The number of coefficients of automorphic \(L\)-functions for \(\mathrm{GL}_m\) of same signs. (English) Zbl 1380.11056
Summary: Let \(\pi\) be an irreducible unitary cuspidal representation for \(\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})\), and let \(L(s, \pi)\) be the automorphic \(L\)-function attached to \(\pi\), which has a Dirichlet series expression in the half-plane \(\mathrm{Re}\) \(s > 1\). When \(\pi\) is self-contragredient, all the coefficients in the Dirichlet series expression are real. In this paper we give non-trivial lower bounds for the number of positive and negative coefficients, respectively.
MSC:
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11F30 | Fourier coefficients of automorphic forms |
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