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On the boundary behavior of automorphic forms. (English) Zbl 1114.11037

The authors study the boundary behavior of modular forms \(f\) on the full modular group. They prove that \(\{ x\in (0,1]:\lim_{y\rightarrow 0^{+}}y^{\frac{k}{2}}|f(x+iy)| \text{ exists}\} \) is contained in a set of Lebesgue measure zero. They also show that the real axis is a natural boundary of definition for \(f\). Finally, by applying the Rankin-Selberg Dirichlet series attached to \(f\), they show that taking the limit over the average over all \(x\in (0,1]\) behaves well.

MSC:

11F11 Holomorphic modular forms of integral weight
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11M41 Other Dirichlet series and zeta functions
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References:

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