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Asymptotic expansions of the non-holomorphic Eisenstein series II(Analytic Number Theory)

Let i = √−1, s = σ + it ∈ C and H be the upper half plane. The nonholomorphic Eisenstein series for SL2(Z) with weight 0 is defined by E(z, s) = y ∑ {c,d} |cz + d|−2s. Here z = x+ iy ∈ H, and the summation is taken over ( ∗ ∗ c d), a complete system of representation of {( 0 ∗) ∈ SL2(Z)} \SL2(Z). The purpose of our talk is to describe an asymptotic formula of E(z, 12 + it) when t → ∞. In applications of the Rankin-Selberg method for the study of automorphic L-functions and the spectral theory, the growth condition of the Eisenstein series perform a fundamental role. It shoud be noted that the t-estimate of the Eisenstein series is a delicate problem. The non-constant terms of the Fourier expansion of E(z, s) are not negligible. Our result may be considered as an analogy of Jutila’s result on the square of the Riemann zeta-function, which is called Voronoi-Atkinson type formula. In order to obtain our asymptotic formula, we need Olver’s works based on the theory of asymptotic solutions of differential equations. The Airy function Ai(τ) is crucial in the uniform asymptotic expantion of the Bessel function. Combining with the estimation of the sum of the divisor function, we have the following type formula. E(z, 12 + it) = 4 √ 2π 1 2y 1 2 ζ(1+2it) N ∑ n=1 n−itσ2it(n) { t − (2πny)2− 1 4 cos(2πnx)τ 1 4 n Ai(−τ 2 3 n ) + (constant term) + (error terms). Here σs(n) is the sum of s-th powers of positive divisors of n, and the parameter τn is defined by y, n and t. It is supposed that t − (4 log t) 3 t 3 ≤ 2πyN ≤ t.