A new zero-density result of \(L\)-functions attached to Maass forms
If \(f\) is a Maass cusp form for \(\text{SL}(2, \mathbb Z)\) which is an eigenform for the Hecke operators \(T(n)\) and the reflection operator \(T: z \mapsto -\overline{z},\) then the associated \(L\)-function is defined as \[ L(f,s) = \sum_{n=1}^{\infty} \frac{\lambda(n)}{n^s} \] where \(T(n)f = \lambda(n) f\) and \(\lambda (1) = 1.\) Denote by \(N_L(\sigma, T)\) the number of zeros of \(L(f,s)\) with real part greater than or equal to \(\sigma\) and imaginary part having absolute value not exceeding \(T.\) The author shows that the result of \textit{A. Sankaranarayanan} and \textit{J. Sengupta} from [Acta Arith. 127, No. 3, 273--284 (2007; Zbl 1137.11035)] to the effect that for \(\sigma \geq \frac{1}{2}+\frac{1}{\log T}\), \[ N_L(\sigma, T) \ll T^{\frac{4(1-\sigma)}{3-2\sigma}}\log ^{26} T \] can be improved for \(\frac{3}{4}\leq\sigma \leq 1\) to \[ N_L(\sigma, T) \ll T^{\frac{(1-\sigma)(8 \sigma - 5)}{-2 \sigma^2 + 6 \sigma -3}}\log ^{57} T. \] As in the result of Sankaranarayanan and Sengupta, the implied constant depends on \(f\).
scientific article; zbMATH DE number 5912925
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