Moments of the Riemann zeta function and Eisenstein series. I
The starting point and motivation of this paper is a comparison between the formulae \[ \int_0^T | \zeta (\sigma +it)| ^2\, dt \sim \zeta (2\sigma )T + \frac{(2\pi )^{2\sigma -1}}{2-2\sigma }\zeta (2-2\sigma )T^{2-2\sigma } \] and \[ \int_0^1 E_{\sigma }(x+iy)\, dx = \zeta (2 \sigma )y^{\sigma } + \pi ^{2\sigma -1}\frac{\Gamma (1-\sigma)}{\Gamma (\sigma )}\zeta (2-2\sigma )y^{1-\sigma }, \] where \(0 < \sigma < 1\) with \(\sigma \neq 1/2\), and \(E_{\sigma }(z)\) is the classical Eisenstein series for \(\text{SL}(2, \mathbb Z)\). Interestingly, the same zeta-values occur on the right in both relations. The authors show that the same phenomenon extends to higher moments of the zeta-function and higher dimensional Eisenstein series if the conjectural formula for the moments of the zeta-function, due to Conrey, Farmer, Keating, Rubinstein, and Snaith, is assumed. For Part II see J. Number Theory 105, No. 1, 175--191 (2004; Zbl 1079.11042).
scientific article; zbMATH DE number 2082499
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