Asymptotic behaviors of the real-analytic Eisenstein series

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Let E (k ; s ; z) be the non-holomorphic Eisenstein series with an even weight k attached to the modular group SL(2, Z). Our first main achievement of the present project is to establish the uniform asymptotic expansion of E (0 ; s ; z) respect to Im (s), which gives the convexity theorem under some conditions. Our second main achievement is to establish its complete asymptotic expansion as Im (z) to infinity, which gives the another proof of the Fourier series expansion and its applications.