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**Coefficients of Maass forms and the Siegel zero. Appendix: An effective zero-free region, by Dorian Goldfeld, Jeffrey Hoffstein and Daniel Lieman.**
*(English)*
Zbl 0814.11032

Let \(f\) be a Maaß form, which is a newform for \(\Gamma_ 0(N)\) with eigenvalue \(\lambda\) and Dirichlet character \(\chi \bmod N\), normalized such that the Petersson inner product is 1. Let \(F\) denote the adjoint square lift of \(f\) to \(\text{GL}(3)\). It is known that the size of the first Fourier coefficients \(\rho(1)\) of \(f\) is closely related to the behavior of the \(L\)-series \(L(s,F)\) near \(s = 1\).

In Theorem 1 the authors show that \[ | \rho(1)|^ 2 \leq c \log (\lambda N + 1) \] with an effective constant \(c\) provided that no Siegel zero occurs, i.e. \(L(s,F) \neq 0\) in a sufficiently large neighborhood of 1. In the appendix the same estimate is derived for all \(f\) which are not lifts from \(\text{GL}(1)\), i.e. the \(L\)-series of \(f\) is not equal to a Hecke \(L\)-series of a quadratic field. A weaker estimate for lifts from \(\text{GL}(1)\) is also included.

The paper goes on with the inequality \(L(1,F) \geq c(\varepsilon) (\lambda N)^{-\varepsilon}\) for all \(\varepsilon > 0\) with an effective constant \(c(\varepsilon)\) and for all \(F\) with one possible exception. This in turn leads to \(| \rho(1)| \ll_ \varepsilon (\lambda N)^ \varepsilon\).

In Theorem 1 the authors show that \[ | \rho(1)|^ 2 \leq c \log (\lambda N + 1) \] with an effective constant \(c\) provided that no Siegel zero occurs, i.e. \(L(s,F) \neq 0\) in a sufficiently large neighborhood of 1. In the appendix the same estimate is derived for all \(f\) which are not lifts from \(\text{GL}(1)\), i.e. the \(L\)-series of \(f\) is not equal to a Hecke \(L\)-series of a quadratic field. A weaker estimate for lifts from \(\text{GL}(1)\) is also included.

The paper goes on with the inequality \(L(1,F) \geq c(\varepsilon) (\lambda N)^{-\varepsilon}\) for all \(\varepsilon > 0\) with an effective constant \(c(\varepsilon)\) and for all \(F\) with one possible exception. This in turn leads to \(| \rho(1)| \ll_ \varepsilon (\lambda N)^ \varepsilon\).

Reviewer: A.Krieg (Aachen)