Real zeros of Eisenstein series and Rankin-Selberg L-functions

Authors: Bauer, C.¹; Wang, Y.²

Source: Acta Mathematica Hungarica, Volume 115, Numbers 1-2, April 2007 , pp. 13-27(15)

Publisher: Springer

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Abstract:

We prove that the Eisenstein series E(z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range <EquationSource Format="TEX"><![CDATA[ $$tfrac{1}{5}$$ ]]></EquationSource> < Im z < 4.94. For very large and very small values of the imaginary part of z, E(z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½.

Keywords: Eisenstein series; Rankin-Selberg L-function; 11F03; 11M36; 11M20

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10474-007-5102-1

Affiliations: 1: Email: cb@dolby.com 2: Email: arithwsun@gmail.com

Publication date: 2007-04-01