+\frac{1}{\log T}\), \[ N^*(\sigma, T) \ll T^{\frac{5(k-\sigma)}{(2k+1-2\sigma)}}(\log T)^C, \] where \(C\) is a positive constant, and \(N^*(\sigma,T)\) denotes the number of zeros \(\rho=\beta+i\gamma\) of \(D(s)\) with \(\beta \geq \sigma\) and \(| \gamma| \leq T\). For \(k-\frac{1}{4}<\sigma \leq k\), \[ N^*(\sigma,T) \ll (M_2^2(M_1^{\frac{1}{2}}(\log T)^{2^{61}})^{\frac{4}{4\sigma-4k+1}}(\log T)^{2^{31}})^{\frac{2(k-\sigma)}{2\sigma-2k+1}}(\log T)^5, \] where \[ M_1=\max\limits_{\frac{T}{2}\leq t \leq 2T}| \zeta(\tfrac{1}{2}+it)| , \quad M_2=\max\limits_{\frac{T}{2}\leq t \leq 2T}| D(k-\tfrac{1}{2}+it)| . \] Also, some results on the density of zeros for more general function as \(D(s)\) with additional conditions are proved.
scientific article; zbMATH DE number 2177286